Visual Quantitative Finance: A New Look at Option Pricing, Risk Management, and Structured Securities 0132929198, 9780132929196

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Visual Quantitative Finance

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Visual Quantitative Finance A New Look at Option Pricing, Risk Management, and Structured Securities Michael Lovelady

Vice President, Publisher: Tim Moore Associate Publisher and Director of Marketing: Amy Neidlinger Editorial Assistant: Pamela Boland Reviewer: Michael Thomsett Operations Specialist: Jodi Kemper Marketing Manager: Megan Graue Cover Designer: Chuti Prasertsith Managing Editor: Kristy Hart Senior Project Editor: Lori Lyons Copy Editor: Krista Hansing Editorial Proofreader: Paula Lowell Senior Indexer: Cheryl Lenser Compositor: Nonie Ratcliff Manufacturing Buyer: Dan Uhrig © 2013 by Michael Lovelady Pearson Education, Inc. Publishing as FT Press Upper Saddle River, New Jersey 07458 FT Press offers excellent discounts on this book when ordered in quantity for bulk purchases or special sales. For more information, please contact U.S. Corporate and Government Sales, 1-800-382-3419, [email protected]. For sales outside the U.S., please contact International Sales at [email protected]. Company and product names mentioned herein are the trademarks or registered trademarks of their respective owners. All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Printed in the United States of America First Printing April 2013 with corrections September 2013 ISBN-10: 0-13-292919-8 ISBN-13: 978-0-13-292919-6 Pearson Education LTD. Pearson Education Australia PTY, Limited. Pearson Education Singapore, Pte. Ltd. Pearson Education Asia, Ltd. Pearson Education Canada, Ltd. Pearson Educación de Mexico, S.A. de C.V. Pearson Education—Japan Pearson Education Malaysia, Pte. Ltd. Library of Congress Cataloging-in-Publication Data Lovelady, Michael Lynn, 1957  Visual quantitative finance : a new look at option pricing, risk management, and structured securities / Michael Lovelady.        pages cm   Includes index.   ISBN 978-0-13-292919-6 (hardback : alk. paper)  1.  Options (Finance)--Mathematical models. 2.  Structured notes (Securities)-Mathematical models. 3.  Finance--Mathematical models. 4.  Risk management.   I. Title.   HG6024.A3L684 2013   332.64’53015195--dc23                                                             2013005466

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Growth in Structured Securities . . . . . . . . . . . . . . . . . . . . . . . 2 Growing Emphasis on Low Volatility and Dividends. . . . . . . 3 Criticisms of Structured Securities . . . . . . . . . . . . . . . . . . . . . 4 Demand for Quantitative Skills. . . . . . . . . . . . . . . . . . . . . . . . 5 Direction of Quantitative Finance . . . . . . . . . . . . . . . . . . . . . 6 When I Realized It Might Be Easier . . . . . . . . . . . . . . . . . . . 8 Try Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 The Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Visualizing the Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 What It Means and Why It Works: A Nontechnical Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 It Doesn’t Get Too Complicated. . . . . . . . . . . . . . . . . . . . . . 18 An Integrated View of Risk Management. . . . . . . . . . . . . . . 18 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 2

Random Variables and Option Pricing . . . . . . . . . . . . . 21 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Building the Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Correcting the Mistake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Optional: Additional Resources. . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 3

An Overview of Option Pricing Methods . . . . . . . . . . . 43 The Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . . 43 Black-Scholes Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 48 The Binomial Option Pricing Method . . . . . . . . . . . . . . . . . 49 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Putting Visual Quant in Context . . . . . . . . . . . . . . . . . . . . . . 52 Additional Reading, Advanced Topics, and Resources . . . . 57 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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Chapter 4

Value-at-Risk and Conditional Value-at-Risk . . . . . . . . 61 How Likely Is Something? . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Multiple Stock VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Stock and Option VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Conditional Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 5

Full Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . 77 Adding Functionality to the Model. . . . . . . . . . . . . . . . . . . . 79 Stock Return Mean (Cell G3) . . . . . . . . . . . . . . . . . . . . . . . . 79 Stock Return Standard Deviation (Cell G4). . . . . . . . . . . . . 82 Discount Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Stock Price Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Summary of New Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . 88 Pricing Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Effects of Assumption Changes . . . . . . . . . . . . . . . . . . . . . . 93 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 6

The Lognormal Distribution and Calc Engine. . . . . . . 97 Definition of the Lognormal Distribution . . . . . . . . . . . . . . 98 The Forward Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Cross Reference: Stochastic Differential Equations . . . . . 100 The Backward Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 The Calc Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Assigning Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Setting the Stock Price Range . . . . . . . . . . . . . . . . . . . . . . . 110 Visualizing Option Pricing As Normal or Lognormal. . . . . 112

Chapter 7

Investment Profiles and Synthetic Annuities . . . . . . 115 What Is a Synthetic Annuity, and How Does It Work? . . . 117 The Investment Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Assigning Probabilities Using Implied Volatility . . . . . . . . 120 Using Options to Reshape the Investment Profile . . . . . . . 123 Adjusting the Profile for Behavioral Finance . . . . . . . . . . . 125 Concentrated Stock Example . . . . . . . . . . . . . . . . . . . . . . . 128 The Synthetic Annuity in Turbulent Markets. . . . . . . . . . . 138

CONTENTS

Chapter 8

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Stock-Only Investment Profile . . . . . . . . . . . . . . . . . . 145 The Purpose and Context of the Model . . . . . . . . . . . . . . . 145 The Stock-Only Investment Profile . . . . . . . . . . . . . . . . . . 146 The Calc Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 The Stock-Only Profit Calculation . . . . . . . . . . . . . . . . . . . 157 Adding the Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Test: Stock-Only Investment Profile. . . . . . . . . . . . . . . . . . 162

Chapter 9

Adding Options to the Model . . . . . . . . . . . . . . . . . . . 167 Long Put Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Short Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Expected Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Black-Scholes Add-In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 The Heading Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Delta Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Time Value and Total Premium Formulas . . . . . . . . . . . . . 176

Chapter 10

Option Investment Profiles . . . . . . . . . . . . . . . . . . . . . 179 Long Call Option Investment Profile . . . . . . . . . . . . . . . . . 179 Short Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Long Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Short Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Chapter 11

Covered Calls, Condors, and SynAs . . . . . . . . . . . . . . 197 Covered Call Investment Profile. . . . . . . . . . . . . . . . . . . . . 198 Put–Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Iron Condor Investment Profile . . . . . . . . . . . . . . . . . . . . . 205 Synthetic Annuity (SynA) Investment Profile. . . . . . . . . . . 209 Adding a Customized Utility Function . . . . . . . . . . . . . . . . 223 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Chapter 12

Understanding Price Changes. . . . . . . . . . . . . . . . . . . 227 Investing in XYZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Attribution: Explaining Why the Option Price Changed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

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Chapter 13

The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 The Option Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Calculating Greeks: Formulas, Models, and Platforms . . . 249 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Theta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Introduction to Chapters 14, “Tracking Performance,” and 15, “Covered Synthetic Annuities” . . . . . . . . . . . 265 Chapter 14

Tracking Performance . . . . . . . . . . . . . . . . . . . . . . . . . 269 Tracking Template. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 TradeStation Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Putting It All Together: Synthetic Annuity Overview . . . . 282

Chapter 15

Covered Synthetic Annuities. . . . . . . . . . . . . . . . . . . . 285 Covered Synthetic Annuity (CSynA) . . . . . . . . . . . . . . . . . 286 Example: Deere & Company . . . . . . . . . . . . . . . . . . . . . . . 289 The Standard CSynA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Supplemental Material: The CBOE S&P 500 BuyWrite Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 BXM Study by Callan Associates. . . . . . . . . . . . . . . . . . . . . 312

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Acknowledgments I would like to express my sincere gratitude to several people who made this book possible. At Pearson/FT Press, Jim Boyd, who believed in the material; Michael Thomsett, who gave the project guidance and direction from beginning to end; Lori Lyons, who served as both patient editor and production manager; Krista Hansing, Russ Hall, and all those who helped with editing, marketing, illustration, and production. I would also like to thank Don DePamphilis at Loyola Marymount University for giving me the idea to write and Cooper Stinson for reviewing early manuscripts and asking all the right questions. Also, my friends and family who gave me encouragement and inspiration, and forgave me for missing tee times: Arnold, Barbara, Barry, Bill, Bobbi, Daniel, David, Ernie, John, Kate, Katy, Kristine, Leslie, Matty, Paul, Steve, and Tony. Above all, for life itself, the Triune God of Creation—I always remember.

About the Author Michael Lovelady, CFA, ASA, EA, works as an investment strategist and portfolio manager, where he specializes in blending traditional and quantitative styles, including reduced-volatility and yieldenhanced option strategies. Michael developed the synthetic annuity and is the author of Profiting with Synthetic Annuities: Options Strategies to Increase Yield and Control Portfolio Risk. Prior to hedge fund management, Michael was a consulting actuary for Towers Watson and PricewaterhouseCoopers, where he worked with employers on the design and funding requirements of plans ranging from defined benefit and defined contribution to hybrid db/dc plans. His experience with retirement income strategies—both as an actuary from the liability side and as a fund manager from the asset side—gives him a unique perspective. Michael has also been involved in teaching and creating new methods for making quantitative investing more accessible to students, trustees, and others interested in investment and risk management. He developed the investment profile—a graphical representation of risk and the basis of a simplified option pricing model, and visually intuitive presentations of structured securities. During his career, Michael has served various organizations, including Hughes Aircraft, Boeing, Global Santa Fe, Dresser Industries, the Screen Actors Guild, The Walt Disney Company, Hilton Hotels, CSC, and the Depository Trust Company. He is a CFA charterholder, an Associate of the Society of Actuaries, and an ERISA Enrolled Actuary. He currently lives in Los Angeles..

Preface Visual Quantitative Finance presents a simplified, but powerful view of financial mathematics. It is written for trustees, investors, advisors, students, and others interested in quantitative finance, risk management, options strategies, structured securities, or financial model building—or for those looking for new ways to explain these topics to someone else. What makes this book different is its visual presentation of formulas and concepts that may be more intuitive, especially for those without quantitative backgrounds. By working directly with the mathematical building blocks of finance—random variables— rather than formulas derived from them, the underlying mechanism of option pricing becomes simple and transparent, creating many advantages: • The Black-Scholes formula can be derived in a few easy steps, with no complicated formulas. • The derivation of the option pricing formula highlights the framework for translating option pricing assumptions into future stock price patterns. • This framework is the key—not only to option pricing, but also to structured securities and risk management in general. • The visual display of random variables emphasizes the simplicity behind quantitative finance, allowing you to look inside the logic of risk metrics and the power of options to reshape riskreward profiles. • You don’t need a prior knowledge of statistical mathematics. Although the tools are developed without stochastic formulas, they may be one of the best ways to learn them. • Metrics that appear complicated when expressed in words or formulas become nothing more than simple lookups in a visual context.

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The book provides an important perspective on options and their value in portfolio management. The material for the book was selected to reflect the change in investor attitudes that began with the 2000–2002 internet bubble and accelerated after the 2008 financial crisis. The change in attitudes has been described in numerous market surveys that indicate investors are (1) tired of traditional portfolios, (2) looking for creative solutions, and (3) not willing to invest in instruments they don’t understand. In response, the use of alternative strategies and the introduction of new funds have grown rapidly, with much of the activity focused on structured securities. Structured securities, ranging from simple covered call strategies to complex institutional hedges, are proving to be more effective than traditional securities at tailoring risk-reward profiles and generating new sources of income. Even though the trends are clear and investor interest has never been higher, the challenge for many investors is to become comfortable with unfamiliar, often seemingly complex instruments. This is especially true for institutional trustees and retail investors who might not have experience with options or the mathematics behind them. One method currently gaining traction is the visual presentation of concepts such as Value-at-Risk, which are more easily communicated in pictures than words. This book extends visual presentation to a variety of topics in hopes of making quantitative finance more accessible to a wider audience.

1 Introduction Visual quantitative finance is a different take on the mathematics of investing. It emphasizes an intuitive view of risk and the interrelationships of option pricing, risk management, and structured securities. This chapter begins with an overview of current investment trends that serve as the backdrop for the material covered in the book. The trends include shifts in investor attitudes and the emergence of new investment alternatives being driven by the application of quantitative finance. I also talk about the personal “discovery” that motivated me to write this book. Like most people involved in asset management, I have struggled often with two things: (1) how to dampen some of the stock market volatility—and losses—that have occurred too often over the last decade, and (2) how to generate higher levels of income in a historically low interest rate environment. Over time, I have become convinced that adding options—not as trading instruments, but as long-term components of portfolios—is the best answer. But unless an investor really understands options, it is hard to fully commit to a strategy involving them. Unfortunately, really understanding options means getting a little technical—sometimes a lot more than a little. I laughed one day when I saw the title of a paper on computational methods (roughly the same subject as this book). The title was “An Introduction to Computational Finance Without Agonizing Pain. If you have tried to approach this subject, you probably know the feeling. I do. My personal “discovery” was not really a discovery in the sense that I uncovered some new truth. For me, it was just one of those light-bulb moments when I saw past the differential equations to a

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simple, beautiful “picture.” What I saw in the picture was an easier way to visualize option pricing. More than that, the picture contained enough information to break down seemingly complex risk metrics and structured securities into basic elements. The picture is a chart of an Excel spreadsheet, shown at the end of the chapter, and used as the framework for most of what is presented in this book.

Growth in Structured Securities According to Bloomberg, investment banks sold $45.9 billion of SEC-registered structured securities in 2011 and another $11.1 billion in the first quarter of 2012. The securities offer customized risk-return and payoff profiles using derivatives based on underlying stocks, bonds, currencies, and commodities, with approximately 60% of these notes tied to equities (including the S&P 500 Index).1 Registered structured products are just the tip of the iceberg. Demand from institutions and retail investors, looking for better ways to invest, is prompting asset management firms and ETF providers to introduce new funds capable of smoothing market volatility and increasing yield. For instance, AQR Capital Management, the hedge fund company, launched four new mutual funds. July 13, 2012: AQR Capital Management announced Monday the launch of four new mutual funds .... The funds seek to provide equity-like returns with lower volatility and smaller drawdowns using an actively managed, risk-balanced approach.2 This is one in a string of announcements. Quant funds are rolling out more creative investment vehicles to meet market demand. Most of these vehicles offer forms of risk management and income features that traditional asset classes do not offer. And structured securities are often the means to do it. Structured securities range from simple covered call strategies to complex institution hedging programs. What they have in common is the ability to tailor risk and reward profiles to match investor objectives in ways that are difficult to do with stocks and bonds.

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On the retail side, more investors than ever use options strategies—not only as trades, but as integral parts of investment portfolios. On the institutional side, allocations to hedge funds and other alternatives using options strategies and structured securities are growing rapidly. Both groups are interested in emerging strategies that combine the explicit use of hedging, insurance, and risk allocations in risk management instead of continuing to rely on traditional portfolio models. Also, in today’s low-interest environment, both groups want access to greater yields, especially those not related to market direction. These investor goals have led to the growing importance of volatility-reducing quantitative methods, particularly methods related to options capable of boosting dividend yields.

Growing Emphasis on Low Volatility and Dividends Some of 2012’s most successful ETFs were funds that combined these themes, including the PowerShares S&P 500 Low Volatility Portfolio (NYSEArca: SPLV) and the iShares High Dividend Equity Index Fund (NYSEArca: HDV). Low-volatility ETFs debuted in 2011 with the launch of the PowerShares S&P 500® Low Volatility ETF (SPLV). Since inception, SPLV has exhibited 69% of the volatility in the S&P 500 Index [and] outperformed its cap-weighted benchmark in terms of absolute returns.3 In terms of relative performance, SPLV has delivered an excess return of 11.7% compared to the S&P 500. Many other ETFs have been introduced with variations on the low volatility strategy seeking to deliver market exposures measured by volatility rather than traditional cap-weighted benchmarks. Other funds, such as the Windhaven Portfolios at Charles Schwab & Co., add dynamic allocation strategies, adjusting portfolio allocations based on changing economic conditions. According to the brochure, this form of proactive risk management “strives to capture much of the up markets and less of the down.”

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Sage Quant Management filed with regulators in summer 2012 to offer a dividend-focused low-volatility fund to be listed as an ETF. In the filing, the company said that the fund might rely on derivatives such as futures and options contracts to “facilitate trading or to reduce transaction costs.” That is consistent with the theme of blending dividends and low volatility. And it is consistent with the work of Roger Clarke at Analytic Investors and others who have argued that it may be possible to pick up 40 to 60 basis points of risk-adjusted return. However, funds offered to retail investors have been reluctant to include derivatives simply because a lot of investors view them as dangerous. It appears that attitude is changing, at least when it comes to more conservative types of derivatives. Personally, I believe that including derivatives in the investment toolkit is a step forward in the nature of the funds offered to the retail investor. After all, derivatives have been utilized in institutional investing for decades to provide exposure to absolute return strategies, long or short, and hedge overlays for pension plans and endowments. Partly, this is in response to the lack of risk management achievable through mean-variance portfolios and their counterparts in the retail space: life cycle and target date funds. It also is partly in response to the need for higher yields to match long-term discount assumptions, which is hard to achieve in a low-yield bond market. Because individual investors face the same challenges, products with derivative components are being offered in more variations by more firms. That is not to say that everything is going smoothly.

Criticisms of Structured Securities Criticism of structured securities is growing as fast as the demand. FINRA, the financial regulatory agency, has looked carefully at how these instruments are designed and marketed. Several published papers have warned investors about complexity, expense, and suitability. Some of the products are so misunderstood that investors are completely unaware of what they own or how much they could lose.

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In 2008, many investors were burned. Structured products that were supposedly “principle-protected” were not. Investors at some of the largest and most respected investment management firms learned that their securities bought to earn income had overnight been “converted” into depreciated stocks. The losses were huge and unexpected. In a Forbes article, “When will FINRA stop this insanity?” Seth Lipner says structured products are too complicated for ordinary investors to understand. They are, in reality, exotic derivatives... Isn’t it clear by now that these newfangled financial products exist just to enrich Wall Street at the expense of naive investors? Isn’t it enough already?4 Other criticisms center not on the danger, but on the fact that they don’t add anything that investors can’t get through simpler and less expensive instruments. One report concluded: These products add nothing to retail investors’ portfolios that can’t be acquired from investments already available in the market in the form of less risky, less complicated, or less costly products.5 Because these products “add nothing,” they may fail even the most basic regulatory “reasonable basis” rules for suitability to sell to retail investors. Despite the criticism, the trend is clear. Investors don’t want the roller coaster that investing has become in the last decade. Financially engineered products are not going away, nor is the demand for people who can build them and explain them.

Demand for Quantitative Skills On the hiring front, recruiters are saying that stock pickers are “out” and quantitative analysts are “in.” The role of quants at hedge

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funds for complex trading has been steady, but it seems the demand for quant talent for more mainstream investing applications is increasing. Firms are changing their emphasis on risk, giving it more weight in the balance between risk and return. Instead of selecting return targets and then minimizing the risk involved in achieving them, the new design order is to determine acceptable levels of risk first and then go for returns. Here is an excerpt from a recent job posting at T. Rowe Price: The T. Rowe Price investment approach strives to achieve superior performance but is always mindful of the risks incurred relative to the potential rewards. The job posting explains the “greatly expanded” capabilities in Quantitative Research, including portfolio analytics and modeling and the outlook for continued growth. These are key areas of focus for the firm where we anticipate a strong growth in demand.6 The job requirements for this T. Rowe Price job listing include a Ph.D.; a CFA; a Master’s degree in quantitative finance, science, engineering, or mathematics; and proficiency with analytic modeling platforms such as MatLab, R, or S-plus.

Direction of Quantitative Finance It might seem that at least one branch of quantitative finance would become less complex as it enters the mainstream, but that is apparently not happening yet. How is it possible for the average investor to understand a security that requires a Ph.D. to design? Paul Wilmott, Michael Thomsett, and many others have advocated the practical use of quantitative methods, emphasizing more transparency in the use of derivatives. In 2008, Wilmott blogged: In my view the main reason why quantitative finance is in a mess is because of complexity and obscurity. Quants are

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making their models increasingly complicated, in the belief that they are making improvements. This is not the case.… finance is not a hard science, one in which you can conduct experiments for which the results are repeatable. Finance, thanks to it being underpinned by human beings and their wonderfully irrational behaviour, is forever changing. It is therefore much better to focus your attention on making the models robust and transparent rather than ever more intricate.7 He describes a “sweet spot” in quant finance. The sweet spot is where models are not too elementary to be of practical use, but not so abstract that even the inventors don’t really understand them. He adds, “I teach on the Certificate in Quantitative Finance, and in that, our goal is to make quant finance practical, understandable, and, above all, safe.” I agree. That is why I have targeted a particular sweet spot in this book: the aspects of quantitative finance that are most helpful in designing and communicating structured securities. This book introduces a new framework to illustrate the mechanics of option pricing. The logic behind option pricing serves as the basis for much of financial engineering, for building structured securities and evaluating alternative investment strategies. What makes the method different is that it uses a simplified spreadsheet to illustrate the “matrix” nature of the building blocks of quantitative finance: random variables. In random variable form, the underlying probabilities are kept transparent and are not condensed into formulas. By keeping the probabilities separate, a number of calculations become much easier to understand, which, in turn, makes the securities evaluated on the same basis easier to understand. I am excited about writing this book because of something that I stumbled across a few years ago that made the entire subject of quantitative finance easier for me. It involved a simple way to replicate complicated formulas. For me, the breakthrough came one night while I was practicing my putting stroke. I had an idea and decided to play with that instead.

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When I Realized It Might Be Easier Starbucks, late. I clearly remember looking at the number: $11.93. It was only 1 cent higher than the number I had gotten using the Black-Scholes formula, $11.92. But I wasn’t using the Black-Scholes formula. I was using a spreadsheet—a simple one. After a few years as a hedge fund manager, I had finally settled into a strategy I felt comfortable with. What I didn’t know how to do was describe it. I didn’t even know what to call it. For lack of anything better, I called it a synthetic annuity. I used synthetic because of the risk-management features that I guessed would qualify as a synthetic hedge, and annuity because it involved selling options to generate monthly income. I knew I needed to devote time to communicating the strategy in a way that the average investor could understand. At a minimum, I needed to put it in context of the various traditional and hedge fund strategies. I struggled with this. Because it involved trading options, I was concerned that it would get the typical bad rap of being too risky or too complex, neither of which I think is true. But it was a form of managed structured security, so I would have to explain the basics of structured securities and how they worked. The previous day, I had been flipping through one of my go-to texts, McDonald’s Derivative Markets, looking for something that might give me a starting point. I saw this: The Black-Scholes formula arises from a straightforward lognormal probability calculation using risk-neutral probabilities. The contribution of Black and Scholes was not the particular formula but rather the appearance of the risk-free rate in the formula. (p. 613)8 I had already been thinking about Black-Scholes, having just reread Peter Bernstein’s beautifully written books on the history and evolution in investment thinking, Capital Ideas and Capital Ideas Evolving. Bernstein referred to options and the pricing model as “the

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9

most powerful financial invention in history.” And I remembered the emphasis Paul Samuelson put on option pricing when he gave his advice to anyone entering the investment field: “Learn the BlackScholes option-pricing model.”9 My immediate interest was more in tailoring risk and reward profiles, but I had reached a point where I needed to construct a reasonable basis for comparing alternative structures. I was skeptical about using the Black-Scholes framework because of its well-publicized limitations, such as not handling fat tails and assuming constant volatility. Then I changed my mind. I wasn’t trying to weigh something precisely, so I didn’t need a very accurate scale. I was measuring the difference in two things, which even an inaccurate scale can do. And using Black-Scholes had the advantage of making the structure approximately hedgeable, which is more important in my work than being precise. So I decided to try what McDonald had suggested: to derive the Black-Scholes formula. I could either start with a differential equation or start with a spreadsheet approximation. I chose the spreadsheet. I was hoping to build something that would fit on a page or two of Excel—and I’m not crazy about differential equations. I began with one of the assumptions used to derive the BlackScholes formula: Continuously compounded returns on the stock are normally distributed. Excel has a built-in function for the standard version of the normal distribution. That was the first step. Then I went through the process of converting it into a stock return distribution and then a stock price distribution. The option payoff was straightforward, as was weighting the payoffs by probabilities. The entire calculation fit in six columns, and there were no complicated formulas. It was symmetric and simple. Too bad it was wrong.

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Try Again That night at Starbucks, I decided to try again. Something was nagging at me. A piece was missing. I knew that volatility affects stock price simulations. The more volatile, the more the distribution of prices is dragged down. But I had not included anything in this spreadsheet to account for that. I was aware of the fact that, in Monte Carlo simulations, adjustments are made to the distribution being sampled so that returns are not overstated. I wondered if that was what I needed to add. I went back to the book. On page 597, McDonald said: [W]e need to subtract 1/2 times the variance. That was the term I had been thinking about. I wrote the following on a napkin: Mean − 1/2 variance = 0.0% − 0.5 × 0.302 = 0 − .5 × .09 = –0.045 = –4.5% Worth a shot. So I plugged that into the spreadsheet. It worked. Then I tried using different assumptions, and it still worked. I still thought it was too easy to be right, but this time I couldn’t show that it was wrong.

The Spreadsheet Because the adjustment (1/2 variance) has important implications, I want to show you what the spreadsheet looked like before the adjustment. In the next chapter, I will correct the mistake and walk through the spreadsheet components step by step. To get a reference point, I priced an option with the BlackScholes formula. I assumed a one-year term, 30% volatility, and $100 for the current stock price and the option strike. I also assumed 0% interest and no dividends. The Black-Scholes price was $11.92. Then I started building the spreadsheet.

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At the top, I entered the pricing assumptions. Then I started filling in the body of the sheet, following the rule that “continuously compounded stock returns are normally distributed.” When dealing with a normal distribution, the usual place to start is with the standard normal distribution. This is just a special case in which the mean or average value is 0 and the standard deviation is 1. Excel has a built-in function, so I filled in the first two columns with an approximate version that fit on two pages. (I divided it into 81 points, ranging from –4 standard deviation to +4 standard deviations in 0.1 increments. To handle the tails, I put everything outside 4 standard deviations in the two endpoints.) I knew it would not be exact, but it would work for a first try. Next, I used a common rule of statistics to transform the standard normal distribution into a normal distribution with a standard deviation of 30% and a mean of 0. I remembered that “continuous compounding” meant using the EXP function. That gave me the stock prices. Knowing the stock price makes it easy to calculate the option payoff. The option payoff is just the difference between the stock price and the strike price, not less than zero. The only thing left to do was weight each option payoff by its probability and add the numbers. The answer was $14.63, shown in Cell F95. I am intentionally showing you the wrong version so that I can focus on the correction in the next chapter. What is important here is the basic format. When I was finished, the spreadsheet looked like Figures 1a and 1b. This spreadsheet describes a simple world. In this world, stock returns, stock prices, and option payoffs are linked to each other, and each of them can be only one of 81 different values. Columns A and B are constants representing the approximated standard normal distribution. Here are the other column formulas: Column C = Column A × 0.30 Column D = EXP[Column C] Column E = MAX[0, Column D − Strike price] Column F = Column B × Column E The option price is the total of Column F.

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Figure 1.1a Option pricing, first attempt

CHAPTER 1 • INTRODUCTION

Figure 1.1b Option pricing, first attempt (Continued)

13

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As an example in reading the spreadsheet, look at Row 53. It is one of the 81 possible outcomes. In this outcome, the stock return is 0%, the stock price is $100, and the option payoff is $0. The probability that this particular outcome will occur is 3.98776%. Similarly, in Row 63 at the one standard deviation point, the stock return is 30%, the corresponding stock price is $134.99 (the fact this differs from $130 is explained later), and the option payoff (the difference between the stock price and the option strike price of $100) is $34.99. The probability that this particular outcome will occur is 2.41971%. Notice that the only positive values for the option payoff are in Rows 54–93. These 40 outcomes are the only numbers factored into the option price. The last column shows the weighted values of the option payoffs. Looking at Column F, the value of the option is concentrated between 0 and 3 standard deviations. The highest contributions occur at around 1 standard deviation, with a weighted value of $0.85. In the tail of the distribution, the payoffs are very high but the probabilities are very low. For instance, even though the payoff goes as high as $232.01 in Row 93, the effect on the value of the option is only 1 cent. The probability at this point is so low that a high payoff has almost no effect.

Visualizing the Result Figure 1.2 is a graph of the spreadsheet. The call option payoffs are shown on the right side of the graph, and the values for these payoffs are on the right axis. The probabilities of the payoffs are shown as the normal distribution curve. The probability values are on the left axis. The option price is the weighted average option payoff, where the weights are the probabilities. In other words, this is Column E (the option payoff) times Column B (the corresponding probabilities), or as shown here in Figure 1.3.

CHAPTER 1 • INTRODUCTION

Figure 1.2 Stock option payoffs with probabilities

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Figure 1.3 Weighted stock option payoffs

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17

The middle section is the curve representing the option price. The sum of these values is equal to the option price. The relative height of the curve shows which stock prices contribute most to its value. In this view, even though the payoff grows large on the right side, the probabilities of those payoffs are growing smaller at a even faster rate, so high payoffs contribute relatively less than payoffs closer to the center of the graph.

What It Means and Why It Works: A Nontechnical Overview At this point, why this works may not be obvious. But assuming that it does, it gives a nice interpretation of option pricing. It is just the weighted average of option payoffs, assuming that stock returns are normally distributed. But why is it logical to assume that stock returns are normally distributed? Normal distributions occur naturally in science and statistics, with some of the earliest work on these distributions linked to observations about purely random events. In fact, normal distributions describe the frequencies of random events. So are stock prices random? The Efficient Market Hypothesis, one of the best-known and most controversial ideas in investing, says they are. The EMH has been tested over decades and against massive amounts of data, and it seems to be just as predictive and controversial today as when it was first introduced. The conclusion of the EMH is that neither technical nor fundamental analysis of stocks helps to predict stock prices in the future. The reason for this is the efficiency of large, liquid markets to absorb and digest new information almost immediately as it becomes known, with stock prices moving to their new price points before investors can take advantage of the information. That is, stock prices reflect all currently known information. The next move in price depends only on information that is not known yet and, therefore, is random. If the EMH is true, stock prices should follow the mathematics of random movements such as Brownian motion, random walks, and

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stochastic processes. And if you can describe stock prices, the option payoffs and option values that depend on them can be described as well. What this means is that only one simple idea is behind the mechanics of option pricing: the unpredictability of stock prices.

It Doesn’t Get Too Complicated The spreadsheet illustrates the basic framework for everything presented in this book. It doesn’t get too complicated. That is the beauty of the method. The underlying assumptions are transparent, and the logic can be broken into simple steps. The challenge to those of you without quantitative backgrounds might be the terminology. In finance, statistics, and stochastic math, the terminology is challenging to everyone. One of the advantages of having a relatively compact spreadsheet is that you can always go back to specific cells and exact formulas whenever you need clarification about what something means.

An Integrated View of Risk Management I have asked myself many times why Paul Samuelson thought studying Black-Scholes was so important. I don’t think it is just for the purpose of pricing options. I think it is because the mathematics of option pricing give us a roadmap to risk management. Risk management, in its simplest terms, is a three-step process: 1. Think about what might happen in the future. 2. Know which of those outcomes will hurt you and how likely they are. 3. Decide what to do about it. Most people weigh the cost of risk management against doing nothing. If the cost of insurance is too high, you can self-insure. But the factors involved are mainly financial.

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In the capital markets, another factor is at work. It is hope, which is related to a historical precedence of mean reversion. Most investors believe that markets that fall will also rise again at some point. If you can suffer the pain, you will be rewarded in the end. Maybe. The turbulence of the 2000–2002 and 2008–2009 markets makes it harder to ignore previous bear markets, and consider the Japanese experience, with a 75% decline in the equity market over a 20-year period. The desire of investors to impose some downside protection is understandable and requires some form of risk management. The three generally recognized ways to manage risk are diversifying, hedging, and buying insurance, and all are related to options. Diversification can be enhanced through options, delta hedging was the most elegant interpretation of option pricing, and put options are the purest form of market risk insurance. The process of defining possible future events, assigning probabilities to those events, and using that information to price risk is the same as the process of pricing options. In that sense, option pricing is the central analytic framework for quantitative finance, risk management, and options-related structured securities.

Endnotes 1. McCarthy, Ed. “Structural Inefficiency,” CFA Magazine, July–August 2012. 2. Satter, Marlene Y. “Top Portfolio Products: AQR Capital Launches Defensive Funds,” AdvisorOne, July 13, 2012. http://www.advisorone.com/2012/07/13/top-portfolioproducts-aqr-capital-launches-defens. 3. “What is driving the performance of low-volatility ETFs?” PowerShares Connection Report, August 2012. http://www. etftrends.com/2012/09/what-is-driving-the-performance-oflow-volatility-etfs/. 4. Lipner, Seth (Zicklin School of Business, Baruch College, City University of New York). “Will FINRA Stop the Structured

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Products Insanity?” http://www.forbes.com/2011/06/16/ finra-structured-products.html. 5. McCann, Craig and Dengpan Luo. “Are Structured Products Suitable for Retail Investors,” Securities Litigation and Consulting Group, Inc. 2006. 6. T Rowe Price Job Description. http://www.globalriskjobs.com/ risk-careers/598926/Quantitative-Investment-Strategist-TeamLead-Region-Baltimore?print. 7. Paul Wilmott Blog. “Actuaries Versus Quants.” http://www. wilmott.com/blogs/paul/index.cfm/2008/11/17/ActuariesVersus-Quants. 8. McDonald, Robert L. Derivatives Markets, 2nd Ed. 2006, Addison-Wesley, Pearson Education. 9. “The Future of Life-Cycle Saving and Investing: The Retirement Phase, Nobel Laureate Panel Discussion, Research Foundation of CFA Institute 2009.” Edited by Zvi Bodie, Laurence B. Siegel, and Rodney N. Sullivan, CFA, available online at www.cfapubs.org.

2 Random Variables and Option Pricing This chapter explains the option pricing spreadsheet from Chapter 1. The explanation involves a mathematical concept called random variables. Random variables are the basic elements of statistics and quantitative finance. They describe the outcomes of uncertain events, such as estimating future stock prices, where it is impossible to know in advance exactly what might happen. If you are familiar with random variables, you may want to skip down to the section “Building the Spreadsheet.” If you are not familiar with random variables, the following introduction contains the basics you need to follow the discussion. In this chapter, I cover only the aspects of the mathematics that apply directly to the pricing model. If you would like a broader introduction to random variables, there are several good online resources such as Wikipedia and Kahn Academy. I know it can be challenging to learn a new subject. It is often the case with quantitative material that you need a solid foundation before you feel comfortable applying it. In this case, however, the quantitative tools and the subject actually complement each other. So rather than feeling like you need to understand random variables completely before tackling option pricing, I think looking at option pricing is a very good way to learn about random variables. The same is true of another technical issue covered in this chapter—the use of the number “e” to calculate “continuously compounded” interest. If you are not familiar with the number “e” or continuous compounding, don’t worry. This is the best context to learn it. In addition, spreadsheets are good tools for working with random variables because they offer a more natural way to show the “matrix” relationships between outcomes and probabilities. They also provide 21

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clarity to the discussion because it is necessary to be exact when translating concepts into spreadsheet formulas. There is one technical issue, however, that is confusing to a lot of people. It is the reduction in the stock returns for the effects of volatility, which I talked about in Chapter 1. Because this issue often leads to mistakes—like it did for me—I want to highlight it by presenting the incorrect version of the pricing model first. Then I will make the correction to clarify exactly what it means in terms of projecting stock prices.

Random Variables In mathematics, a variable is used in an equation to describe some type of relationship. For instance, x and y are variables in the equation y = x + 2. To calculate y, you add 2 to x. When x is 2, y is 4. The relationship is exact, so no uncertainty is involved. Random variables are a little different. They are used when uncertainty does exist. For example, if you flip a coin, you don’t know whether it will be heads or tails. A random variable describes this uncertainty by (1) defining the possible outcomes, and (2) assigning a probability to each outcome. In the case of a coin flip, the two possible outcomes are heads and tails. The corresponding probabilities are 50% for heads and 50% for tails. Random variables typically are displayed as two columns: Outcome

Probability

Heads

50%

Tails

50%

Notice that the probabilities of the two events total 100%. This is a property of all random variables. For a random variable to be properly defined, it must include every possible outcome. Therefore, the probabilities of those outcomes must add up to 100% because no other things can happen.

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As another example, let’s say you are interested in the number of heads in two coin flips. The possible outcomes are 0, 1, and 2, based on the following four possible scenarios: Scenario 1: Flip 1 = Tails, Flip 2 = Tails (number of heads = 0) Scenario 2: Flip 1 = Tails, Flip 2 = Heads (number of heads = 1) Scenario 3: Flip 1 = Heads, Flip 2 = Tails (number of heads = 1) Scenario 4: Flip 1 = Heads, Flip 2 = Heads (number of heads = 2) These are the only possible outcomes. Each outcome is equally likely, with each having a 25% chance of happening. Because the number of heads equal to 0 or 2 can happen in only one way, the probability of each is 25%. Because the number of heads equal to 1 can happen in two ways, the probability is 25% + 25% or 50%. The random variable (number of heads) can be described in two columns: Outcome

Probability

0

25%

1

50%

2

25%

In summary, any time the outcome of an event is uncertain, a random variable can be used to describe it. A random variable has only two elements. The first is the possible outcomes, and the second is the probability of each of those outcomes occurring.

Mean (Expected Value) The mean, also referred to as the expected value, of a random variable is the weighted average of its possible outcomes, where the weights are the probabilities. For example, for the random variable defined as the number of heads in two coin flips, the mean is 1.0, calculated as follows:

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Outcomes

Probability

Weighted Outcomes

0

25%

0.0

1

50%

0.5

2

25%

0.5

Total

100%

1.0 = Mean or expected value

The first weighted outcome is 0.0 (0 × 25%). The second weighted outcome is 0.5 (1 × 50%), and the third weighted outcome is 0.5 (2 × 25%). The mean or expected value is the sum of these numbers, or 1.0. Admittedly, terminology can be confusing. In general, when I refer to the random variable itself, I am talking about two columns of numbers that define the random variable: the possible outcomes and the probabilities of those outcomes. When I refer to weighted outcomes or weighted values, I am talking about the column of numbers obtained by multiplying the outcomes by their probabilities. And when I refer to the average value or the mean or the expected value of the random variable, I am talking about the single number that is the sum of the weighted outcomes.

Cumulative Probabilities The cumulative probability of a particular outcome is the probability of that outcome and all outcomes that are smaller. For the example, the cumulative probability of 0 heads is 25%, as before. The cumulative probability of 1 head is 75% (25% for 0 heads plus 50% for 1 head). The cumulative probability of 2 heads is 100% (25% for 0 heads plus 50% for 1 head and 25% for 2 heads). In other words, the cumulative probabilities are calculated by adding the individual probabilities up to and including the specific outcome, as shown here: Outcome

Probability

Cumulative Probability

0

25%

25%

1

50%

75%

2

25%

100%

Total

100%

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The reverse is also true. If you know the cumulative probability, you can “back into” the individual probabilities. For instance, the probability of getting exactly 1 head is equal to the difference in the cumulative probability for 1 head (75%) and the cumulative probability for 0 heads (25%), or 50%. Outcome

Cumulative Probability

Probability

0

25%

25%

1

75%

50%

2

100%

25%

Total

100%

Because the cumulative probability can be used to back into the individual probabilities, it is just another way of describing the probability distribution. It contains exactly the same information in a different form. When working with random variables that have a finite number of outcomes, describing them using individual outcome probabilities is more common. However, the cumulative approach is easier when working with random variables that have an infinite number of outcomes.

Discrete and Continuous Random Variables There are two types of random variables: discrete and continuous. The number of heads is an example of a discrete random variable. Discrete refers to the nature of the outcomes. Sometimes this concept is stated in terms of whether you can count the outcomes. For the coin flip random variable, the answer is yes—there are three possible outcomes: 0, 1, and 2. In the case of other random variables, the answer is no. The most common random variable, the normal random variable, has outcomes occurring on a continuum between minus and plus infinity. For a random variable with an infinite number of outcomes, placing a probability on any particular point is impossible. Any particular number can be subdivided into smaller and smaller intervals. For instance, if you wanted to know the probability that the number –2 occurs in the normal random variable, the answer cannot be determined. The reason is

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that the probability of the number –2 occurring “exactly” is zero. No matter how precise you are, there is another, more precise decimal point that can be considered. Extending the decimal points to infinity (–2.00000000000000000...) makes it impossible to state exactly the probability of –2. Even though you cannot exactly measure the probability of –2, you can determine the probability of “–2 or less.” For continuous random variables, cumulative probabilities are typically used to answer questions about how likely something is. By taking the difference between cumulative probabilities at two points, you can determine the probability of the random variable having a value within that range. This is similar to the procedure described previously for the discrete random variable, in which we “backed into” the probability of a particular discrete point. In the case of continuous random variables, however, the probability corresponds to a “range of outcomes” rather than a particular outcome.

The Normal Distribution and NORMSDIST Note Because random variables are often associated with the shape of their probabilities when plotted on a graph, they are also referred to as distributions. For example, “normal random variable” means the same as “normal distribution,” and “standard normal random variable” means the same as “standard normal distribution.” The normal distribution is sometimes referred to as the ND, and the standard normal distribution is sometimes referred to as the SND.

The normal distribution has the shape of the familiar bell curve. It is symmetric, meaning that it is a mirror image of itself on either side of the middle. Only two parameters are needed to describe it. The mean is its average value, or middle, and the standard deviation measures how it spreads out.

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The standard normal distribution is just a special case of a normal distribution where the mean is 0 and the standard deviation is 1. Any normal distribution can be “standardized,” or converted into a standard normal distribution. This is a common practice, in order to compare normal distributions with different means and standard deviations to one another. Because the standard normal distribution is used so often in statistics and finance, published tables report both the cumulative probabilities and the probabilities associated with specific outcome ranges. Excel includes a built-in function called NORMSDIST that provides cumulative probabilities for the standard normal distribution.

Random Variables in the Option Pricing Spreadsheet Figure 1.1, the option pricing spreadsheet in Chapter 1, contains four random variables, each of which is a discrete approximation to the following continuous random variables: 1. The standard normal random variable 2. The stock return random variable 3. The stock price random variable 4. The option payoff random variable The values (possible outcomes) of these random variables appear in Columns A, C, D, and E, respectively. Because only one column includes probabilities, you might wonder where the associated probabilities are. Actually, all four random variables share this one set of probabilities. That happens because the values of the random variables are derived from each other. For instance, when the stock price is $134.99, the option payoff is $34.99. The option payoff is equal to $34.99 only when the stock price is $134.99. So, the likelihood of an option payoff of $34.99 is the same as that of a stock price of $134.99. As another example, if the stock price of $103.05 has a 3.96789% chance of occurring, an option payoff of $3.05 (that is derived from the stock price) has exactly the same chance of occurring.

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Building the Spreadsheet As mentioned earlier, I would like to continue with the incorrect version of the spreadsheet for now. In the incorrect version, the option price is $14.63, not the correct Black-Scholes price of $11.92. I want you to see the mistake I made. To highlight the mistake and keep the stock return and stock price assumptions clearly separated, Figure 2.1 includes two new cells and a new column. The two new cells, G3 and G4, contain the mean and standard deviation for the stock return random variable. The stock return random variable is the primary link between the pricing assumptions and the pricing logic, so it is helpful to break down this information and show exactly how it flows into the body of the spreadsheet. Column G shows the weighted stock price. The total of Column G, shown in Cell G95, is the mean of the stock price random variable. The new column provides a check on the overall movement of stock prices going forward. The figure also includes the column formulas. Even though the option price is not correct in this version of the spreadsheet, the formulas in the body of the spreadsheet, Rows 13–93, are correct and will not change. All the functionality of the full Black-Scholes formula in later chapters will flow into the model through the stock return random variable. The numbers in Figure 2.1 have not changed from those in Figure 1.1. The following is a step-by-step explanation of how to program the spreadsheet.

Step 1: Create a Discrete Version of the Standard Normal Distribution SND The Black-Scholes assumption is that “continuously compounded stock returns are normally distributed.” Therefore, the first step is to build a normal distribution. A common way to do this is to start with the standard normal distribution and then convert it into a normal distribution with the correct mean and standard deviation.

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Figure 2.1 The Formulas—Incorrect Option Price

Because the standard normal distribution is continuous and extends from negative to plus infinity, it does not fit on a spreadsheet. Instead, we can approximate the distribution by breaking it into discrete intervals or points. The spreadsheet approximates the standard normal distribution by converting the continuous version into a discrete version with 81 points. The 81 points start in Row 13 of Column A. The first point is –4.0 standard deviations. The next point is –3.9. Column A increases in 0.1 increments until Row 93, where the last point is 4.0 standard deviations. Each row represents an interval of the continuous distribution. In total, about 99.9% of the total probability of the standard normal distribution is between –4 and +4 standard deviations.

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There was nothing exact about why I picked 81 points in the original version. I wanted enough points so that it looked continuous on a graph but not so many points that it wouldn’t fit on two pages. It seemed to work well, so I kept it. It could just as easily be expanded to 101 points running from –5 to +5 standard deviations, or 121 points running from –6 to +6 standard deviations. The probabilities of each of the 81 points were calculated using Excel’s NORMSDIST function. Because the interval between points is 0.1, the points represent a range of values extending 0.05 above and 0.05 below the midpoint. For example, the point 3.0 is actually a range extending from 2.95 on the downside to 3.05 on the upside. The two endpoints (–4 standard deviation and +4 standard deviation) are different. Except for the endpoints, NORMSDIST is used to determine the probabilities as follows: NORMSDIST(Midpoint + 0.05) – NORMSDIST(Midpoint – 0.05) For the two endpoints, the entire tail of the distribution is included. For the point –4.0, NORMSDIST(–3.95) gives the probability of a value less than –3.95. This includes everything in the lower tail of the distribution. For the point 4.0, the formula 1 – NORMSDIST(3.95) includes everything in the upper tail. As a check to ensure that the entire distribution is included, Cell B95 verifies that the total probabilities add to 100%. Going forward, there is no need to assign these probabilities again. In fact, after they have been calculated once, you can save these values as constants. The only reason to recalculate them is if you wanted to build a more precise model with more than 81 points or if you wanted to assume a distribution other than the normal distribution. In the rest of the book, Columns A and B are constants and do not change. Columns A and B represent a discrete approximation to the standard normal random variable. This random variable has a mean of 0 and a standard deviation of 1.0, just like the standard normal distribution. Unlike the continuous version, it can be one of only 81 values.

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Step 2: Create the Stock Return Random Variable in Column C The next step is to convert the discrete standard normal random variable into an appropriate normal random variable. From the pricing assumptions: Volatility = 30% Risk-free rate = 0% Volatility is defined as the annualized standard deviation of the stock return random variable. In this case, the option term is one year, so the standard deviation is equal to the volatility. Therefore, Cell G4 (the standard deviation of the stock return random variable) is equal to Cell C8 (volatility). When the option term is anything other than one year, the two numbers are different. The mean of the stock return random variable is related to the risk-free rate. The mistake I made was to assume that they were equal. Continuing with this mistake for now, let’s leave Cell G3 equal to the risk-free rate of 0%. Now we use the mean (0%) and standard deviation (30%) to build the stock return distribution. We know that Column A has a mean of 0% and a standard deviation of 1.0. Note that a standard deviation of 1.0 can also be expressed as 100%. The next step is to convert Column A, a standard normal distribution (SND), into a distribution with the right parameters. This is no problem because the standard normal distributions can be scaled to produce a normal distribution (ND) for any specified mean and standard deviation using the following formula: ND = SND × Standard deviation + Mean In other words, we multiply Column A by the desired standard deviation and add the desired mean to create Column C, a normal distribution with the correct mean and standard deviation. For example, the stock return on Row 16 is: 1.11 = –3.7 × 0.30 + 0

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Or equivalently, in percentage terms: –111% = –370% × 30% + 0% In cell references, the formula is: C16 = A16 × G4 + G3 Here, G3 and G4 are the assumed mean and standard deviation, respectively, for the stock return random variable. As mentioned, a random variable consists of two sets of numbers. The first set is the possible outcomes. The second set is the corresponding probabilities. Because each number in Column C was calculated using the number in Column A from the same row, the probabilities are the same. In other words, Column C is 90% only when Column A is 3.0 or 300%. So the stock return random variable is completely described by the values in Column C and the probabilities in Column B.

Step 3: Calculate the Stock Price Random Variable in Column D Because we know the stock return in Column C, all we have to do now is translate the stock return into a stock price. Using (1 + i) to develop Column D is tempting. However, we are told that the BlackScholes assumption is that continuously compounded stock returns are normally distributed. What does this mean, exactly? To answer this, it might be helpful to briefly review interest compounding. Say that you have $100 in a bank account, and you want to estimate how much you will have at the end of one year, assuming an interest rate of 12%. The answer depends on the number of compounding periods. The simplest case is one compounding period (annually), where interest is credited once at the end of the year. With one compounding period, $100 at 12% grows to $112, as follows: $100.00 × (1 + 0.12) = $112.00

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Now assume that the compounding period is quarterly. This is a little more complicated because it is necessary to credit interest each quarter at 1/4 of the annual rate. Here is how the account grows at the end of each quarter: $100.00 × (1 + 0.12/4) = $103.00 at the end of quarter 1 $103.00 × (1 + 0.12/4) = $106.09 at the end of quarter 2 $106.09 × (1 + 0.12/4) = $109.27 at the end of quarter 3 $109.27 × (1 + 0.12/4) = $112.55 at the end of the year Because interest is applied at steps along the way, quarterly compounding earns interest on the interest credited earlier in the year. At the end of the year, you have 55¢ more with quarterly compounding. Fortunately, we can use the following formula to calculate the value in one step: $100 × (1 + .12/4)4 = 112.55 Monthly compounding is the most common method. For banks, crediting interest at the same frequency at which balances are reported makes sense. With monthly compounding, balances are credited with [1/12] the annual interest rate at the end of the month. Balances are updated and rolled forward each month. With monthly compounding, the account balance grows to: $100 × (1 + 0.12/12)12 = $112.68 With monthly compounding, the account balance is 13 cents higher than with quarterly compounding. It is possible to extend this process with more compounding periods. For instance, we could look at compounding done every day, hour, minute, and so on. The following mathematical expression helps determine what the maximum account balance would be, even if the compounding period is infinitely small—and it uses the number e: ex = The limit (as n approaches infinity) of (1 + x/n)n This is exactly the process we were working through. So to find out the maximum amount that we could make at 12%—that is, by

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compounding continuously—we raise the number e to the power of 0.12. e0.12 = $112.75 Notice that, by compounding infinitely fast, the account increases by only 7 cents compared to monthly compounding. Diminishing returns are realized as the rate of compounding increases. Figure 2.2 summarizes the results for different compounding periods.

Figure 2.2 Compounding period formulas

Now we have a definition of continuous compounding. To calculate the stock price random variable, we multiply the current stock price by the number e raised to the power of the stock return. Stock price = Current stock price × eStock return For example, in Figure 2.1, the stock price in Cell D18 is $34.99, calculated as: $34.99 = $100 × e–105% or, equivalently: $34.99 = $100 × e–1.05 In Excel, the number e raised to the power of X is EXP(X). In cell reference terms, the number in Cell D18 is: D18 = C4 × EXP[C18]

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One interesting and important aspect of this definition is that stock prices can never be negative. Even at very negative stock returns, stock prices are greater than zero. This is true because e–infinity approaches zero; it cannot go lower. This property of continuous compounding creates asymmetry in stock prices. At lower return rates, the values resulting from continuous compounding are fairly close to the values from simple interest. But at higher return rates, the differences become significant. This is especially true at high negative rates.

Step 4: Calculate the Option Payoff Random Variable in Column E The call option payoff is the difference between the stock price at expiration and the option strike price, not less than zero. For all stock prices below the option strike price, the payoff is zero. Column E shows the values of the option payoffs. For example, in Row 23, where the stock price is $40.66, the option payoff is zero. In Row 83, where the stock price is $245.96, the option payoff is: $145.96 = $245.96 – $100 Or in cell reference terms: E23 = MAX(0,D23 – C5) The probabilities in Column B apply to all rows. That is because the option payoff of $145.96 occurs only when the stock price is $245.96. The probability of this payoff is 0.0004447, from B23.

Step 5: Calculate the Weighted Option Payoff in Column F The weighted option payoff in Column F is the option payoff from Column E multiplied by its corresponding probability in Column B. The total of Column F, shown in Cell F95, is the weighted average of the option payoffs. It is also referred to as the mean or expected value of the option payoff random variable. It is also the option price.

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One of the advantages of the spreadsheet is the ability to interpret the option pricing formula. The option price is the weighted average of option payoffs.

Correcting the Mistake The mistake I made occurred in Column C. I overstated stock returns. Initially, one of the things I liked about the look of the values in Column C was that they were symmetrical: –120% at the top, +120% at the bottom, and 0 in the middle. As it turns out, Column C is not supposed to be symmetrical. The problem is that continuous compounding creates asymmetry in the pattern of stock prices. To counteract this effect, the mean of the stock return distribution must be reduced. In Chapter 1, I talked about the effect of volatility: There was a missing piece. I knew that volatility affects stock price simulations. The more volatile, the more the distribution of prices is dragged down. But I had not included anything in this spreadsheet to account for that. So, I went back to the book where McDonald pointed out: [W]e need to subtract 1/2 variance. (p. 597) That was the mistake. It is an important point and one that has interesting implications. It creates hidden side effects of volatility, such as creating a drag on returns and pushing the median stock price down. Before getting into the details of the adjustment, let’s talk briefly about the intuition behind the adjustment.

Volatility Reduces the Mean Looking at Figure 2.1, you can see that the stock returns in Column C are symmetrical. For instance, in Cell C23, corresponding to –3 standard deviations, the stock return is –90%. In Cell C83, corresponding to +3 standard deviations, the stock return is +90%. That is

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true for all the stock return values. The stock return for any standard deviation point is the negative of its mirror image. The problem is reflected in Column D. When we convert a stock return into a stock price, we lose the symmetry. In Cell D23, where the return is –90%, the stock price is $40.66. This is obviously not a 90% loss; it is more like a 60% loss. On the other end, in Cell D83, where the return is +90%, the stock price is $245.96. This is not a 90% gain—it is much more. If you made a bet with someone at equal odds, in this situation, you would expect to lose $90 or win $90. In fact, the odds of a 90% gain (Column B) are the same as those of a 90% loss, so the odds are equal. But the payoffs are not. The loss is not as much as you would expect, and the gain is more than you would expect. Why is this? It is a simple consequence of continuous compounding, the way e works. One reason for assuming continuous compounding is that it has a nice mathematical feature: It does not go below zero. No matter how large the loss, the answer is always zero or above. So “e” raised to a very large negative return, even minus infinity, will never go below zero. The same is true of stock prices. Because stock prices are bounded on the downside but not on the upside, stock prices in the model cannot be symmetrical. Otherwise, when the stock return is –120%, the stock price would be –$20, a result that makes no sense. We cannot change the way stock prices are calculated under the assumptions, so we must change the stock returns. Currently, they are overstated. It can be shown mathematically that the adjustment is 1/2 the variance (variance is defined as the square of the standard deviation). To verify the effect of the adjustment, I added a column to the spreadsheet to calculate the weighted stock price. Now the question is exactly how the adjustment should be applied. The answer: The adjustment is applied to the mean of the stock return distribution in Cell G3.

The Correct Version of the Spreadsheet In Figure 2.3, the stock return mean has been changed from 0% to –4.5%. This number, –4.5%, is the volatility drag, the amount by

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which stock returns are reduced to account for the asymmetry in stock prices. To see that it works, first notice that it reproduces the Black-Scholes option price, at least within 1 cent: $11.93. Second, notice the mean stock price in Cell G95. Before, it was $104.63; now it is $100.00.

Figure 2.3a The spreadsheet—correct option price

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Figure 2.3b The spreadsheet—correct option price (Continued)

Why is the stock price mean important? I talk about this much more later, but briefly, it has to do with the drift in the stock price. Drift refers to the tendency of the average stock price to move up or down over time. For example, a common assumption is that stock prices increase over time, to reflect the risk involved in owning the stock. However, in the Black-Scholes model, the drift in stock prices is not related to the riskiness of the stock; instead, it relates to the return that could be achieved on a risk-free investment, often approximated

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by the rate of return on a Treasury security. This rate is the risk-free rate shown in the pricing assumptions. The Black-Scholes assumption, the risk-free rate, is related to the drift but is not equal to the drift. For instance, when the drift is assumed to be zero, the stock price, on average, is not supposed to move up or down. Because the stock price is currently $100, the mean value should also be $100 at the end of the year. Otherwise, there is a built-in upward or downward bias in the price movement. In the previous version, Figure 2.1, the average stock price at the end of the year was $104.63. This alone should have alerted me that something was wrong. To highlight the stock price mean going forward, it is displayed in the heading in Cell G5. Notice that, in the corrected version, the mean is $100.00, as it should be with no drift. The effect of the adjustment to the mean is that every value in Column C has been reduced by –4.5%. The formulas in the body of the spreadsheet, however, are the same as before.

Note on Monte Carlo Simulation Monte Carlo simulation is a popular way to model securities. The method involves drawing random samples from a distribution to produce “what if” scenarios. The samples are drawn from the stock return distribution. To make the draws, you specify the mean and standard deviation of your distribution. If you are using a projection program, the volatility drag is probably calculated for you. But if you are performing simulations in Excel, make sure that the stock price does not include a drift you did not intend. In this case, to avoid the upward drift, the mean of the sampling distribution should be –4.5%, and the standard deviation should be 30%. This can also be stated as follows: The stock return random variable is normally distributed with a mean of –4.5% and a standard deviation of 30%. If you do not have experience with random variables or with using the number “e”, this chapter has covered a lot of new material. However, the only thing I did was develop the mechanics. There was no real thought involved. I just followed the cookbook, a statement of one assumption used in deriving the Black-Scholes equation, namely:

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“Continuously compounded returns on the stock are normally distributed” Let’s look back through the steps: 1. Knowing that I needed a normal distribution, I set up Columns A and B, the standard normal distribution—the typical starting point for building a normal distribution. 2. The standard method for creating a generalized normal distribution is to scale the standard normal distribution. So, Column C scaled the standard normal distribution (mean = 0, standard deviation = 1) into the particular normal distribution I was interested in (mean = –4.5%, standard deviation = 30%). This is the stock return distribution. 3. To get Column D, I followed the instructions to use “continuous compounding,” which meant using the number e to convert stock returns into stock prices. 4. Column E is just the option payoff. 5. Column F simply weights the payoffs by their probabilities in Column B. That is all there is to it. For a respectable quant, it is a ten-minute job. Even I, a wimpy quant making mistakes, did it in a couple of hours. In Chapter 5, we will complete the logic to price options under any range of assumptions. However, the basic framework of the model doesn’t change.

Optional: Additional Resources Introduction to Random Variables http://www.khanacademy.org/math/probability/independentdependent-probability/old_prob_videos/v/introduction-to-randomvariables

Introduction to the Normal Distribution www.khanacademy.org/math/statistics/v/introduction-to-thenormal-distribution

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3 An Overview of Option Pricing Methods This chapter provides an overview of common option pricing methods. It primarily serves as background to show how the spreadsheet or “visual” method fits into the mix. The chapter also provides the Excel code for the Black-Scholes formula used later in model building. Some of the material in this chapter is theoretical and is not used in later parts of the book. The more technical sections are only to give readers who are already familiar with option pricing methods a frame of reference. Please don’t worry if some of this doesn’t sink in. However, if it sounds interesting to you and you would like more information, there are many good resources available. For general options information, see The Options Institute, at the Chicago Board Options Exchange (CBOE) website (http://www.cboe.com/learnCenter/ OptionsInstitute1.aspx); Wikipedia; and Investopedia. For more detailed information and webinars, see the MathWorks/MatLab website (http://www.mathworks.com/products/matlab/). For online calculators and practical guidance on when to use various methods, see Hoadley.net (http://www.hoadley.net/options/options.htm). At the end of the chapter, I have listed additional resources that address practical limitations and theoretical issues.

The Black-Scholes Formula Fischer Black and Myron Scholes developed the Black-Scholes formula (in collaboration with Robert Merton) and published it in 1973, within days of the opening of the Chicago Board Options Exchange (CBOE). The formula, developed from the point of view 43

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of an options market maker, allowed trading firms to price and adjust exposures to market risks. To give you an idea of the importance of the formula, when legendary educator and Nobel Laureate Paul Samuelson was asked about advice for young people, he said: “Learn to use your computer. Learn your calculus. Learn the Black-Scholes option-pricing model”1 I think Samuelson emphasized Black-Scholes because it encourages a certain way of thinking. The Black-Scholes formula contains more than just the option price. I think Samuelson was talking about the information and logic behind the number. Any financial decision is really an evaluation of payoffs and probability distributions. These two pieces of information enable you to value an investment and measure its risk. This principle applies to every type of investment— not just stocks and options, but also bonds, commodities, real estate, derivatives, and structured securities.

History of the Black-Scholes Formula After Fischer Black earned his Ph.D. from Harvard in 1964 in applied mathematics and artificial intelligence, he met Jack Treynor at a firm called Arthur D. Little. Treynor was working on the Capital Asset Pricing Model (CAPM), and Black became fascinated. I worked on the Capital Asset Pricing Model because I wanted to discover the truth. ...the cruel truth. ...To get higher expected gain, you must take more risk.2 Black thought there must be a way to apply the CAPM to warrant pricing. (He worked with warrants rather than options because, at the time, warrants were traded on the stock exchanges, whereas options were traded only over the counter. This was before the opening of the CBOE.) He assumed that both stock and warrant prices would obey the model, and according to the “cruel truth” of pain and gain, the return of the stock and the warrant would be proportional to their risk as measured by beta, or relative volatility.

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Black translated his views about CAPM and the risk of warrants compared to stock into mathematical terms using a stochastic partial differential equation. But he couldn’t solve it. I applied the Capital Asset Pricing Model to every moment in a warrant’s life, for every possible stock price.... I stared at the differential equation for many, many months. I made hundreds of silly mistakes that led me down blind alleys.3 He said nothing worked. The value of the warrant was not related to the stock’s expected return or the expected return of any other asset. Then he began to work with Myron Scholes. Scholes had been working on the same problem, so they decided to join forces. As their work progressed, both were surprised to find that although CAPM focused on the tradeoff between risk and expected return, equations for solving the warrant pricing problem involved neither risk nor return. It seemed that they canceled each other out. Two stocks, each selling for $50 today, might have very different expected prices five years from now, each with its own risk profile. But in setting today’s price, investors have already weighed the less risky stock with a narrower price distribution against the more risky stock with a wider price distribution and concluded that each is worth $50 now. After Black and Scholes realized that the expected gain was not relevant to the pricing of the warrant or option, they made progress. In early 1970, Scholes talked to Robert Merton, his good friend at MIT, about what he and Black were trying to do. Merton had already been working with Paul Samuelson on warrant pricing and had broken new ground by introducing Ito’s lemma and other stochastic math advances. He wasn’t convinced that the CAPM was the way to approach the problem. The discussions and differences in opinion created a great atmosphere for achievement. In the end, Merton offered a second, more elegant derivation of the formula based on an arbitrage argument.

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Many years later, Black said: “A key part of the option paper I wrote with Myron Scholes was the arbitrage argument for deriving the formula. Bob gave us that argument. It should probably be called the Black-Merton-Scholes paper.”4 Even though the result is one of the most famous financial formulas ever developed, Black and Scholes couldn’t get it published. Journals at the University of Chicago and Harvard turned down the paper. Merton Miller eventually stepped in with a recommendation, and the published paper appeared in the May/June 1973 edition of the Journal of Political Economy, two and half years after the first draft had been rejected. It was incredible timing. The CBOE opened in April 1973, within a month of the publication. The combination of formula and marketplace has transformed financial practice ever since. Eugene Fama called the options pricing model and the innovation stemming from it “the biggest idea in economics of the century.” The explosive growth in the number of products and strategies related to the derivatives market doesn’t seem to be slowing down. By some estimates, the notational amount of derivatives now exceeds $400 trillion.

What Does the Black-Scholes Formula Look Like? Figure 3.1 is the Black-Scholes formula from Wikipedia. See http://en.wikipedia.org/wiki/Black–Scholes.

Note This form of the Black-Scholes formula does not price options for dividend paying stocks. Wikipedia also presents the extension for dividends. Also see Hoadley.net for discussion and tools for quarterly dividends.

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Figure 3.1 The Black-Scholes formula

Using this formula, we can price the option from the last chapter. Following the notation from Figure 3.1 gives: • The option term is one year, or time to maturity: T – t = 1 • The current or spot price of the stock: S = $100 • The option strike price: K = $100 • The risk-free rate: r = 0%, and • The volatility of returns: X = 30% Figure 3.2 provides the Excel code. Plugging the numbers into the formula gives a call option price of $11.92.

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Figure 3.2 Black-Scholes Excel code

Black-Scholes Assumptions The Black-Scholes formula makes certain assumptions, including: 1. Stock prices are continuous, meaning that there are no “jumps” in stock prices. But, in reality, stock prices do jump. Jumps are common after earnings announcements, M&A activity, product announcements, and other significant news. 2. Volatility and interest rates are set at the point of calculation and remain unchanged until expiration. This is also unrealistic, especially for longer-dated options. 3. Options can be exercised only at the expiration date. This is also referred to as European-style options. However, most options traded on exchanges are American style, meaning that they can be exercised at any point during the option term. 4. Continuously compounded returns are normally distributed. Evidence indicates that this assumption probably does not correctly describe the actual prices observed on exchanges.

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These assumptions create limitations in the practical use of the formula. The binomial method and Monte Carlo simulation are two commonly used alternatives to address some of the limitations.

The Binomial Option Pricing Method The binomial model, also known as the Cox-Ross-Rubinstein model, is a discrete-time numerical method for valuing options. It is often used to value American options and exotic options. Unlike Black-Scholes, the binomial option pricing model does not have a closed-form solution. Binomial model option pricing generates a pricing tree in which every node represents the price of the underlying financial instrument at a given point in time. The pricing tree handles options with nonstandard features such as path dependence, look-back payoffs, and barrier events. Pricing an option with the binomial model involves building the pricing tree, projecting stock prices at points along the tree, and then working backward through the tree to discount the value of various outcomes. Because the description of the underlying instrument occurs over a period of time instead of at a single point, the model prices certain types of options that are difficult with other models. For example, it is more accurate when valuing American options that are exercisable at any time during the option term, as well as with more exotic options that are exercisable at specific instances of time. It is also more accurate for longer-dated options on dividend-paying stocks. The binomial method is an important tool, but it is not particularly helpful in comparing structured securities in general. Although it is possible to extract explicit probabilities from the model, it is tedious compared to the visual method.

What Does the Binomial Method Look Like? Figure 3.3 is a screenshot from Hoadley.net showing the first ten steps of a binomial pricing for an option (one year, 30% volatility, zero

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rates). The model prices the call at $11.89, compared to $11.92 for the Black-Scholes and $11.93 for the visual method.

Figure 3.3 Hoadley.net online calculator

Relationship of Binomial Method to Black-Scholes The binomial model and the Black-Scholes model use similar assumptions, and the binomial model provides a discrete time approximation to the continuous process underlying the BlackScholes model. In fact, for European options without dividends, the binomial model converges to the Black-Scholes formula as the number of time steps increases. In terms of the probability distribution,

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the binomial model assumes that movements in price follow a binomial distribution. For a large number of trials, the binomial distribution approaches the normal distribution. For options with several sources of uncertainty (such as real options) and for options with complicated features (such as Asian options), binomial methods are less practical due to several difficulties; Monte Carlo option models are commonly used instead in these cases.

Monte Carlo Methods In some ways, the Monte Carlo method is the simplest of all pricing methods. It is also one of the most powerful. To get an idea of how the method works, consider this situation. Assume that you don’t know the chances of getting a head or tail on a coin flip. The Monte Carlo method is to flip a coin many times and then use the number of heads and tails to figure out the probabilities. If you flip the coin 1,000 times and get 490 heads and 510 tails, you use 490/1,000 as an estimate of the probability of heads. But if you already know that the probability of a head is 50%, you don’t need to flip the coins. The same is true with stock prices. If you know in advance what the probabilities are, you don’t need to go through a simulation to estimate them. The visual method calculates the probabilities directly. Monte Carlo methods of replicating Black-Scholes option pricing assume that you don’t know the probabilities and must estimate them by sampling from a distribution. The method works as follows: 1. Draw a random number from a uniform distribution. 2. Use the normal distribution inverse function to convert the random number into a standard deviation point along the SND. 3. Convert the SND into the appropriate stock return by adjusting for the mean and standard deviation. 4. Convert the stock return into a stock price.

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This is similar to the approach already described, except that we don’t need to sample from the distribution to figure out the probabilities—we already know what they are. In this case, we can derive the probabilities from manageable formulas in the spreadsheet. The real value of Monte Carlo is the valuation of options where the payoffs are complicated and where it may be difficult to derive probabilities from first principles. For example, we may want to project stock prices and vary the level of future volatility based on whether the stock price has trended up or down. Figure 3.4 illustrates Excel code for stock price sampling.

Figure 3.4 Excel simulation

Putting Visual Quant in Context The logic behind the option pricing approaches is similar, but the methodology is different. Each has pros and cons. Having all of them available is a good idea, depending on what you are trying to do. The Black-Scholes formula is easy and quick. The binomial lattice prices options where the stock price path is important (such as barrier options). Monte Carlo simulation is flexible and produces an explicit probability distribution. The visual approach has similarities to all of them.

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Compared to Black-Scholes The difference between the Black-Scholes formula and the visual method is the difference between an integral and a sum. The spreadsheet prices options with 81 points, but it could be expanded to a larger number of points. As the number of points grows larger, the spreadsheet becomes more accurate. The limit of a sum as the number of points increases to infinity is the definition of an integral. So the spreadsheet “becomes” the Black-Scholes formula as the number of points approaches infinity. Practically, however, the answers from the spreadsheet are already within a penny or two of the exact answer, so the incremental improvement is very small. Mathematicians like integrals for two reasons. First, they are more exact than sums. Second, integrals can be expressed as closed-form solutions. The Black-Scholes formula shown in Figure 3.1 is a closedform solution. It is elegant. And it is convenient. When the CBOE opened, floor traders needed a formula that could be programmed so that five or six numbers were entered on one side and the option price came out the other side. However, a closed-form solution mixes all the pieces together. That is, it obscures the probability distribution. It was not designed to add visibility to the calculation process; it was designed for efficiency. The visual approach stops at the summation stage before the pieces are combined. Sums may not be as mathematically elegant as integrals, but that is fine. Personally, I like the spreadsheet format because it “looks” like a matrix. Seeing two columns, with the outcomes and probabilities presented separately, reminds me exactly what a random variable is: two sets of numbers representing a particular “view” of reality. It is transparent. You can see exactly what is being assumed. So the visual approach is the same as the Black-Scholes approach, except for the step of transforming it into an integral and stating it as a closed-form formula. In mathematical terms: The limit of the visual method as the number of points approaches infinity is the Black-Scholes formula.

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Compared to the Binomial Method The binomial method assumes that a stock price can be only one of two values at the end of one time period. To get more realistic results, you can assume more time periods. For every new period, the possible outcomes increase by one. The method is very mechanical. It has some visual appeal, in that it emphasizes the diffusion of stock prices. The visual is not accurate, in that the diffusion appears to be linear with time, not the actual square root of time process. (MatLab “bends” the lattice to look like a square root function.) The visual method has 81 points. But if we reduce the number of points, it would be converged to the binomial method. On the other hand, a binomial method set to 80 periods (producing 81 possible outcomes) becomes the visual method. The visual method reduced to two points is the binomial method. Looking at it the other way, the visual method might be described as an 81-nomial method. In most texts, the binomial method is shown to approach BlackScholes as the number of steps increases to infinity. In this context, the visual method is a middle ground between the two points of the binomial method and the infinite number of points in Black-Scholes. The advantage of having a method between the binomial and BlackScholes is that you avoid the tedious stepping process of the binomial method and the differential equations needed to work with the continuous time Black-Scholes formula.

Compared to Monte Carlo The Monte Carlo method is really two steps. The first step is the production of the probability distribution. The second step is the calculation of the option price as the weighted average of payoffs using the distribution from the first step. The visual method is different in the first step but identical to Monte Carlo in the second step. With Monte Carlo, a large number of random draws are made from a normal distribution representing the stock return random variable. Then the results are summarized into

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a histogram to approximate the probabilities of the individual price points. The visual method produces these probabilities directly. In other words, the visual method assigns the probabilities based on known parameters of the normal distribution instead of sampling from known distribution and then estimating. As the number of simulated Monte Carlo draws approaches infinity, the Monte Carlo method converges to the visual method. In terms of the second step, both the visual method and the Monte Carlo method make it clear that option prices are simply the “probability weighted” average of option payoffs, which is often obscured by Black-Scholes.

The Dividing Line: Stochastic Partial Differential Equations As quantitative finance has evolved, a dividing line has developed between those who can work with stochastic equations and those who can’t. These equations, also called partial differential equations (PDEs), are not well known. That is why investment banks and hedge funds would rather hire physicists and math Ph.D.s and teach them investing than hire experienced investment strategists and teach them PDEs. As mentioned earlier, even Fischer Black had trouble with the PDEs: I stared at the differential equation for many, many months. I made hundreds of silly mistakes that led me down blind alleys. Nothing worked....” Differential equations can be difficult to work with. Sometimes making simplifying assumptions that don’t reflect reality is necessary just to get the formula into a manageable form. For instance, assuming that stock returns are “continuously compounded” is helpful, in that it avoids negative stock prices. It is also helpful because the number e has nice properties when used in differential equations and closed-form solutions. But this doesn’t mean it represents reality.

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One of the advantages of keeping random variables in matrix or spreadsheet form is the flexibility you gain in modifying the underlying assumptions. By keeping the components transparent, you might be able to build models that are not restricted by closed-form solutions or the difficulties of working with differential equations. For instance, changing the probabilities in Column B of the spreadsheet to reflect fat tails is trivial, but in the context of closedform solution, it could be messy or impossible. Another example is using utility functions to transform actual gains and losses into how investors perceive gains and losses.

The Purpose: Pricing versus Comparing Alternatives If the visual method is so closely related to Black-Scholes, doesn’t it suffer from the same limitations? Yes it does, at least in the form presented here. Because the visual method keeps the component parts of option pricing separate, it is possible to do things that are hard to do with closed-form solutions or lattices. In other words, the visual method can be expanded to handle special situations such as fat tails, customized utility functions, and returns that are not continuously compounded. If you are pricing OTC options with specialized payout structures or you are using an updated form of Black-Scholes that includes price jumps or stochastic volatility, you probably have specialized software available designed for that application. The purpose of the visual method was not to price options that could just as easily be priced with another method. It was to build a framework. Compared to Black-Scholes, the spreadsheet is very transparent. The assumptions can be traced to the actual numbers that make up the random variables. You can see exactly what information is being used and how. As we move forward in the second half of the book, the framework will be used as a simplified way of assigning probabilities to structured security payoffs. The objective is to measure the relative performance of two investment alternatives. Precise accuracy is not as important

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as making the right decision and being able to communicate why the decision was made. So the limitations applied to different option pricing methods are not as important. This is similar to the accuracy of a weighing scale. If you are selling gold, precision is critical. But if you just want to know the difference in the weights of two objects, the scale can be inaccurate and still give the right answer. In comparing two alternative investment structures, using a scale consistent with the price of hedging is more valuable than exactly pricing the underlying risk for either structure. Because the framework was developed as part of the option pricing model, it is consistent with option pricing and with the instruments available for hedging.

Additional Reading, Advanced Topics, and Resources During the financial crisis, the Dow Jones Industrial Average (DJIA) dropped 54 percent, from a high of 14,164 on October 9, 2007 to a low of 6,547 on March 9, 2009. Financial crises are not unusual. Banking and currency crises have occurred globally many times. Two excellent books on the nature and frequency of crises are: • Reinhart, Carmen M.; Rogoff, Kenneth. This Time Is Different: Eight Centuries of Financial Folly. Princeton University Press (2011). • Allen, Franklin; Gale, Douglas. Understanding Financial Crises (Clarendon Lectures in Finance). Oxford University Press (2009). Even though we know they are coming, predicting when the next crisis will happen and at what magnitude is notoriously difficult. So what does this have to do with option pricing? Several things. First, it raises questions about the underlying model—the random walk—and its ability to represent reality. In the following article, Zhou and Zhu discuss the relatioship between economic risks and the random walk model. An interesting finding is that “in terms of

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investing in the stock market, long-term investors should have prepared for a market drop of more than 50 percent.” The models show that a drop of 50% or more (50% drawdown from a high point) in the market is almost certain over longer periods of time. “Is the Recent Financial Crisis Really a ‘Once-in-aCentury’ Event?” Zhou, Guofu; Zhu, Yingzi. Financial Analysts Journal, Volume 66 Number 1, p. 24. ©2010 CFA Institute Abstract: The most common model for projecting stock prices into the future is the random walk model. Some argue that the random walk model is too simplistic to account for enough risks in the economy. For an interesting interpretation, see the long-run risks model of Bansal and Yaron (2004), one of the more viable models that explains simultaneously the equity risk premium puzzle; the risk-free rate puzzle; the high level of market volatility; and other facts about the stock market, consumption, and dividend-to-price ratios. Second, there are well-known patterns in option prices that deviate from those predicted by random walks. Understanding those patterns has the potential to help option market participants anticipate periods where it is better to be more conservative. For instance, there is evidence that information about equity markets can be detected earlier in the options markets. This raises interesting questions about how and when this information appears. As an example of this type of research, see the following article: “Implications for Asset Returns in the Implied Volatility Skew” Doran, James S; Krieger, Kevin. Financial Analysts Journal, Volume 66 - Number 1, p. 65. Abstract: Equity and option markets are distinct entities. Securities within each market are traded at different times and locations. Despite the physical constraints, the markets are highly integrated, and information that one market reveals should be seamlessly reflected in the other. Black (1975) and others have suggested that informed traders will first go to the option markets to use the leverage of option contracts to achieve higher returns. Consistent with that notion, many

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recent studies have shown that information contained in option prices has implications for both the returns and the volatility of the equity markets. These results suggest that “information spillover” occurs from the option markets to the equity markets. Our intent is to explain and summarize a subset of the information contained in option volatility and prices, specifically the information contained in the implied volatility skew and its relationship with future returns. Third, more advanced models of stock returns are available that account for such variations as jumps in stock prices that might occur because of earnings or unexpected annoucements. See for example, the jump-diffusion, CEV (constant elasticity of variance), and Heston models. Fourth, as an example of how corporate earnings announcements are reflected in volatility and the shape of the stock price distribution, see the following article: “Adjustments for Anticipated Days of Higher Volatility” R. L. McDonald, Derivatives Markets, 2nd ed. (Boston: Pearson Education, 2006). Abstract: When a firm announces earnings, volatility will be higher than on ordinary days. You can show that this is true by comparing the volatility of returns on earnings announcement days against that on other days. This effect is also apparent in option prices, which implies a higher volatility before an earnings announcement than afterward. This finding suggests that, in addition to the use of increasingly sophisticated mathematical pricing models, careful option pricing requires data sets that identify anticipated days of unusual volatility. Although the mathematics behind this material is beyond the scope of this book, I wanted to include the topics here to emphasize the difference between reality and the simplicity of the model underlying the Black-Scholes formula. As mentioned in Chapter 1: “Introduction,” comparing two similar instruments using simplistic models helps to eliminate the estimation limitations more than analyzing a single security.

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Endnotes 1. The Future of Life-Cycle Saving and Investing: The Retirement Phase, Nobel Laureate Panel Discussion, Research Foundation of CFA Institute. 2009. Edited by Zvi Bodie, Laurence B. Siegel, and Rodney N. Sullivan, CFA, available online at www.cfapubs.org 2. Peter L. Bernstein, Capital Ideas (Hoboken, New Jersey: John Wiley & Sons, 1992). 3. Ibid. 4. Ibid.

4 Value-at-Risk and Conditional Value-at-Risk Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are widely used measures of the risk of loss on an asset or portfolio of assets. These metrics measure the potential loss in value over a defined period for a given confidence interval. The Securities and Exchange Commission (SEC) in 1980 began requiring financial services firms to estimate losses that might be incurred over a 30-day period with 95% confidence. The SEC wanted to make sure these firms were holding enough capital to cover potential losses. This required an estimate of probability distributions, usually calculated from historical data across asset classes. Capital requirements continue to evolve in regulations such as Basel II and Basel III, for example, in the case of banks. In a broader context, VaR and CVaR can be applied to any financial asset, and definitions of time period and confidence intervals are flexible. In this chapter, to illustrate the concept, we will look at the calculation of VaR and CVaR on the stock used for option pricing in Chapter 2, “Random Variables and Option Pricing.” The option pricing spreadsheet gives visibility into the underlying probabilities of stock returns, stock prices, and gains or losses. So it is possible to answer a wide range of questions about these random variables, such as how likely a particular outcome is or how likely is it that a result will be larger than a specified amount.

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How Likely Is Something? How likely is it that a random number drawn from a normal distribution is within 2 standard deviations from the mean? About 95.5%. Within 1 standard deviation? About 68.3%. One of the things I realized after building the spreadsheet was that I don’t have to remember these numbers anymore. If I want to know how much of the normal distribution is contained within one or two or three standard deviations, I can just look at the probabilities in Column B and add them up. For example, in Figure 4.1a and 4.1b, the probabilities associated with the points starting at –2 standard deviations and ending at +2 standard deviations are dragged over to Column E and summed in Cell E95. Because the spreadsheet is only an approximation, it is not exact, but it’s still pretty close. The approximate probability is 96% (0.9596356 in Cell E95). If you want to be exact, you can calculate the probabilities in Excel using the same function used earlier, NORMSDIST. For example, to calculate the exact number corresponding to a range of –2 standard deviations to +2 standard deviations, use the following: NORMSDIST(2.0) – NORMSDIST(–2.0) The first term is the cumulative probability up to 2.0 standard deviations. The second term is the tail distribution below –2.0 standard deviations. The difference includes the area between +/–2 standard deviations: 0.97724987 – 0.02275013 = 0.95449974 The exact number is 95.45%, versus the spreadsheet’s 95.96%. The reason for the difference is that the spreadsheet is measuring a slightly different range. Because each point in the spreadsheet extends above and below its midpoint, the range selected is actually –1.95 to +2.05 standard deviations. The range is a little larger, and so is the probability. In my work, this is not normally a problem, but if you

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want more precise numbers, you can modify the approach to exclude half the probability in the endpoints to get a comparable range.

Figure 4.1a How likely is something?

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Figure 4.1b How likely is something? (Continued)

In this case, the two endpoints each contain 0.0054058. Excluding half of each from the previous total, you get this: 0.9596356 – 0.5 × 0.0054058 – 0.5 × 0.0054058 = 0.9542297 Now the answer is very close, 95.42% versus 95.45%.

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In the approximate world, we have access to individual probabilities, and we can add them for any particular range of stock prices or gains and losses we are interested in calculating.

How Likely Is a Loss Greater Than $25? In Column F, “How Likely is Stock Loss > $25?,” the probabilities for all stock prices less than $75 are dragged over and added. The answer is 19.77%, as shown in Cell F95. This works because there is a one-to-one relationship at each step in the process of determining stock prices. Reading across any row, the probability of a particular standard deviation point is the same as the probability of the return and the stock price shown on that row. An example is –1.0 standard deviations. The probability that a randomly drawn number from the SND is –1.0 is 2.41971%. This is also the same probability that the stock return is –34.5% and that the stock price is $70.82. Generally, you can answer any question about the chances of an asset prices and gains and losses by selecting which of the 81 outcomes should be included and adding them.

How Likely Is an Outcome in the Tail of the Distribution? Let’s say you are interested in looking at what stock price (or stock loss) might occur with 5% or 10% probability. The easiest way to do this is to use the cumulative probability function. Column G shows the cumulative probabilities, calculated by adding the probabilities of all rows up to and including the current row. The closest number to 5% is 4.947% in Cell G36. It corresponds to a standard deviation point of –1.7. Reading to the left, the corresponding stock price is $57.41, which translates into a stock loss of $100 – $57.41 = $42.59. This means that, approximately 5% of the time, the stock loss will be $42.59 or higher.

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Any time you are asking a question about the left or lower tail of the distribution, cumulative probabilities are convenient. The previous question “How likely is a stock loss > 25%?” is about the tail. We just answered this question by looking at the specific points that should be included and adding them, but because it is a lower tail question, we can also answer it by looking in the stock price column that would produce a loss of greater than $25 and reading the answer directly from the cumulative column. In this case, the first stock price producing a loss greater than $25 is $72.98 in Row 44. Reading across, the cumulative probability is 19.77%, the same as before.

Value-at-Risk Value-at-Risk (VaR) is directly related to the questions about how likely something is to happen. It is also an example of how a topic can sound complicated when expressed in words, but very easy when expressed visually in a spreadsheet. The definition of VaR from Wikipedia (http://en.wikipedia.org/wiki/Value_at_risk) follows: For a given portfolio, probability and time horizon, VaR is defined as a threshold value such that the probability that the loss on the portfolio over the given time horizon exceeds this value (assuming normal markets and no trading in the portfolio) is the given probability level. For example, if a portfolio of stocks has a one-day 5% VaR of $1 million, there is a 0.05 probability that the portfolio will fall in value by more than $1 million over a one day period if there is no trading. Informally, a loss of $1 million or more on this portfolio is expected on 1 day in 20. A loss which exceeds the VaR threshold is termed a “VaR break.” Thus, VaR is a piece of jargon favored in the financial world for a percentile of the predictive probability distribution for the size of a future financial loss. VaR has five main uses in finance: risk management, financial control, financial reporting and computing regulatory capital. VaR is sometimes used in non-financial applications as

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well. Important related ideas are economic capital, backtesting, stress testing, expected shortfall, and tail conditional expectation. The definition says that “for a given portfolio, probability and time horizon” VaR is a “threshold value.” Let’s apply the definition to the earlier example. In that case, the portfolio is one share of stock purchased for $100. The probability is 5%, and the time horizon is one year. We are looking for the threshold value. Looking at Figure 4.1 the 5% tail occurs at Row 36. That is, 5% of the time, a stock price of “$57.41 or less” will occur. This means that a stock loss of “$42.59 or higher” will occur 5% of the time. The threshold number is $42.59. Does this mean VaR is equal to $42.59? Yes. VaR is nothing more than the question we asked earlier. The spreadsheet enables us to interpret VaR in simple terms. In spreadsheet form, VaR is a no-brainer. You just look it up. The entire process is a matter of looking at the tail probability you are interested in, such as 1%, 5%, or 10%, and reading to the left to see what the stock price is; from the stock price, you can calculate the gain or loss.

The Formula Approach The mathematical expressions for VaR can be complex. The typical approach to VaR is to start with an expression for the lognormal probability of a particular stock price and work backward through the logic that we went through in building the spreadsheet. Following the logic produces a page of expressions, ending with a number such that the level of loss represented by that number will be exceeded 5% of the time. If we go through that process here, the answer would be approximately $42.59. A note on accuracy: At 5%, the difference between the cumulative probability in Row 36 (4.947%) and (5.000%) is very small. At 10%, the closest number is in Row 40 (10.565%). If you want more precise numbers, you can interpolate between the results shown here or add more standard deviation points to the model. I have never needed to do this. Part of the reason for this is the very unstable nature of VaR.

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Later, in the discussion of regime shifts, you will see how drastically VaR can change under adverse conditions.

Multiple Stock VaR The VaR of two or more stocks is normally less than the sum of the VaRs of the individual stocks. This is due to diversification effects. As long as the stocks are not perfectly correlated, they might not experience losses (or heavy losses) at the same time. The extent to which the stocks move together is reflected in the correlation coefficient, and this information is used in the standard formulas for computing multiple stock VaR. The correlations are usually the same as those used in developing mean-variance optimized portfolios. This book does not cover this topic, for a couple reasons. First, it is traditional and well known, and thousands of articles and books cover the subject. Second, meanvariance optimized portfolios lack a mechanism for providing effective downside protection in market crashes. In other words, it doesn’t work when you need it most. I am not saying that a portfolio should not be diversified or that mean-variance is not an elegant way of building a portfolio. It is. And it works well to capture risk premiums 90% of the time. But in terms of risk management, it is important to recognize that correlations break down under stress. Under extreme stress, asset correlations go to 1.0. When correlations go to 1.0, multiple stock (or asset class) VaRs become nothing more than the sum of individual VaRs. In practical terms, this is a matter of what you want to accept as true in market turbulence. If you believe that diversification effects fail in crashes and you are interested in what happens during a crash, you can ignore the mathematics of correlated assets and the diversification benefits that come from them. Look at the individual VaRs and add them up.

Stock and Option VaR One of the objectives of structured securities is to strengthen the diversification benefits in ways that do not disappear when you need

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them. For example, a stock and a short call on that stock provide diversification even in extreme events. By definition, the values of these two securities move in opposite directions, the ultimate in diversification. In later chapters we examine how to calculate structured security VaR as a standard component of the model. Using the spreadsheet method, calculating a combination stock and option VaR is no harder than doing so for a stock VaR alone. If you try to solve this problem using formulas, it gets complicated quickly.

Conditional Value-at-Risk You can see from the example what a simple concept VaR is. Many think it is too simple, and it has been widely criticized, as follows (from Wikipedia, http://en.wikipedia.org/wiki/Value_at_risk): VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 1997 debate between Nassim Taleb and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR ignored 2,500 years of experience in favor of untested models built by non-traders. Was charlatanism because it claimed to estimate the risks of rare events, which is impossible, gave false confidence and would be exploited by traders. More recently David Einhorn and Aaron Brown debated VaR in Global Association of Risk Professionals Review. Einhorn compared VaR to “an airbag that works all the time, except when you have a car accident.” He further charged that VaR led to excessive risk-taking and leverage at financial institutions, focused on the manageable risks near the center of the distribution and ignored the tails, created an incentive to take “excessive but remote risks,” and was “potentially catastrophic when its use creates a false sense of security among senior executives and watchdogs.” To address some of these problems, CVaR evolved as a new metric. CVaR is also referred to as Expected Shortfall and Average Value at Risk, among other names.

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Expected shortfall (ES) is a risk measure, a concept used in finance (and more specifically in the field of financial risk measurement) to evaluate the market risk or credit risk of a portfolio. It is an alternative to value at risk that is more sensitive to the shape of the loss distribution in the tail of the distribution. The “expected shortfall at q% level” is the expected return on the portfolio in the worst % of the cases. Expected shortfall is also called conditional value at risk (CVaR), average value at risk (AVaR), and expected tail loss (ETL). ES evaluates the value (or risk) of an investment in a conservative way, focusing on the less profitable outcomes. For high values it ignores the most profitable but unlikely possibilities, for small values it focuses on the worst losses. On the other hand, unlike the discounted maximum loss even for lower values of expected shortfall does not consider only the single most catastrophic outcome. A value often used in practice is 5%. Expected shortfall is a coherent, and moreover a spectral, measure of financial portfolio risk. It requires a quantilelevel, and is defined to be the expected loss of portfolio value given that a loss is occurring at or below the quantile. To give you an idea of how messy this version of VaR looks in formula terms, Figure 4.2 shows the Wikipedia page. I would rather not try to explain this form of the definition. Instead, I would like to go back to the spreadsheet. By making a minor addition, we will be able to calculate CVaR. But first, it is helpful to look at the concept of conditional expectations. This concept makes understanding CVaR easier.

Conditional Expectations If I told you I’d flipped a coin and the outcome was not heads, you would know that it was tails. The mathematical expression for this looks something like this: Prob[Tails | Not Heads] = Prob[Tails] / Prob[Possible Outcomes] = 50% / 50% = 100%

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Figure 4.2 Wikipedia definition of VaR

Here, the vertical line in the first bracketed term is read “given that” or “conditional on.” The expression in words is: The probability of tails, given that the result is not heads = The probability of tails / The probability of all possible outcomes This adjustment scales up the probability of a tail, “given that” it is not a head or “conditional on” it not being a head. Given that it is not a head, the probability of a tail jumps up from 50% to 100%.

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Walking through a more complicated case may be helpful. Look at what happens when we count the number of heads in two coin flips. From the earlier example, we know the three possible outcomes and their probabilities: 0 Heads, with 25% probability 1 Head, with 50% probability 2 Heads, with 25% probability Now assume that we are told the answer is not 1 head. That means we are left with two possible outcomes, 0 heads and 2 heads. We need to scale up the probability of each, given the new information, so that the new possibilities add up to 100%. Prob[0 Heads | Not 1 Head] = Prob[0 Heads] / Prob[Possible Outcomes] = 25% / 50% = 50% The same applies to 2 heads. It also has a probability of 50%. In effect, this takes one random variable (the number of heads in two tosses) and defines a new, related random variable based on some additional information (the number of heads in two tosses, conditional on the answer not being 1 head). The new random variable is: 0 heads with 50% probability 2 heads with 50% probability Now let’s look at the spreadsheet for an illustration of CVaR at the 5% level. We are interested in the expected value of the conditional distribution—that is, conditional on the fact that the outcome is in the 5% tail. Before, we didn’t know where the outcome was, so we were dealing with the entire length of the distribution form –4.0 to +4.0 standard deviations. Now we are told that the outcome has already occurred and that it was bad. It is somewhere in the lower 5% tail. Figure 4.3 calculates the revised probabilities of each point, given the new information.

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Figure 4.3 Conditional Value-at-Risk

The first four columns are the same as in Figure 4.1. Column E is the tail probabilities we are interested in. The values in Column F are the tail probabilities divided by 0.0494715 (the total probability of the possible outcomes). This scales up each outcome so that they now represent the total universe of possible outcomes, which means they add to 100%, as shown in Cell F38. Column G is the weighted losses [($100 – Column D) × Column B]. The sum of the weighted losses is $48.30. This number is the CVaR.

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Comparing VaR and CVaR Here is a comparison of the VaR and CVaR statements: 5% VaR is $42.59 means: Given no information about the outcomes, 5% of the time, the stock loss will equal or exceed $42.59. 5% CVaR is $48.30 means: Given that the outcome is in the 5% tail, the average loss is $48.30. CVaR is a more conservative number because it already assumes a bad outcome. But CVaR suffers from the same instability in stock prices during market turbulence that VaR does. Just because it is a more complicated formula does not mean it is a better indicator of risk. After all, there is no new information in the spreadsheet. By that I mean we didn’t add any new numbers—we used the same numbers in a different way to calculate CVaR. Proponents point to the fact that CVaR is intended to give better information when the shape of the distribution is not normal. It is true that, with fatter tails, CVaR does give larger numbers proportionally than VaR.

Conditional Value-at-Risk with Fat Tails Figure 4.4 presents the extreme case. In this example, the entire 5% tail probability is placed at –4.0 standard deviations. At this point, the stock price is $28.79 and the loss is $71.21. Because this is the only outcome, CVaR is also equal to $71.21. This illustrates that CVaR weights the outcomes by the relative probabilities in the tail. The point here is that we already knew that the worst case was a stock price of $28.79, which means a loss of $71.21. The issue with VaR, CVaR, and fat tails is what you do with the information. The worst case is $28.79 under a particular set of assumptions and this pricing model (which uses “continuous compounding of interest”). The real worst case is a stock price of $0 and a loss of $100. In the end, risk metrics are only as valuable as the decisions tied to them.

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Figure 4.4 Conditional Value-at-Risk with fat tail

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5 Full Black-Scholes Model In this chapter, the option pricing spreadsheet is expanded to add full Black-Scholes functionality. Full functionality refers to the ability to price options under any set of assumptions, including option terms, stock prices, strike prices, risk-free rates, dividend rates, and volatilities. To get started, let’s review the relationships between random variables, discussed in Chapter 2, “Random Variables and Option Pricing.” Figure 5.1 is an updated version of Figure 2.1, where the Stock Return Mean has the correct value of –4.50%. In the figure, Column A contains the values of the discrete version of the standard normal distribution, and Column B shows the corresponding probabilities of those values. The combination of Column A and Column B defines the first random variable: the discrete version of the standard normal distribution (or just the standard normal distribution, where we keep in mind that it is an approximation to the continuous version). The next random variable stock return is defined by Columns C and B. Column C contains the individual values of stock returns, and Column B contains the corresponding probabilities. The random variables stock price and option payoff are similar, with all four random variables sharing the probabilities from Column B. To summarize, the four random variables are: 1. Standard normal fistribution (Columns A and B) 2. Stock teturn (Columns C and B) 3. Stock price (Columns D and B) 4. Option payoff (Columns E and B) 77

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Figure 5.1 Modified Figure 2.1: the spreadsheet formulas

As shown in the figure formulas, the standard normal distribution is converted into the stock return random variable with the correct mean and standard deviation by multiplying by the standard deviation and adding the mean: Stock Return = SND × Standard deviation + Mean Then, the stock return random variable is converted into the stock price random variable using continuous compounding of returns: Stock Price = eStock Return

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Next, the option payoff random variable is calculated: Option Payoff = Stock Price – Strike Price, not less than zero Finally, the option price is the sum of the weighted average option payoffs. In other words, the option price is the expected value of the option payoff random variable. This process traces the numbers in the body of the spreadsheet given the numbers in Cells G3 and G4. What we have not done yet is to talk about how to calculate Cells G3 and G4.

Adding Functionality to the Model In the earlier examples, we assumed the option term was one year, the risk-free rate was zero, and there were no dividends. Limiting the assumptions made it easier to highlight the relationships between the random variables. Now we are ready to add full functionality so that the spreadsheet will calculate option prices under any set of assumptions. It may be a little surprising, but we can extend the model to handle any term, stock price, strike price, interest rate, dividend rate, or volatility just by expanding the definition of the mean and standard deviation of the stock return random variable in Cells G3 and G4. All the relationships in the body of the spreadsheet stay the same, so it is not necessary to change anything. The formulas in Figure 5.1 are still correct. There is one other minor change: an interest discount step. When we used zero interest, the value of the option payoffs at option expiration were the same as those at time 0. Now, with a nonzero interest rate, it is necessary to apply an interest discount to the payoffs at expiration to discount them back to today’s values.

Stock Return Mean (Cell G3) The stock price might be the more visible random variable, but the stock return random variable is the driver of the model. Of the

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six pricing assumptions, four of them flow directly into the model through the stock return random variable. The stock return mean is a function of: 1. Time in years 2. Risk-free rate 3. Dividend rate 4. Volatility The complete formula for the mean is shown in Figure 5.2.

Figure 5.2 The Stock Return Mean formula

In this example, the mean is –4.5%. This was calculated earlier as a reduction to the mean, to account for the effects of volatility and the asymmetry of stock prices. If we refer to this reduction as the volatility drag, we can restate the formula in Cell G3 as: Stock Return Mean = (Risk-free rate – Dividend rate – Volatility drag) × Time in years

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Where the volatility drag = 0.50 × C8^2 Or in cell formula terms: G3 = (C6 – C7 – 0.50 × C8^2) × C3 In terms of how each element affects the direction of the mean, the risk-free rate (C6) increases the mean; the dividend rate (C7) and the volatility drag, both with negative signs, decrease the mean. The option term (C3) scales the mean in proportion to time. If you cut the time in half, you cut the mean in half. This is different from the standard deviation, where the option price is a square root function of time, as discussed in the later section, “Stock Return Standard Deviation (Cell G4).”

Interpreting the Stock Return Mean The way I think of these variables is in terms of how they affect the stock price drift. The drift is a measure of the tendency of the stock price to go up or down between now and option expiration. The drift in stock price depends on the relationship between the stock return mean (G3) and the volatility drag: If G3 is equal to the volatility drag, the drift is zero. If G3 is greater than the volatility drag, the drift is up. If G3 is less than the volatility drag, the drift is down. The risk-free rate increases the drift. This makes sense because the risk-free rate can be thought of as the return on a competing asset. If Treasury rates go up, stock prices need to increase over time to keep up. The dividend rate decreases the drift. If a stock pays dividends, the price should go down by the amount of dividends paid. Similarly, the volatility drag decreases the drift. The higher the volatility, the lower the drift, meaning there is a stronger tendency for the stock price to trend down over time. In the previous example, the formula is: Mean = (0 – 0 – 0.5 × 0.30^2) × 1 = (–0.5 × 0.09) × 1 = –0.045, or –4.5%

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By changing the terms, you can see what happens to the mean. For example, if the option term were shorter, the mean would be lower. If the term were 6 months instead of a year, the mean would be: Mean = (0 – 0 – 0.5 × 0.30^2) × 0.5 = (–0.5 × 0.09) × 0.5 = –0.0225, or –2.25% The mean is cut in half when the option term is cut in half, as expected. If the risk-free rate were 4% instead of 0%, the mean would be: Mean = (0.04 – 0 – 0.5 × 0.30^2) × 1 = –0.5% This makes sense if you think about the competing asset. If a Treasury pays 4%, the stock would need to “drift” up by the same amount to keep pace. If the stock pays a dividend of 4%, what happens? Notice that the dividend term appears in the same expression as the drift, but with a negative sign. If the risk-free rate is 4% and the dividend yield is 4%, the two cancel each other out and the mean remains –4.5%. A dividend acts as a negative drift on the stock price. This makes sense because the dividend tends to decrease the overall value of the company compared to a company that does not pay a dividend. Of course, the total value is not affected. The dividend plus the remaining company value are the same in either case.

Stock Return Standard Deviation (Cell G4) The definition of volatility is the annualized standard deviation of the stock return random variable. In the case where the option term is one year, the volatility in Cell C8 is equal to the stock return standard deviation in Cell G4, both of which are 30% in the prior example. Now, we can generalize the formula for standard deviation by defining Cell G4 to be:

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Standard deviation = Volatility × Square root (Time in years) Or in Excel terms, as shown in Figure 5.3: G4 = C8 × C3^0.5

Figure 5.3 The stock return standard deviation

You can see from the formula that standard deviation is proportional not to time, but to the square root of time. This is a property of random walk diffusion processes. The square root behavior of diffusion processes has interesting implications for option pricing. For example, the price of a four-month option is not four times the price of a one-month option; it is only two times the price. This can be thought of as a time-volume discount in option pricing. A technical point is worth mentioning here: Normally, the volatility drag is stated as half the variance, where variance is equal to “volatility squared.” The technical distinction is between Cell C8, the assumed annual volatility, and Cell G4, the standard deviation of the stock return random variable. When the option term is one year, Cell C8 and Cell G4 are the same. But when the option term is anything

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other than one year, the two numbers are different. Keep in mind that the volatility drag is calculated using Cell G4, not Cell C8. This is a potential point of confusion because Cell C8 is referred to as the volatility.

Discount Factor Option payoffs occur at the time of expiration. For example, if the option term is one year, Column E displays the payoffs that will happen one year from today. However, in option pricing, we want to know what those payoffs are worth today, so we need to discount them with interest for one year. In the earlier examples, the assumed interest rate was zero, so we didn’t need to include the discount. Now we do, so Cell G96 includes the discount factor, calculated as: Discount factor = 1/e(Risk-free rate × Time in years) Figure 5.4 shows this in Excel terms.

Figure 5.4 The discount factor

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As an example, if the risk-free rate is 4% and the option term is one year, then: Discount factor = 1 / EXP(0.04) = 1 / 1.04081 = .96079 In today’s rate environment, the risk-free rate is fairly low, especially for shorter-term options. For monthly options and the near-zero current level of interest rates, this aspect of option pricing is almost meaningless. This is the final step in option pricing. Applying the discount factor to the payoffs occurring at option expiration puts the option value in today’s dollars.

Stock Price Median There is another metric I like to track, the stock price median value. It is related to the volatility drag and is a predictable side effect of volatility. Specifically, increased volatility pushes the distribution of stock prices down. And as stock prices become more volatile, losses occur more frequently. The median value of a distribution is the value where 50% of the time the result is larger and 50% of the time the result is smaller. The median stock price is rarely mentioned. When most analysts talk about future projections of asset prices, they talk about mean and variance, not median. But median is important. Figure 5.5 illustrates what happens to the median stock price in future years, depending on the volatility of returns. The bold black line tracks a fixed-income instrument such as a bond as it accumulates in value over a ten-year period at 5% return. The bond worth $30 today grows to almost $50 after ten years.

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Figure 5.5 Median stock price distribution over multiple periods

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Now consider a stock worth $30 today, with an annual expected return of 10%, or 5% more than the bond. This 5% can be viewed as the risk premium or extra return in exchange for taking more risk. But how should we think of this risk? What does it look like? In terms of average future stock prices, or the mean stock price, volatility has no effect. For all levels of volatility, the expected stock price after ten years is equal to $81.55. Relative to the bond, we pick up about $32 in equity risk premium if everything goes according to plan. More risk, more return. Right? Well, maybe. But what do you have to do to get that extra return? How comfortable is it over the ten-year period? In other words, what would happen if we simulated the outcome of the stock prices assuming different levels of volatility? The graph gives us the median stock price. The first line shows the case for 0% volatility, which is obviously unrealistic, but we can still interpret the results. When volatility is 0% every year, the return is 10%. In that case, the mean and median are both equal to $81.55, which is the only value the stock price can be. Now look at what happens when volatility increases to something more realistic in the stock market, such as 30%. In that case, the median value at the end of ten years is $52. That means that, half the time, the ending stock price would be above $52, and half the time it would be below $52. Even though the expected value is still $81.55, the halfway point, the median, drops by almost $30 and is now roughly the same as the bond. As an investor, if you hold the bond, you know you will have about $50 at the end of ten years. You might have invested instead in the “hopefully” much higher-yielding equities. Nevertheless, half of the time, you will have taken much more risk but not outperformed the bond. You are led to believe that the longer you hold equities, the better your chances are of capturing the risk premium. That may be true, but the range of outcomes is highly affected by volatility. As a more extreme case, look at the 60% volatility line: Half the time, the stock price after ten years will be below $13.48. This is an extraordinary result and one that most investors don’t realize when investing in highly volatile securities. How many people would be willing to accept an investment worth less than half of what they paid

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for it roughly 50% of the time? Most people think about the 10% expected return. The mean value of $82 sounds great, but to get it with 60% volatility, you should expect—half the time—to have less than $14. As McDonald (2004) points out, “More than 50% of the time, a lognormally distributed stock will earn below its expected return. Perhaps more surprisingly, if volatility is large, the stock will lose money more than half the time!”1 In the spreadsheet, the stock median value is easy to determine. It is always equal to Cell D53. That is because Row 53 is the center of the distribution. It corresponds to the zero point of the SND, meaning that half the distribution is above and half the distribution is below this point. Because the probability distribution in Column B also applies to the other Columns, Row 53 is the center or median value of the other random variables as well.

Summary of New Formulas Figure 5.6 summarizes the new formulas for Cells G3–G7 and the discount factor in Cells F96 and G96.

Pricing Put Options So far, all the focus has been on call options. We can also price put options by adding two Columns to calculate the put option payoff and weighted put option payoff. Another way is to change the Call Option Payoff Column to a Call (and Put) Option Payoff Column, and price both calls and puts at the same time. Figure 5.7 shows this setup. It works because, at any particular strike price, the payoff can be positive for only one of the options. If the call has a positive payoff, the put payoff is zero, and vice versa.

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Figure 5.6 The stock return standard deviation

In Figure 5.7, Column E has been changed to equal the difference between the option strike price and the stock price. When the payoff is negative, it is the amount of the put payoff. When it is positive, that is the call payoff. Columns F and G pick up the appropriate number, as follows: Column F = Max(0, Column E) × Column B Column G = -Min(0, Column E) × Column B In this case, the prices are the same because the interest rate is zero and the strike price is equal to the current stock price.

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Figure 5.7 Call/put combination

Example With the changes to Cells G3 and G4 and the addition of the discount factor, the spreadsheet now calculates Black-Scholes option prices for any values of the input assumptions. Figures 5.8a and 5.8b show the full spreadsheet for the new current assumptions: Assumptions

Prior

Current

Time in years

1 year

3 months

Stock price

$100

$100

Option strike price

$100

$105

Risk-free rate

0%

4%

Dividend rate

0%

2%

Volatility

30%

20%

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Figure 5.8a Pricing example

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Figure 5.8b Pricing example (Continued)

Cell G8 is the actual Black-Scholes formula option price. In this case, the answers are the same, $2.21.

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Effects of Assumption Changes Notice there were no changes in Columns A and B. These numbers will not change (as long as we assume a normal distribution). In fact, in this spreadsheet, Columns A and B are constants. Column C is a normal random variable determined by the mean and standard deviation in Cells G3 and G4. In other words, Cells G3 and G4 completely determine the values in Column C.

The Stock Return Mean (Cell G3) Let’s compare the mean to the previous value. Start with the volatility drag. It indicates how much the mean is reduced solely due to the level of volatility. Volatility drag = 0.5 × Variance Previous: 0.5 × 0.302 = 0.045 = 4.5% New: 0.5 × 0.202 = 0.020 = 2.0% Note: By definition, the variance is equal to the standard deviation squared. Stock return mean = (Risk-free rate – Dividend rate – Volatility drag) × Time in years Previous: 0% – 0% – 4.5% = –4.5% New: 4.0% – 2.0% – 2.0% = 0% In this case, the mean happens to be zero. The risk-free rate increased the mean, the dividend return decreased it, and the volatility drag decreased it. The net effect is a distribution in Column C centered at zero.

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The Stock Return Standard Deviation (Cell G4) The stock return standard deviation is: Standard deviation = Volatility × Square root(Time in years) Previous: 30% × Square root(1.00) = 30% New: 20% × Square root(0.25) = 20% × 0.50 = 10% The option term is only 1/4 of what it was. However, because the standard deviation is proportional to the square root of time, this factor is not reduced by a factor of 4; it is reduced by only a factor of 2.

The Stock Return Random Variable To get a feel for what is going on with the Stock Return random variable in general, certain numbers act as good metrics. Start with the stock returns and stock prices at –1 standard deviation and +1 standard deviation. In this case, the numbers are: –1 standard deviation: –10% and $90.48 +1 standard deviation: +10% and $110.52 From the probabilities in Column B (or the rule of thumb), about 68% of the outcomes will be between these numbers. At –/+2 standard deviations, the prices are $81.87 and $122.14, which will contain about 95% of the outcomes. This gives you a good idea of the spread of the price distribution. To get a feel for how much drift is in the projection, look at the stock price mean in Cell G5. The stock price mean in this example is $100.50, which indicates a sight upward drift in the stock price during the three-month term.

Visualizing the Assumptions As mentioned earlier, the primary driver of the model is the stock return random variable. And the stock return random variable is completely determined by its mean and standard deviation. You can think of Column A, the standard normal random variable, as corresponding

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to an assumption set producing a mean of zero and a standard deviation of 100% as shown in the top panel of Figure 5.9. The pricing assumptions transform this distribution into a distribution specific to the option being priced. The transformation adjusts both mean and standard deviation. For the mean, the adjustment is related to four of the pricing assumptions: 1. The option term (the mean is proportional to time) 2. The risk-free rate (the mean moves in the same direction) 3. The dividend rate (the mean moves in the opposite direction) 4. Volatility (the mean is reduced by the amount of drag) For standard deviation, the adjustment is related to only volatility and option term (the standard deviation is proportional to the square root of time). The bottom panel presents a picture of the adjustments, where the term of the option is a half-year and the volatility is 30%. The width is narrowed to reflect standard deviation of around 20% (instead of 100%), and the mean has been shifted to the left 2.25% to account for volatility drag. Option pricing is primarily a matter of defining the center point and width of stock return distribution (Cells G3 and G4), which define Column C. After that, the framework of converting the return into the price and then option payoff is straightforward. To visualize this process, look at the top panel of Figure 5.9. This is a graph of the SND in Columns A and B, where by definition, the mean is 0.0 and the volatility is 1.0 (or 100%). Now we adjust the top panel in two ways. First, we apply the factor in Cell G4 to modify the spread for the particular option being priced. Second, we apply the factor in Cell G3 to move the distribution to the right or left. The resulting distribution in the bottom panel shows up in Column C and drives the entire option pricing calculation. In the end, all we are doing is defining the appropriate “picture” of the stock return random variable to reflect the pricing assumptions. In the next chapter, we will use this framework to start building a structured security model.

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Figure 5.9 The stock return random variable

Endnote 1. McDonald, Robert L. Derivatives Markets, 2nd Ed. 2006, Addison-Wesley, Pearson Education.

6 The Lognormal Distribution and Calc Engine The lognormal distribution is the most commonly used random variable in finance to describe stock and other asset prices. Unlike the normal distribution, it is bounded by zero on the downside, which highlights the fact that stock prices cannot go negative. Whenever you look at a graph of stock price probabilities, it is more than likely a lognormal distribution. The distribution is also frequently used in the derivation of the Black-Scholes formula and Value-at-Risk (VaR). But if it is so important, why hasn’t it been used here? In a way, it has. It is implied in the option pricing spreadsheet. And if you change the way stock prices are spaced, you can see it. The lognormal distribution is more of a mechanical adjustment than anything new or different from what you have seen so far. It doesn’t add information—it is just a different format. Because the lognormal display is more common and familiar, the graphs in the rest of the book are in lognormal form. There is a side benefit to this as well. By deriving the relationship between the normal and lognormal distributions—and how to switch from one to the other—you can better understand two basic formulas: the forward and backward equations. After developing the forward and backward equations, we will build the Calc Engine. The Calc Engine is the first module in the structured security model. The module calculates stock prices in both normal and lognormal formats and makes the relationships between these formats transparent. Note that this chapter covers only the aspects of the lognormal distribution that are directly relevant to the spreadsheet model. For a 97

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more thorough overview of the lognormal parameters, its probability density function, and its mean and standard deviation definitions, a variety of good resources are available, including these: http://en.wikipedia.org/wiki/Log-normal_distribution www.riskglossary.com/link/lognormal_distribution.htm McDonald, Robert L. Derivatives Markets (Pearson Eduction, 2004)

Definition of the Lognormal Distribution Normal and lognormal distributions are closely related. In fact, the lognormal distribution is defined in terms of the normal distribution, as follows: If X has a normal distribution, then Y has a lognormal distribution if Y = eX. In the option pricing spreadsheet, the stock price random variable was defined as: Stock Price = eStock Return Since stock return is a normally distributed random variable, then by definition, stock price is a lognormal random variable. But in the spreadsheet, there is only one probability distribution, the one in Column B, and it was used for all the random variables, including stock prices. Column B is the probabilities for the normal distribution. So the probabilities of stock prices in Column D are the normally distributed probabilities in Column B. But if stock prices have normally distributed probabilities, aren’t they normally distributed? Yes. So which is it, normal or lognormal? Actually, it can be either, or both. It is more a matter of presentation than of content. And it is possible, without much complication, to move between the normal version and the lognormal version of the stock price random variable. Because it is sometimes easier to

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work with one form over the other, the Calc Engine calculates both. Illustrating both methods will clear up any confusion about the differences in the two formats.

The Forward Equation In the option pricing spreadsheet, we used two steps to calculate stock prices. Stock returns were calculated first, and then stock prices were calculated in a second step. The forward equation is simply the combination of the two steps: 1. Stock Return = SND × StdDev + Mean 2. Stock Price = S0 × eStock Return This becomes: Stock Price = S0 × e(SND × StdDev + Mean) In cell reference terms: Column D = C4 × e(Column A × G4 + G3) S0 refers to the current stock price in Cell C4; SND refers to the values of the standard normal distribution in Column A; and Mean and StdDev refer to Cells G3 and G4, respectively. Figure 6.1 shows the Excel formula for the forward equation.

Figure 6.1 Forward equation

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Example: The two-step process in Row 13 first calculates the stock return in Column C as –124.50%. Then the stock price is calculated using the stock return in Column D as $28.79: 1. C13 = A13 × G4 + G3 = –124.50% 2. D13 = C4 × EXP(C13) = $28.79 Combining the two formulas into the forward equation gives the same answer in one step: E13 = C4 × EXP(A13 × G4 + G3) = $28.79

Cross Reference: Stochastic Differential Equations This section is for those who are familiar with or interested in the stochastic differential equations related to option pricing. In Chapter 1, I talked about the underlying meaning of option pricing in nontechnical terms. The main point of that discussion was the unpredictability of stock prices, as reflected in the random nature of their changes. Given this unpredictability, stock prices follow a process called geometric Browian motion, which is defined by the stochastic differential equation: dS = Rdt + XdZ S where the symbol R in the first term is the assumed drift and X in the second term is the standard deviation. Sometimes, you will see references to a Weiner process, where W is used in place of Z. dSt = RStdT + XStdWt The solution to these equations is the forward equation. Figure 6.2 provides a cross-reference to the more formal mathematical symbols for the Black-Scholes assumptions and the forward equation.

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Figure 6.2 Stochastic formula cross-reference

Often, different terminology and symbols are used. The three imposed formulas in the body of the spreadsheet in Figure 6.2 are examples. Depending on the text, you might see any of these. In any case, the solution assumes constant drift and volatility and is derived by applying Ito’s lemma to the preceding stochastic equations. Here are some observations: 1. The solutions shown here apply to a non-dividend-paying stock. For a dividend-paying stock, the stock return mean contains an extra term (Cell G3 contains a reference to Cell C7). 2. Because the mathematics of Brownian motion were already known at the time the Black-Scholes formula was developed, the real contribution of the formula was the fact that the drift is equal to the risk-free rate, as shown in Cell D6. 3. The expression inside the brackets of the solutions is the random variable stock return, which has values shown in Column C and probabilities in Column B. You can see how it is derived from the standard normal random variable in Columns A and

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B, adjusted for the mean and standard deviation in Cells G3 and G4. 4. The stock price random variable is the beginning stock price, S0, in Cell C4 multiplied by EXP[stock return]. Please note that this section did not add any new information. It is simply a cross-reference to translate the stochastic formulas into the spreadsheet column definitions. One important aspect of working with these stochastic formulas initially is visualizing how Z (or W) interacts with the assumptions through time. By varying the time until expiration in Cell C3, you can model the behavior of these random variables with respect to time. In a broader context, you can also use the spreadsheet tool to see what the initial distribution looks like across any of the pricing assumptions. For example, you can change the drift or volatility to see the effects of different conditions on the stock return and stock price distributions. These applications will become more apparent as we build the model and use it to illustrate structured securities under different market conditions and with the passage of time.

The Backward Equation The backward equation is the inverse of the forward equation. Instead of starting with the standard normal distribution values in Column A to get stock prices, we start with stock prices and calculate their corresponding standard normal distribution values. In other words, given a particular stock price and what we know about its distribution, it is possible to identify where that stock price falls on a standard normal distribution curve. In Figure 6.1, the stock price corresponding to a standard deviation value of –4.0 is $28.79, calculated by the forward equation as follows: E13 = $28.79 = $100 × EXP(–4.0 × 0.30 + –0.045)

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For the backward equation, we want to perform the reverse. Given a stock price of $28.79, we want to know where it falls on the standard normal curve. To answer this question, we can rearrange the forward equation to solve for StdDev. Starting with the forward equation: Stock Price = S0 × e(SND × StdDev + Mean) First divide by S0 and take the ln of both sides (where = X), which gives:

ln[eX]

ln(Stock Price / ln[S0]) = SND × StdDev + Mean Then subtracting Mean from both sides and dividing by StdDev gives: (ln[Stock Price] / ln[S0]) − Mean) / StdDev = SND This is the backward equation. Figure 6.3 shows the backward equation in cell formulas:

Figure 6.3 Backward equation

If the stock price is $28.79, then the formula produces the associated standard deviation value of –4.0, as follows: F13 = –4.0 = (LN[$28.79 / $100] + 0.045) / 0.30

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We are back to where we started. That verifies the formula works and gives us a convenient way to map any stock price to a point along the standard normal distribution. Then we can calculate the associated probability of that price. Having the ability to assign probabilities to any set of stock prices allows us to build stock price graphs any way we choose, including with evenly spaced price intervals. “Evenly spaced intervals produce lognormal graphs.” That means the relationship between the normal and lognormal versions of stock prices is just about spacing. Earlier, the question was whether stock prices were normal or lognormal. The answer is both, depending on how you want to look at it—literally. When graphing stock prices, if the x-axis is evenly spaced with respect to standard deviation values (as in the option pricing spreadsheet, where the points were –4.0, –3.9, –3.8, and so on), the stock price appears as a normally distributed random variable. When the x-axis is evenly spaced with respect to stock prices, the graph appears as a lognormal distribution.

The Calc Engine The Calc Engine is the first module in the structured security model. The Calc Engine calculates stock prices and associated probabilities in both normal and lognormal form, as shown here in Figures 6.4a and 6.4b. Columns A and B are the same as before, a discrete version of the SND. Column C contains the values of the stock price random variable. These values are calculated using the forward equation. So Column C is the same as Column D of the earlier option pricing spreadsheet. By combining two steps, the stock return column is omitted and the stock price column moves from Column D to Column C.

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Figure 6.4a The Calc Engine

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Figure 6.4b The Calc Engine, (Continued)

Evenly Spaced Stock Prices Column D is new. It is a column of evenly spaced stock prices. For this example, the beginning and ending points of Column D are the same as in Column C. The first number in Row 13 is equal to

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$28.79, the same number in Column C. Likewise, the ending number in Cell D93 is the same as Cell C93, $317.40. The difference in the two columns is that, between the endpoints, the numbers in Column D are evenly spaced. The difference in any two adjacent cells in Column D is $3.6076. Three cells in the heading control the prices in Column D. Cell G5 is an input that lets you specify the low stock price in the range to be graphed. Cell G6 is the input for the high stock price. Cell G7 is a calculated cell for the spacing between points, based on 80 steps between the first and last prices: G7 = (G6 − G5) / 80 = ($317.40 − $28.79) / 80 = $3.6076 In contrast, in Column C, the difference in the dollar amounts between adjacent cells varies. For instance, the difference in Cell C13 and C14 is: $29.67 − $28.79 = $0.88 The difference in Cell C93 and C92 is: $317.40 − $308.02 = $9.38 In Column C, the spacing is even with regard to standard deviation points in Column A, where any two adjacent cells are different by 0.1 standard deviations. Now the question is how to figure out the probabilities that are associated with the prices in Column D.

Assigning Probabilities You may remember the discussion in Chapter 2, “Random Variables and Option Pricing,” about the range associated with the discrete values of the standard normal distribution. In that discussion, we talked about needing to assign a range to each value in Column A for purposes of calculating probabilities. The convention was to look at each point as the midpoint of a range that extends 0.05 above and below that midpoint. For example, the probability of the standard

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deviation point +1.8 is determined by the the cumulative probabilities at +1.85 and +1.75, as: Column B = NORMSDIST(1.85) − NORMSDIST(1.75) An exception arises for the two endpoints, where the probability includes the full tail of the distribution: B13 = NORMSDIST(–3.95) B93 = 1 − NORMSDIST(+3.95) Now we want to calcuate the probabilities associated with Column D, the evenly spaced stock prices. By switching to even spacing along the x-axis, the standard deviation points no longer have a 0.10 range. In fact, the range will be different for each pair of adjacent points and for each assumption set. But the procedure is the same. We figure out the upper and lower bounds of the range and use the cumulative distribution function to determine the probability within that range. First, we use the backward equation to calculate the midpoint of each of the intervals. The midpoints are the numbers shown in Column E. The midpoints are averaged to determine the upper and lower boundaries of the range. As before, NORMSDIST gives us the cumulative probability for the boundaries. Taking differences gives us the probability of the interval. As an example, look at Columns D, E, and F of Rows 13–15: Price

StdDev

Probability

$28.79

–4.000

0.00007

$32.40

–3.607

0.00023

$36.01

–3.255

0.00068

The upper and lower boundaries for the stock price equal to $32.40 are: Lower bound = (–4.000 + –3.607) × 0.5 = –3.805 Upper bound = (–3.607 + –3.255) × 0.5 = –3.430

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In other words, we are just finding the midpoint between any two values and using that as the boundary, as follows: NORMSDIST(–3.430) − NORMSDIST(–3.805) = 0.00023 The probability of this range is the probability assigned to the stock price of $32.40. By continuing this process of referencing the average value of the surrounding cells, we can fill in the probabilities of Rows 14–92 with the formula: F14 = NORMSDIST([E14 + E15] × 0.5) − NORMSDIST([E13 + E14] ×0.5) The two endpoints complete the column: B13 = NORMSDIST([E13 + E14] × 0.5) B93 = 1 − NORMSDIST([E92 + E93] × 0.5) Figure 6.5 summarizes the spreadsheet formulas.

Figure 6.5 The probability formulas

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At this point, we have two different ways of expressing the same information. In both views, there are 81 possible stock prices with assigned probabilities. Even though the stock prices and probabilities are different, they both have the same expected values. You can verify this by calculating the weighted averages.

The Graphs Figure 6.6 is a graph of the two versions. Notice that the range of stock prices in the two graphs is the same, both going from $28.79 to $317.45. The difference is that the top graph is evenly spaced on the x-axis with regard to standard normal distribution points, and the bottom graph is evenly spaced along the x-axis with regard to stock price. The top graph is normal. The bottom graph is lognormal. The top graph represents Column C (unevenly spaced stock prices) and Column B (normal distribution probabilities). The bottom graph represents Column D (evenly spaced stock prices) and Column F (lognormal distribution probabilities). For both Column B and Column F, the total probabilities add to 100%, as they should for any random variable. And in both cases, the expected values or weighted values of the stock prices are equal to the current stock price adjusted for drift. Because the same basic data was used for both the normal and lognormal graphs, this illustrates that there is no difference in the information content—just the presentation format.

Setting the Stock Price Range In the model, you can decide how much of the stock price range to display by setting the high and low prices for Column D. The earlier example used the same range as in Column A, but that is not necessary. You may set a narrower range to highlight the parts of the stock price spectrum you are more interested in. This will not affect the calculation unless the range is so narrow that too much of the total distribution is included in the two tail points.

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Figure 6.6 The Stock Price distribution: normal and lognormal 111

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You can specify whatever range of prices you want to display by entering the starting price in Cell G4 and the ending price in G5. Cell G6 is the difference between the starting and ending points divided by 80, the total number of steps to fill out Column D. The difference in shapes between the two distributions is related to the term of the option. In this example, the option term was one year. For shorter periods such as monthly stock price graphs, the normal display and lognormal display will look more similar to each other.

Visualizing Option Pricing As Normal or Lognormal In Chapter 1, “Introduction,” the option pricing spreadsheet was graphed using the normal version of the stock price distribution. Figure 6.7 is a reprint of Figure 1.2. The call option payoffs are shown on the right side of the graph, and the payoff values are on the right axis. The probabilities of the payoffs are shown as the normal distribution curve, with the values on the left axis. Notice that the option payoff in Figure 6.7 is not a straight line. It slopes up. That is because the x-axis is evenly spaced for standard deviations, not for stock price or option payoff. The result is the graph shape with probabilities that are normally distributed and option payoffs that are upward sloping. The more conventional way to display a stock price or option payoff curve is to plot stock prices and option payoffs at equally spaced dollar amounts, as opposed to equally spaced standard deviation points. Figure 6.8 shows what this looks like. It is the bottom part of the earlier figure, with an overlay of the option payoffs (not shown in the Calc Engine). With an evenly spaced dollar x-axis, the option payoffs increase dollar for dollar with the stock price, so the option payoff becomes a straight line.

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Figure 6.8 Stock option payoffs—lognormal distribution

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Figure 6.7 Stock option payoffs—normal distribution

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By changing the x-axis to equal dollar amount increments instead of equal standard deviation increments, a normal distribution becomes a lognormal distribution. In the Calc Engine, both versions are available, so you have a choice of which to use. The normal distribution is easier for certain calculations. For display, the lognormal distribution creates a more conventional look.

7 Investment Profiles and Synthetic Annuities Before continuing with the mechanics of model building, it might be helpful to provide some practical context for structured security investing. Chapter 7 is a reprint of Chapter 2 that originally appeared in Profiting with Synthetic Annuities: Option Strategies to Increase Yield and Control Portfolio Risk (Pearson Education, 2012). Synthetic annuities are a form of structured security that combine options and management rules to customize the risk/return profile of investments. Options are used to create a synthetic risk-smoothing mechanism and annuity-like cash flows. The management rules are designed to mitigate risk and maximize income over the long term. Because Profiting with Synthetic Annuities is primarily concerned with how to build and manage portfolios, it offers a practical perspective on two topics that are important to the remainder of this book, the investment profile and how options are used to reshape it. Investment profiles are visual representations of investment positions such as stocks, options, and structured securities. They are graphs of the underlying random variables. Hopefully, the background in this chapter will make some of the material to follow more relevant as we go into the mechanics of constructing investment profiles.

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A Note on Terminology In the excerpt that follows, I use the terms payoff curve and profit curve interchangeably. In this more technical book, I want to point out that the two terms are not the same. In the chapters on option pricing, we looked at option payoffs or payoff curves. Going forward, we look primarily at profits and profit curves. The difference is that profits are adjusted for the amount paid or received from an option transaction. The profit is also referred to as the net payoff.

Begin Excerpt: Chapter 2 from Profiting with Synthetic Annuities This chapter provides an overview of the synthetic annuity (SynA) and how options are used to achieve the design objectives outlined in the preface, including: • The use of hedging, insurance, and risk allocations in risk management instead of reliance on traditional portfolio models • The desire for greater yields not related to market direction • A recognition of behavioral influences on investor performance • The growing importance of volatility-reducing quantitative methods, particularly those related to stock options • The desire of many investors for annuity-like income streams The presentation is somewhat unique in that there are no formulas. Structured securities, such as a SynA, become very complex and difficult to communicate with formulas, but the same structures are fairly easy to understand when translated into pictures. The device used to translate the SynA into graphs is the investment profile, which is simply an enhanced standard payoff curve. The only difference between a payoff curve and the investment profile is that the investment profile also includes the probability that each payoff will occur. Adding probabilities turns the payoff curves into “random variables,” the basis of stochastic math and option pricing.

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In a sense, the investment profile is the “fractal” version of quantitative finance. The information contained in the investment profile also makes it possible to quantify the value of various payoff curves to answer questions about the tradeoffs involved. For instance, how do the gains and losses of a SynA compare to the gains and losses of the underlying security? How much current income does it generate? Is it efficient in providing a shock-absorbing effect to volatility? From a behavioral finance point of view, does it increase the value of the holding to the investor? And, is it effective at managing the risk of an investment position? The first example in this chapter is the investment profile of a stock. Then options are added to the stock to show how the payoff curve is reshaped to achieve the design objectives. Next, a case study for an investor with a concentrated stock position illustrates a real-life context. The case study also talks about behavioral issues, including the difference between actual gains/losses and the perceived value of those gains/losses. This chapter ends by extending the single-period investment profile to a multiple period example. The example looks at the actual performance of a SynA on Apple Inc. during the 2008 calendar year, including the financial crisis. The example documents how the Apple SynA operated in calmer periods to generate income and reduce cost basis, and in turbulent periods to protect principal.

What Is a Synthetic Annuity, and How Does It Work? A SynA is a combination of an underlying security and options on the underlying security. The underlying security can be a stock, a stock index, an ETF, or a futures contract. The purpose of the SynA is to create annuity-like cash flows and a risk-management framework. Options are used to tailor the investment profile of the underlying

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security, to customize risk/reward preferences. The options can be used for defensive purposes in some instances; at other times they can help with offense. They can be used in a standard setup as a default security structure, or they can be used contingently as conditions change. At setup, a typical SynA looks similar to a covered call position, with two differences. The first difference is the use of staggered strikes. The second is that some of the cash received from selling the call options is used to purchase put options. The net effect is to transform the stock position into a related security that is less volatile—and produces higher levels of current income. Of course, nothing is free— the tradeoff is some upside. The steps to create a typical SynA are: 1. Buy the underlying security. 2. Sell in-the-money covered call options on a portion of the position. 3. Sell at-the-money covered call options on a portion of the position. 4. Sell out-of-the-money covered call options on a portion of the position. 5. Buy out-of-the-money put options, using part of the money from the call options. Normally, all options are for the near-month contract. For example, for Apple, Inc., the SynA might look like this: 1. Buy 1,000 shares of AAPL. (For the example, assume a current price of $550.) 2. Sell 300 shares (three contracts) of the near-month call option, with a strike of $530. 3. Sell 400 shares of the near-month call option, with a strike of $550.

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4. Sell 300 shares of the near-month call option, with a strike of $570. 5. Buy 300 shares of the near-month put option, with a strike of $500. The option transactions reshape the stock-only investment to decrease the chances of large losses, increase the value of the likely outcomes around the current stock price, and, in exchange, give up the chances for large gains. In broad strokes, from a defensive point of view, the SynA adds shock absorbers to smooth the volatility of stock price movements. From an offensive point of view, the SynA generates income through the sale of options on the underlying stock. Set up this way, a SynA is less volatile and less risky than a stock position. However, you can be aggressive as well. Within an options framework, there are several ways to create leverage or high levels of current income without violating the risk management discipline. In fact, using options to make tactical trading decisions is often less risky than expressing the same views through security purchases. Several offensive strategies are discussed in the chapters on setup and management of the generalized SynA. The objective of the generalized SynA is to provide a flexible architecture to design risk-management strategies, so the degree to which options are used depends on your objectives and your current view of market risk. The Apple SynA presented above is more appropriate for moderately trending or stable markets. In dynamic markets, you can choose to be more or less defensive by varying the type, number, and strike prices of the options.

The Investment Profile The investment profile is a visual way of describing a security. As mentioned, the only difference between an investment profile and a

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standard payoff diagram is the addition of a probability distribution. The probability distribution indicates how likely each possible payoff is.

Note Mathematically, whenever you specify each possible outcome of an unknown event and the probability of that outcome, you have defined a random variable. Random variables are the basis of stochastic math and the Black-Scholes option pricing formula. In these terms, the investment profile is a graph of the investment gain-loss random variable.

Figure 7.1 is an example of an investment profile for a stock investment. In this example, an investor has bought 10,000 shares of XYZ at $45 a share. The graph shows the gain (loss) as the straight line labeled “stock utility” and the probabilities of each gain (loss) as the imposed lognormal distribution. The investment profile contains a great deal of information. Using just the information in this picture, it is possible to calculate many riskrelated metrics, such as value-at-risk, conditional value-at-risk, and expected gain and expected loss. It is also possible to use an investment profile to compare the value of various investment alternatives.

Assigning Probabilities Using Implied Volatility Probabilities can be assigned to the investment profile in different ways, but one method has both practical and theoretical advantages: using the Black–Scholes option pricing model and implied volatility (IV) from the options market. In pricing options, several input items are used, such as the stock price, the strike price of the option, the term of the option, and the risk-free interest rate. All these items are known at the time the option is priced.

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Figure 7.1 Investment profile of stock-only position

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Only one variable is not known: volatility. Volatility must be estimated in some way. Given an estimate of volatility, you can price the option. Volatility and option price have a one-to-one relationship. For a given level of volatility, you get a certain option price. You can also look at this relationship in reverse. For a given option price (knowing all the other pricing variables), you can “back into” the volatility that corresponds to the option price. When volatility is calculated in this way, it is called implied volatility (IV). Most trading platforms calculate IV; but if not, you can use the option price on the exchange and back into the volatility that corresponds to the observed price. When you know the IV, you can use it to assign the probabilities to the investment profile. A theoretical advantage of using IV is that it is consistent with currently traded options. A practical advantage is that the future stock prices predicted in the model are consistent with the options and futures instruments that can be used to hedge the stock price risk. In addition, because of higher leverage available by using options, knowledgeable investors often use the options markets first rather than the cash equity markets to make directional bets; thus, it can be argued that the options markets contain useful information about future stock prices. To create an investment profile, you have to make one more assumption about the exact definition of volatility that relates to how the volatility is translated into the shape of the probability curve. A convenient and commonly used definition is that of normally distributed stock returns used in the Black–Scholes option pricing formula. Figure 7.1 was based on an implied volatility of 32% and the Black– Scholes definition of volatility.

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Using Options to Reshape the Investment Profile What if you are interested in changing the profile—for example, if you want to decrease the chances of large loses? If you are willing to give up some upside, you can change the payoff curve to accomplish that by using options. Figure 7.2 shows a typical SynA profile as the white line. Notice that the gain–loss of the stock has been reshaped so that losses are smaller. Consider another positive benefit for prices around the current share price of $45: The line has moved up, and the breakeven point has decreased, or moved to the left. On the other hand, you give up possible large gains. The crossover point of the two lines occurs close to $47. Below that price, the SynA is better. Above that amount, the stock is better. Because the gain–loss values and the probabilities of both the stock and the SynA are known, it is possible to compare them. One way to do this is to calculate the expected values (or weighted outcomes) of the gain–loss profiles, as shown in the table in Figure 7.2. The ratio of the SynA to the stock is 1.24. That means, based on expected or weighted values, on average, the SynA is 1.24 times, or 24%, more effective in meeting the risk/reward objectives. Intuitively, you can see how this might be true. By focusing on the most likely outcomes—for example, the range –1 and +1 standard deviations, which occurs about 68% of the time—of the 21 possible stock prices, all but 4 are actually more favorable under the SynA. Because the more favorable outcomes are concentrated in the most likely areas of the distribution, the SynA performs better. This example assumes that investors value $1 of gain and $1 of loss equally. According to behavioral finance research, however, most investors don’t think that way. They tend to dislike large loses more than they like large gains. It is possible to reflect attitudes about gains and losses through the use of utility functions.

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Figure 7.2 Investment profile of stock only position and SynA

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Adjusting the Profile for Behavioral Finance The purpose of a utility function is to express how investors feel about various levels of gain and loss. According to generally accepted patterns based on research, people’s perception of the value of large gains tends to decrease as the gain becomes larger. Also, people tend to dislike losses at a relatively fast pace as the losses pile up. In fact, as losses become worse, it is not unusual for an investor to feel the loss two or three times as bad as the actual dollar amount. Figure 7.3 shows a typical pattern for a utility function. The darker area of the payoff line is the same as in previous figures; it is just the dollar amount of the gain-loss. The lighter curved line represents the investor’s perceived value of the gain-loss. Notice that the utility function curve lies under the gain-loss line at all points. On the ride side of the chart, as gains become larger, the line is slightly lower than the actual gain-loss because according to behavior finance, most investors do not appreciate increasingly larger gains on a dollar-for-dollar basis as much as smaller gains. Here, the discount at the far right is around 15%. On the left side of the chart, the deviation is much more exaggerated. As losses approach –2 standard deviations, the utility curve is about 2[1/2] times the gain-loss. By using utility functions, you have a choice of two ways to look at gain-loss profiles. The first is to view the gain-loss as the actual dollar amount. The second is to view the gain-loss in terms of emotional impact. Combining the information in Figure 7.2 with the utility function in Figure 7.3 produces the revised, utility-based random variables shown in Figure 7.4.

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Figure 7.3

Investor utility curve

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Figure 7.4 Comparison of utility curve for stock-only position and SynA

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The right side of this chart, where the maximum discount is 15%, is not that different from the actual dollar version. However, the left side is noticeably different, as losses are measured on the much stricter utility scale. Because loss reduction is more important on this scale, it is not surprising to see that the SynA is relatively more attractive also, as shown in the revised table on the preceding page. Using the utility curve—that is, perceived gain-loss rather than actual gain-loss—the SynA has a weighted average score of 1.64 compared to the stock-only position. In other words, the SynA is 64% more effective at balancing this particular risk/reward preference.

Concentrated Stock Example My friend Steve’s company was recently purchased. Under the terms of the deal, he will receive stock in the acquiring company. He called me to talk about alternatives to selling the stock. He mentioned selling call options to generate income. He also wants some downside protection. He told me, “When it drops a half-point, I think about how much I lost.” This situation is common among investors who have concentrated stock positions. It provides a good place to start in terms of describing synthetic annuities. The acquiring company was the model for XYZ earlier—that is, the stock is currently trading for $45, the number of shares is 10,000, and the annualized implied volatility is 32%. A typical SynA is constructed by selling options at various strike prices, with relatively more sold at-the-money. A portion of the money received is used to buy put options. Here are the number of options purchased (long) or sold (short) and the strike prices: • 3,000 short call options with strike price = 42.50 • 4,000 short call options with strike price = 45.00 • 3,000 short call options with strike price = 47.50 • 4,000 long put options with strike price = 40.00

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The following table is a summary of the cash flow from the option transactions.

The net proceeds from setting up the SynA were $18,530. These proceeds may be viewed as a type of dividend. In the context of the SynA, I sometimes refer to it as a virtual dividend. One important effect of the virtual dividend is to reduce the cost basis in the position. Of course, by selling the call options, some of the potential upside is given up, at least over the single period. Over multiple periods, the management rules discussed later in the book are designed so that it is not necessary over longer periods of time to give up large upside potential. Above a certain price level (strike plus premium earned), it would have been better to hold the stock rather than the SynA. In terms of downside protection, the gain/loss profile at lower price levels changes because of the 4,000 put options. If the stock price drops below $40, the SynA is better for two reasons. First, the cost basis is reduced by $18,530. Second, purchasing the put options provides downside protection on 4,000 shares for prices below $40. The net effect of the SynA setup can be seen in Figure 7.5 (also shown earlier in Figure 7.2). Notice that although the stock crosses the x-axis at $45.00, the SynA crosses the x-axis at $43.15. The breakeven point for the SynA has shifted to the left because the cost basis was reduced. The shift is equal to $1.85, the per-share virtual dividend ($18,530 ÷ 10,000 shares).

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Figure 7.5 Investment profile of stock-only position and SynA

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On the left side of the graph, the SynA line becomes less steep as the stock price goes below $40. This is the effect of the put options, which removes the loss below $40 on 4,000 shares. On the right side, at prices above $46.85, the stock line is higher than the SynA. In setting up the SynA, one gain-loss profile has been exchanged for another. In moving from the stock profile to the SynA profile, a portion of potential profits is exchanged for improved gains around the current price and better loss protection at lower prices.

Comparing a Stock-Only Position to a SynA The table in Figure 7.5 contains enough information to answer several questions about the relative attractiveness of the two alternatives. A common method of comparing two investment alternatives is to look at the weighted average outcomes, or expected values, of each. The calculation is straightforward: Multiply each gain or loss amount by its corresponding probability and add the terms. The calculation is done using all values between –4 and +4 standard deviations rather than just the values between –2.1 and +2.1 standard deviations shown here. The full range is not displayed because only 4% of the possible stock prices fall outside this range. The expected value of stock position is $0, and the expected value of SynA is $1,586. In the following summary, the expected values are shown in the first row. In this summary, it is assumed that actual gains and losses are the same as perceived gains and losses. The second row shows the effects of behavioral preferences. The next two items, the expected value of perceived gain and the expected value of perceived loss, are conditional expectations, calculated in the same way as a normal expectation, except in the case of the expected value of perceived gain where only the gains times are multiplied by their respective probabilities. The calculation is similar for expected value of perceived loss.

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The stock has a ratio of 1.00 and the SynA has a ratio of 1.24. Therefore, ratio of the ratios (shown in the center bottom of the preceding table) is 1.24. This means that, from a risk/reward standpoint, the SynA is 24% more attractive than the stock. Two other metrics are not shown in the table: VaR and CVaR. The period of the VaR must match the period used in the construction of the probability distribution. This example uses one month, so the VaR period is also one month. It uses the 5% level, meaning the loss that will be exceeded 5% of the time. Using a normal distribution, this happens at a standard deviation of –2.1. The stock loss at –2.1 standard deviations is $75,922. For the SynA, it is $47,023. The results for CVaR are similar. Because the SynA is a more conservative strategy, it performs better on potential loss measures. What about gains? How will Steve feel if the stock goes up and he misses out on the move because he sold the upside to someone else? This is a very real concern and the reason many investors do not like to sell call options. For a speculative trade with a catalyst that might cause the stock price to move up quickly, the disadvantages will probably outweigh the advantages. Steve wants to hold the stock longer term and believes that the price moves will be relatively smooth. He is aware of the potential to give up gains, but he also realizes that the stock is highly correlated with his future income. This is an important point for diversification. You might be wondering how selling options that expire in one month can accommodate an investor who wants to hold a stock for the longer term. The options will be repurchased before they expire and rolled out to the next month. The options will be considered to be a permanent part of the security going forward.

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Behavioral Finance Adjustments I wanted to get an idea of how Steve would react to different levels of gain and loss. We just talked about a few points on the curve, and Steve told me how he would feel about those outcomes. Using these points of reference, I adjusted a general risk-averse pattern to construct the full utility curve. Usually, investors can simply look at the curve and have confidence that they have communicated accurately how they feel about various possible outcomes. For purposes of utility curve, I have extended the option period to one year so Steve can see a wider range of outcomes. Steve’s utility curve is the one presented earlier and is shown again in Figure 7.6. The lighter area extending down the left side of the graph represents Steve’s utility. At the far left, the stock price was $23. At a price of $23, the actual loss per share was $22 ($45 – $23). Steve told me that an actual loss of $23 would feel to him like a loss of $60, so I extended the actual loss line to the perceived loss of $60. On the far right side of the graph, the new line fell below the old line also, but on a percentage basis, it was not as dramatic as losses. The curve is consistent with a risk-averse profile in which each additional dollar of gain is appreciated at a declining rate. In Steve’s case, he appreciated large gains for another reason, but not as much as the actual dollar amount: because of the correlation between the stock price and the health of the industry in which he works. If the stock price is up, both the economy and Steve’s job security are likely better than if the stock price were down. This correlation also helps to explain the steepness of the dashed line curve on the left side of the graph. Steve said that a large loss, like that depicted on the left side, would feel almost three times as bad as it really was. Losses of this magnitude could threaten his retirement goals.

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Figure 7.6

Investor utility curve

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How is this information translated into the SynA? First, the cost basis has already been reduced from selling call options. Second, the put options provide some absolute protection against large losses on some of the shares. The next step is to set a limit on how much price can drop before you start making adjustments. The more risk averse you are, the tighter the limit on the price drop should be. Your primary metric is cost basis. The difference in position value and the cost basis of the position determine when and how much of an adjustment you need to make. In terms of the SynA setup, the put option helps you prevent large losses. At this point, you might wonder why you didn’t just purchase the put options without selling the call options. Why not just buy the insurance without giving up the upside? That’s a good question. The short answer is that buying put options alone is expensive. A more complete answer has to do with the longer-term expected value of profits.

Applying the Utility Curve to the SynA Profile Now that each loss has been translated into a “perceived” loss, it is possible to apply the generalized utility curve to the SynA in the same way it was defined it for the stock position. In other words, the losses are mapped in the same way, so if a $20,000 loss was mapped to a $30,000 perceived loss in the stock position, it will be mapped the same way in the SynA, even if the $20,000 loss occurs at a different stock price. The utility function is related to the loss amount, not the stock price. One other question needs to be answered before evaluating the SynA. What numbers do you use in comparing the stock profile to the SynA profile? Should you use the actual gain (loss) or the perceived

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gain (loss)? This question has no single correct answer. The conservative answer is to use the perceived gain (loss). In terms of protecting against loss of capital, perceived gains (losses) both understate the gains and overstate the losses, creating a bias toward less risky investments. Steve asked that trading decisions be based on the utility curve (perceived), not the actual gain (loss). As you can see from Figure 7.7, the utility function causes exaggerations of the left side of the graph, reflecting greater aversion to loss. As before, the probability that the stock price will close within the two orange markers (–1 to +1 standard deviation) is 68.3%. The probability that the stock price will close within the two blue markers (–2.1 to +2.1 standard deviations) is about 96%. Corresponding to the new graph is the new summary table shown in Figure 7.7. In this version, both the stock only value and the SynA expected value are lower than before. Because losses are considered to be far worse than corresponding gains, the expected value of the stock and SynA is negative, with the stock-only position having an expected value of –$17,169 and the SynA having an expected value of –$2,663. On a relative basis, however, the SynA has become even more attractive. When measured on actual gains and losses, the SynA is 24% more valuable. On a utility-adjusted basis, the SynA is 64% more valuable.

A Multiple-Period Perspective The previous analysis compared a SynA to a stock investment over a single period. Over longer periods of time, the management rules of a SynA describe how to adjust the options positions. The details of the management rules will be presented later, but I wanted to point out how the SynA is intended to work over time in calm and turbulent markets by looking at an actual example.

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Figure 7.7 Utility curve for stock only position and SynA

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In calmer markets, the only adjustments to the options positions are often the rollouts of the options that occur at expiration. In more turbulent markets, tactical adjustments may be required between expiration dates. In other words, if the stock price remains within certain bounds, the setup could remain unchanged for the entire month, with the first change happening at the options expiration date when you roll out the options on the SynA for the next month. If the stock price moves outside a predetermined price boundary during the month, you make adjustments to keep the effective market exposure within target ranges. The intra-month adjustments and inter-month rollouts are part on the overall SynA design, which is long-term in nature. The following section illustrates how a SynA works over multiple periods to generate income and control risk.

The Synthetic Annuity in Turbulent Markets Apple might be the greatest company in history, but it is not immune to market turbulence. During 2008, its stock price was cut in half. In one portfolio, I bought Apple in December 2007 for around $200. As the financial crisis unfolded in 2008, the stock price fell below $90. In my opinion, risk management—and, in particular, a sell discipline—is the most important aspect of portfolio management. Many sell disciplines exist, ranging from adopting a “never sell” mentality to setting stop-losses at some predetermined amount, such as limiting losses to 5% to 10% of the purchase price. The problem I had with many sell disciplines is that in volatile markets, the only way to control risk was to act like a trader rather than an investor. But principal risk is related to both price and cost basis. After you purchase a stock, you cannot control the price. But you can, at least

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to some extent, control cost basis. The advantage of a SynA is that you can reduce your cost basis without having to sell the underlying security. A SynA has two components that smooth volatility in an effort to avoid having to make cost basis adjustments. The first is a form of delta hedging (through the short options) that reduces volatility. The second is partial insurance protection provided by the put options. If these are not enough and the position loss exceeds an explicit risk budget, then the adjustment rules kick in. Even then, the risk budget does not force the sell of the security; it acts as a signal that it is time to adjust the cost basis. In Apple’s case, during 2008 there was a risk budget of approximately 10% of the cost basis. Depending on the macroeconomic outlook and fundamental health of a particular company, some flexibility exists in how to apply the risk-management tools. However, when the loss on a position exceeds a predefined amount, the adjustment rules require a reduction in cost basis. Figure 7.8 shows how the cost basis compared to the price of Apple stock through 2008. As the year began, the cost basis was close to the stock price, both around $200. During the year leading up to September, the cost basis fell significantly below the price, which was needed by the end of the year, as the price dropped to about $90. As the stock price moved down with the overall market in reaction to the Lehman bankruptcy, the cost basis continued to be reduced, to keep it close to the position value. Because the real risk of the position is the difference between value and cost basis, at no time during 2008 was there exposure to a large unrealized loss. The derivatives created a shock-absorbing effect and the management rules enforced discipline. By contrast, if the position had been managed as buy-and-hold, the potential loss (unrealized loss) at the end of the year would have exceeded $100 per share.

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Figure 7.8

Cost basis compared to Apple stock price 2008

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During stressful market periods, the focus of the synthetic annuity is risk control. This was true during the second half of 2008 and the first quarter of 2009. At other times, when the market is more stable, either trending or oscillating, the focus shifts to income generation, which also steadily reduces the cost basis. For Apple, an example is the second quarter of 2008, a relatively stable price environment. Figure 7.9 zooms in the period from April 1, 2008, to June 30, 2008, and again compares stock price to cost basis. On April 1, the difference between Apple’s stock price ($150) and the cost basis ($127) was $23. By the end of the quarter, the difference had widened to $55. An important aspect of the strategy is the ability to increase this spread, even when the stock price is not increasing. The basic idea behind this part of the strategy, during relatively stable times, is to be paid for waiting. Option time decay means the position is increasing in value just by the passage of time. The reduction in cost basis over time contributes to overall risk control. For example, if you buy a stock for $100 and you get a $5 dividend during the first year, your “real” principle exposure is $95, versus a nondividend-paying stock for which your exposure stays at $100. A SynA uses the same logic; it just provides the opportunity to reduce principle exposure at a much faster rate. When a stock has a period of stability or a moderate uptrend, it is possible to recapture the entire cost basis in three to seven years.

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Figure 7.9

Cost basis compared to Apple stock price, 2nd quarter 2008

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In addition to reducing the cost basis, the put option component acts as partial insurance protection. Put options are a simple and effective way to insure against large losses, but they are also expensive. The cost of complete protection on most portfolios can average more than the investment returns. For insurance to work and be affordable, either you have to time the market to have it only when you need it, or you need to finance it in some way that doesn’t put too much of a drain on investment returns. A SynA is set up so that it has partial put protection. As cost basis is reduced and put options become less expensive at lower strikes, the sell of call options finances larger amounts of put protection, with the intent of establishing full principal protection over time.

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8 Stock-Only Investment Profile This chapter explains how to build the stock-only investment profile. As described in the last chapter, this profile combines information about profits with an underlying probability distribution. The idea behind investment profiles is to add richness to the description of an investment, in terms of both the underlying calculations and the graphics that are helpful in visualizing what might happen in the future. It is a more rubust framework that incorporates information from the options markets and volatility into future stock price estimates. Chapter 8 is the first of three chapters on the structured security model. This chapter covers the display page and the links to the Calc Engine for a stock-only investment. Chapter 9, “Adding Options to the Model,” adds option profit calculations and option pricing to the model. Chapter 10, “Option Investment Profiles,” completes the basic model. The model will enable you to track several aspects of structured securities over time, including the effect of price and volatility. The model highlights the changes in the underlying probability distribution—the key to understanding the behavior of structured securities.

The Purpose and Context of the Model In practice, I use different types of models for different purposes. For day-to-day portfolio tracking and management (such as adjusting risk exposures and theta levels), I use the TradeStation platform and cost basis tracking software. These practical tools are introduced in Chapter 14, “Managing Positions,” and Chapter 15, More on Synthetic Annuities.” 145

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The model presented in this book is the one I use to think about structured security design. It enables me to evaluate alternative structures that fit my risk and return targets. The model helps me focus on the metrics more visually and intuitively. In its basic setup, the model compares a stock-only investment to a structured alternative, which gives me a way to think about the tradeoffs that happen as options are layered into the structured security to change its behavior, such as producing higher yield, more leverage, or more downside protection.

The Stock-Only Investment Profile Building a stock-only investment profile is a fairly simple process. The Calc Engine from Chapter 6, “The Lognormal Distribution and Calc Engine,” gives us most of what we need: stock prices and their probabilities. The rest involves getting started setting up the spreadsheet, defining metrics, and creating the chart. In the previous chapter, Figure 7.1 illustrated a stock-only investment profile. The graph included a profit curve for a stock investment and its corresponding probability distribution. Figure 8.1 is a screenshot from the spreadsheet that produced the graph. For this example, assume that you just bought 100 shares of a stock for $100 per share. Also assume that the time horizon is one year, the risk-free rate is zero, the dividend rate is zero, and the annualized volatility is 30%, as shown in the figure. Even though no options are involved at this point, you can model stock behavior based on volatility implied in the options market. In this case, assume that the implied volatility of an actual one-year option being traded on the exchange is 30%. This volatility can be used to project the shape of probability distribution for stock prices at the end of the year. You can think of the probability distribution in this example as being implied by the market (consistent with option prices) rather than assumed or based on historical data.

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Figure 8.1 Stock-only investment profile

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The Heading: Assumptions, Descriptions, and Metrics The heading sections in Rows 1–12 above the graph in Figure 8.1 contain input items, the description of the structured security, and a few calculated metrics. The input items include the option pricing variables and the stock price range. The desciption of the structured security includes the number of shares of stock and number of options, along with the option strike prices. The calculated metrics include the level of market exposure, a measure of option-generated yield, and VaR at the 5% level. Input items in Columns B and C are shown at two points in time, labeled as Time 0 and Now. Time 0 refers to the purchase date; Now can be any time between the purchase date and the option expiration date. Columns B and C, Rows 3–7 include the pricing assumptions, such as 365 days until expiration, risk-free rate equal to 0%, dividend rate equal to 0%, and annualized volatility of 30%. Row 8 indicates that the price range to be graphed is between $28.79 and $317.40, the same as in previous examples. The next two items, Option Time Value and Annualized Average Theta, do not apply to stocks, so those numbers are 0 for the stock-only example. The next item in Row 11 is labeled Crossover/Probability. On the right side of the heading, the structured security is defined, including the number of shares of stock and the number and strike prices of any options. In this case, the number of shares of stock is 100, and the number of options is 0. The information in Rows 5–11 is calculated by the model and used to track the security’s gains and losses, market exposure, and risk levels.

Crossover/Probability Crossover (Cell C11) is the stock price at which a stock-only position has higher profits than the modeled structured security. In this case, because we are looking at only the stock itself, crossover is not defined. For the purposes of this example, I set the crossover point to be the stock purchase price simply to illustrate how the Calc Engine looks up probabilities.

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For stock values above the stock purchase price, the stock position has a gain. The probability that the stock price will be above the purchase price is 45.71%. The fact that the probability of a gain is only 45.71% might be a little surprising. The fact that it is below 50% is due in part to the assymetry of the stock price distribution. It is also partly due to the use of “risk-neutral” probabilities, which we talk about in more detail as we go through the examples.

VaR 5.0% VaR at the 5.0% level for one year is $4,250. If you buy this stock for $100 and the assumptions are correct, then 5% of the time, the loss will be $4,250 or higher. In other words, if you think about running a large number of simulations, each time making random draws from the distribution, then 5% of the time, the loss will be $4,250 or higher.

The Stock Price Range Cells B8 and C8 can be changed to model the price range you are most interested in viewing. To illustrate how this works, in the next example, the current scale will be changed from $28.79 at the low end to $40, and from $317.40 at the high end to $240. The narrower range zooms in on the more likely outcomes. Anything beyond –/+3 standard deviations is not that graphically interesting because the probabilities associated with points beyond 3 standard deviations are so low. Figure 8.2 is set to the narrower range of $40 to $240. This is the basic layout of the display page of the model. If you want to begin building the spreadsheet model at this point, you can type in the input descriptions, the assumptions, and the stock price range in Columns B and C. For now, leave the chart area blank. We’ll return to the chart after adding the Calc Engine and Profit Calculator sections.

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Figure 8.2 Stock profile, price range = $40 to $240

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The Calc Engine The next step is to copy the Calc Engine from Chapter 6 into Columns L through R of the spreadsheet. Then link the Calc Engine pricing assumptions to their corresponding values on the display page, as follows: N3 = C3, N4 = C4, N5 = C4, N6 = C5, N7 = C6, N8 = C7 Also link the stock price range by setting R5 = B8 and R6 = C8. Columns L through R of your spreadsheet should now look like Figures 8.3a and 8.3b, except for the formatting changes, which were made to highlight points in the following discussion. It is not important to make the formatting changes, but the numbers should be the same. As discussed in Chapter 6, the Calc Engine provides two different views of the stock price random variable. The first three columns are stock prices based on evenly spaced standard deviation points (–4 to +4 in 0.1 increments). The next three columns are stock prices based on evenly spaced dollar increments ($40.00 to $240.00 in $2.50 increments). The last column is the cumulative probability for the evenly spaced dollar version, indicating how much of the total distribution is included up to that point. As with all cumulative distributions, the number in the first row is equal to the first point, and the number in the last row is 1.00, or 100%.

VaR and CVaR VaR at the 5% level is $4,250. Now you can see where the number came from. If you read down Column R (cumulative probability), you do not see an exact 5.0% level. However, the first number above 5% is 5.222% in Row 20. That means the stock prices in Rows 13–20 have a combined probability of 5.222%. Actually, this is VaR at the 5.222% level, but that is close enough to 5% to serve as a rough estimate and walk through the logic.

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Figure 8.3a Calc Engine

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Figure 8.3b Calc Engine (Continued)

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Given that the price is in the 5.222% tail, there are eight possibilities for the stock price, $40 to $57.50. VaR is based only on the highest of these values, $57.50. When the stock price is $57.50, loss per share is $42.50, or $4,250 for 100 shares. That means that, 5.222% of the time, the loss will be $4,250 or higher, which is the definition of VaR. VaR is a simple lookup, but CVaR is the weighted average of losses occurring in the tail of the distribution—given that the result is in the tail. To calculate CVaR, first compute the weighted average of the eight possible loss amounts, as follows: $249.91 = $4,250 × 0.01375 + $4,500 × 0.01106 + ... + $6,000 × 0.00253 Now adjust this answer by dividing it by the probability that the outcome will be in the tail, which we know is 5.222%. So CVaR is $249.91 / 0.05222 = $4,785.49, the average loss in the tail of the distribution.

Another Way to Get the Same Answer Using Standard Deviations You can get the same answers by taking a different approach. A one-to-one relationship exists between tail probabilities and standard deviations. For a normally distributed random variable, the 5% tail always occurs at –1.645 standard deviations. Likewise, a 1% tail always occurs at –2.326 standard deviations. In fact, each possible tail value can be translated into standard deviations. For instance, Cell S20 indicates where a stock price in Column R falls on the standard normal distribution curve. Cell S20 is –1.695, the closest standard deviation point to –1.645. So it is no coincidence that the 5% tail is bordered by this row. Column A can be used to calculate VaR. I did not include a cumulative probability column for the normal version, but it is true that –1.645 standard deviations is the 5% tail. Looking at Cells A36 and A37, you can see that the stock price around the midway point is close to $58. That is within rounding error of the result using the lognormal

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version. Here are a few points of cross-reference for various levels of VaR and standard deviations: VaR 5% occurs at –1.645 standard deviations. VaR 1% occurs at –2.326 standard deviations. At –2 standard deviations, the level of VaR is 2.28%. At –3 standard deviations, the level of VaR is 0.13%.

Crossover/Probability In this example, the crossover point for the stock has been set to the purchase price of $100. If the stock price at the end of one year is equal to or above $100, there is a gain or breakeven. Otherwise, there is a loss. We can figure out how likely a stock price of $100 or over is by looking at Column T. Notice the highlighted cells in Rows 37–93. These are the probabilities for each stock price point equal to or greater than $100. Cell T97 is the sum of these probabilities, equal to 45.71%. That number is shown in the heading in Cell C11. The numbers in Cells B11 and C11 give us the crossover point and the probability that the stock price will be equal to or greater than the crossover point. In later examples, this information is more meaningful in the context of comparing two alternative investments. In this case, it simply indicates that a gain occurs at any stock price of $100 or above; this happens 45.71% of the time under these assumptions. As a fine point, the entire probability of the point where the stock price is exactly $100 was included in the total. Whenever a stock price value is exactly equal to the crossover point, as it is in this case, we could split the probability of this particular cell in half. Technically, half the “range” is above the breakeven and half is below the breakeven. If you do that, the probability drops to 44.06% (Cell T99). Let’s look at the same question using the normal version of the stock price random variable. We can get the probabilities for the normal version from Column P. The cells are highlighted where the stock price in Column Q is greater than $100. In this case, no stock price is

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exactly equal to $100, so the probabilities do not need to be adjusted. The answer in Cell P97 is 44.04%. This is very close to the evenly spaced version when we divide the probability by 2 for the exact cell. Given that the answer is close to 44%, what exactly does it mean? It means that, only about 44% of the time, if these assumptions turn out to be accurate, you would make a profit. One point to keep in mind, however, is that there is no assumption of an upward drift in the stock price. An upward drift would make a difference in the answer. The issue of assumed drift is an important technical point related to a mathematical shortcut in option pricing. Instead of calculating actual probabilities and discount factors, the shortcut discounts future payoffs at the risk-free rate and, at the same time, alters the probabilities to something called “risk-neutral” probabilities. The implication is that probabilities in the model are based on the risk-free rate instead of a higher drift calculated using the Capital Asset Pricing Model or some other estimate of security returns. This is not a big problem when the option pricing framework is used to compare investments with similar profit curves. However, when we ask questions about the probability of a gain on a standalone security, the probability of a gain might be understated.

Median Stock Price The median stock price is the price at which 50% of the time the stock price at expiration will be higher and 50% of the time the stock price at expiration will be lower. For symmetric distributions like the normal distribution, the mean and median is the same. For lognormal distributions, however, the mean and median are not the same. This is one time when it is easier to use the normal version of the stock price random variable. In that version, the distribution is symmetric, so the median price is always in the middle, or at zero standard deviations. To determine the median price, just look at Cell Q53, which, in this case, is $95.60. As volatility goes down, the median value approaches $100. The higher the volatility, the lower the median price. This is consistent

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with the earlier discussions on the effects of volatility on the probability of making a profit.

The Stock-Only Profit Calculation The only necessary piece of the model missing at this point is profits, or gains and losses. Figure 8.4 shows the spreadsheet section called the Profit Calculator, which includes columns for these calculations. In this chapter, we go through stock-only examples; we add the options and structure profits in the next two chapters. In the layout for the Profit Calculator in Figure 8.4, the heading pulls in information from the input screen (Rows 3 and 4) and also displays calculated security prices (Rows 5 and 6) and deltas (Rows 7 and 8). For the stock-only example, we look at only Columns T and U.

Figure 8.4 Profit Calculator

Column T pulls in the stock prices from the Calc Engine. To continue building the model, set the stock prices in Column T to their

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corresponding values from Column O. Define T13 = O13, and copy this formula down to T93. Also set the stock prices at T = 0 and Now to the input screen values: U5 = B4 and U6 = C4. The Stock Gain (Loss) in Column U is equal to the difference between the stock prices at expiration in Column T and the purchase price of the stock in U5 multiplied by the number of shares in U3. The Excel formula for Row 13, which can be copied down through Row 93, follows: U13 = (T13 − U$5) × U$3 When the stock price is $40, the loss per share is $60, so the total loss on 100 shares is –$6,000 (Cell U13). If the stock price is $42.50, the loss is –$5,750, and so on. At the highest possible stock price, the profit is $14,000 (Cell U93).

Stock Gain (Loss) Random Variable Stock gain (loss) is a new random variable. As in previous examples, the probability of any particular gain (loss) is the same as the probability of the stock price used to generate it. In this model, these probabilities are in Column Q of the Calc Engine. So the probability of any value in Column U is the same as the corresponding row in Column Q. For example, the stock price in Row 13 is also the probability for the stock gain (loss) in Row 13. Under the current scenario, 100 shares of a stock are purchased for $100; one year later, there are 81 possible values of the stock price. These prices range from $40 to $240, as shown in Column T. Those prices have the probabilities shown in Column Q. Likewise, the gains and losses have 81 possible values and Column Q probabilities. This set of values and probabilities is the stock gain (loss) random variable. Sometimes this is also referred to as the stock profit random variable. Note that, by setting the stock price range to $40 to $240, the maximum profit is lowered just by lowering the top of the range. The maximum loss is lowered because the price in Row 13 increases from $28.79 to $40. The probabilities have also been adjusted to account

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for the difference. Usually this doesn’t cause a problem, as long as the range is wide enough to include –/+3 standard deviations.

Adding the Chart Insert the chart below the heading section by selecting Insert, Chart, Area from the Excel menu. Next, drag the sides of the chart area to match the figure. Then add the chart data by selecting Chart, Add Data for each of the four columns that appear in the chart, as follows: The x-axis is in Column O (O13:O93). The probability distribution is in Column Q (Q13:Q93). The stock-only profit values are in Column U (U13:U93). The structured security profit values are in Column Z (Z13:Z93). The probability distribution is displayed on the primary (left) axis, and the two profit curves are displayed on the secondary (right) axis. See the charting tips in the following sidebar for selecting the axis. If your chart doesn’t line up on the y-axis—that is, the right and left sides have different zero values—you can adjust the minimum and maximum scales on one of them to make them line up. Try a few minimum and maximum scale numbers on the profit curves until you get them to sync.

Tips on Creating Charts Here a couple tips and references to help you with the chart. For a general overview of charting, see: http://office.microsoft. com/en-us/excel-help/create-a-chart-HP001233728.aspx#BM6b. Tips on Primary and Secondary Axis Excel charts give you the choice of using two y-axes. When adding data, you can specify whether you want the data to appear on the

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left y-axis or the right y-axis. The left axis is referred to as the primary axis, and the right axis is the secondary axis. Figure 8.5 shows the Format Data Series screen.

Figure 8.5 Format Data Series screen

To get this box to appear, click on the chart so it is highlighted. Then move the cursor over the chart area until you see “Series stock only” below the cursor and click. Now, go to the main menu and select Format and click on the option “Format Data Series....” The display screen graph plots the probability distribution on the left, or primary, axis, and plots the two gain/loss values on the right, or secondary, axis. For specific examples of creating a chart with two y-axes, see http://lytebite.com/2009/02/24/how-to-create-excel-chart-with2-y-axis-or-x-axis/.

Standard Deviation Markers In Figures 8.1 and 8.2, the charts also have markers indicating the –3, –2, –1, +1, +2, and +3 standard deviations points. These points are based on lookups from the standard deviations in Column P (P13:P93). You can either determine the first point that crosses a boundary (–3, –2, –1, +1, +2, +3) or the closest point to the boundary. These markers can be assigned to either the primary or the secondary axis. You might want to insert a section of additional columns to figure out where to put the markers. Figure 8.6 shows a sample format.

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Figure 8.6 Standard deviation markers

In this example, Columns AC, AE, and AG contain lookups on Column AB (Column AB is equal to Column P) to determine where a standard deviation marker should go. For instance, Column AC contains 1 when a 1 standard deviation boundary is crossed; otherwise, it returns 0. Column AD is used to scale the size of the marker. You can choose whether to display “closest to” or the first point that “crosses” a standard deviation marker. You can also choose whether you want to display standard deviation markers or VaR markers. To add a marker for 5% VaR, use the point corresponding to –1.645 standard deviations. After defining the markers you want to display, choose the Add Data feature under Chart and select Rows 13 to 93 to add the marker. You must indicate whether you base this on the primary or secondary axis. Depending on which you choose, you want to scale the indicator to be the right size. That is why the scaling factor is included in Row 7. In this case, $2,000 creates a reasonably sized marker using the scale specified for the secondary axis.

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Adding markers is optional. Setting the standard deviation markers at –/+1, 2, and 3 seems to work well, but you might prefer others, depending on what you are most interested in seeing on the chart. To include standard deviation markers shown in the figures, add the data in Rows 13–93 of Columns AD, AF, and AH to the chart using the Add Data feature. The Transparency of the profit curves was adjusted so that probability distribution can be seen through the curves, and Drop Lines were added. This is a matter of how you want the chart to look. Don’t worry if your graph doesn’t look exactly like the one shown here: You might be using a different version of the software or different color settings, for example. The main thing is to make the numbers match and have the profit curve look similar.

Test: Stock-Only Investment Profile Before going further, test the spreadsheet to see if it works properly and has the correct links. The spreadsheet should be arranged as follows: Columns A–J Column K Columns L–R Column S Columns T–Z

Display Screen Blank Calc Engine Blank Profit Random Variables

Optional: Column AA Columns AB–AH+

Blank StdDev/VaR Markers

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Try the assumptions and input from Figure 8.1, the stock-only investment profile. Make sure you get the same information through Row 9 of the heading and that the probability and profit curve look similar. Also make sure that the Calc Engine and the stock gain (loss) numbers match. As a second check on links and formulas, change the assumptions in Columns B and C of the display screen to match Figure 8.7. The scenario is almost the same at setup, except for a nonzero risk-free rate. The model looks ahead nine months under the conditions that the stock price has dropped to $85, the risk-free rate has gone down (a typical reaction in the bond market to equity market turbulence), and an increase in volatility. If your display page looks like Figure 8.7, things are working properly.

The Calc Engine Figures 8.8a and b show the new entries in the Calc Engine.

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Figure 8.7 Test of stock-only investment profile

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Figure 8.8a Calc Engine Test

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Figure 8.8b Calc Engine Test (Continued)

9 Adding Options to the Model The next step in building the model is to complete the options section of the Profit Calculator. The following is a review of the standard profit formulas for options. In the chapters on option pricing, option payoffs were used instead of option profits. The difference is that option profits are adjusted for the premium paid or received. In the following formulas, the stock price at expiration is S and the option strike price is K. • Long call option: The payoff of a long call option (purchased call option) is given by the expression max{S − K, 0}. The profit is max{S − K, 0} − premium paid. • Short call option: The payoff of a short call option (written, or sold call option) is given by –max{S − K, 0}. The profit is –max{S − K, 0} + premium received. • Long put option: The payoff of a long put option (purchased put option) is max{K − S, 0}. The profit is max{K − S, 0} − premium paid. • Short put option: The payoff of a short put option (written, or sold put option) is max{K − S, 0}. The profit is –max{K − S, 0} + premium received. Figure 9.1 is an excerpt of the profit calculation section incorporating these formulas. In this example, the number of shares of stock is 100. Strike prices are $100, and the prices of the options are all equal to $11.92. The Excel formulas for option profits corresponding to the previous definitions for Row 13 are:

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V13 = V$3 × (MAX(0,V$4 – $T13) – V$5) W13 = W$3 × (–MAX(0,W$4 – $T13) + W$5) X13 = X$3 × (–MAX(0,$T13– X$4) + X$5) Y13 = Y$3 × (MAX(0,$T13 – Y$4) – Y$5) Z13 = SUM(U13:Y13) To complete the section, copy these formulas down through Row 93.

Figure 9.1 Profit Calculator

Long Put Profit The long put gain (loss) is another new random variable, similar to the stock gain (loss) random variable of the last chapter. The 81 possible values of the random variable are in Column V, and the probabilities of those values are from Column Q of the Calc Engine. As mentioned earlier, probabilities follow the row. That is, the probability

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of any calculated number in Rows 13, 14, ... has the probability from the corresponding row in Column Q of the Calc Engine. The Black-Scholes formula price of this put option is $11.92. For now, hardcode this number in Cells V5 and V6. Later, in the section on the Black-Scholes formula, we will add the calculation and replace the hardcoded number with a new cell reference. A long put option gives you the right to sell the underlying security for the strike price. It pays the difference between the strike price and the stock price at expiration, if the stock price is below the strike price. It pays nothing if the stock price is above the strike price. In Row 13, for instance, the stock price is $60 below the strike price, so the long put option pays $60 per share, or $6,000 in total based on 100 shares. The payoff is $6,000, but the profit must be adjusted for the cost of the put option. In general, the profit on the transaction is the payoff minus the amount paid for the option. The profit is $4,808 ($6,000 − $1,192). That is the number in Cell V13. Similarly, when the stock price is $45, the profit is $4,308, and so on.

Short Put With a short put option you have given someone else the right to sell you the underlying security at the strike price. The payoff to the holder is the difference between the strike price and the stock price at expiration, when the stock price is below the strike price. The payoff to the holder is nothing if the stock price is above the strike price. For Row 13, the stock price is $60 below the strike price, so the short put option requires you to pay $60 per share, or $6,000 total, to the option holder. The short put profit is adjusted for the proceeds you received from selling the option. For example, in Row 13 the $6,000 loss is adjusted for the proceeds from selling the put option of $1,192. The net effect is a loss of $4,808 shown in Cell W13. The calculations for short and long calls are similar.

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Expected Values At this point, you may be wondering how the Profit Calculator relates to the option pricing work we did earlier. This section answers that question by looking at weighted or expected values. This is not part of the model, but if you would like to reproduce the figures, copy the Profit Calculator and paste it into Columns AB through AH. Then multiply the profits in each column by the probabilities in Column Q and add the results in Row 95, as shown in Figure 9.2.

Figure 9.2 Profit random variables expected values

The answer is zero, or very close, for both stock and options. The answers are slightly off because the model is not exact. This is a reminder that the model assumes a world in which only 81 possible stock prices exist, so there are 81 possible payoffs for the stock and the options positions. Even so, the answers are close. The stock outcome of zero makes sense because there is no assumed drift in the stock price under these assumptions. If the drift

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were 5%, expected profit on the stock would also be 5%. The options outcome of zero also makes sense. This is because we are valuing the options under the same assumptions used to price them. If there were a built-in gain or loss from simply buying or selling an option, it would be an arbitrage opportunity. If the option is priced correctly, adding the option to your portfolio does nothing to change the value of the portfolio at the moment you buy it. Of course, many people don’t think options are correctly priced, and they buy or sell them to take advantage of the mispricing. But that implies that the “real” probabilities or some other aspect of the assumptions are different from those assumed in pricing. That may be a great reason to trade, but for the model, it is a good sign that the expected values are zero. To see why the answers are zero, let’s change the formulas in the Profit Diagram to exclude the option premiums. In other words, let’s set the values to the payoffs instead of profits. Actually, that puts the calculation back to option pricing, where the price of an option is the expected value of the option payoff random variable. Figure 9.3 summarizes this.

Figure 9.3 Payoffs

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For this figure, the Profit Calculator columns were copied and pasted into Columns AJ through AP. Then the option prices in Rows 5 and 6 were set to zero. By doing this, the module calculates the payoff rather than the net payoff (or profit). This is similar to the procedure for option pricing, which means it should produce the same result, and it does. Figure 9.4 is the same as Figure 9.3, except that the payoffs are multiplied by the probabilities.

Figure 9.4 Payoff expected values = option values

This illustrates, once more, that option prices are the weighted payoffs or the expected values of the payoff random variables. The only difference is that the answers refer to the total value of the options, given the number of options specified in the heading. Earlier, the option price was for one option. Here the value of the options represents 100 options. If you set Row 3 equal to 1 instead of 100, it would give you the price for one option.

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With the number of options set to one, the price is $11.91 for the put options and $11.90 for the call options, compared to the BlackScholes formula answers of $11.92. The mathematical notation for expected value of the random variable X is E[X]. Using this notation, the relationship between the profit and payoff random variables is: E[Stock Profit at Time 0] = Assumed stock price drift E[Option Profit at Time 0] = $0 E[Option Payoff at Time 0] = The value of the options And when the number of options is 1 E[Option Payoff at Time 0] = The option price The last three exhibits were included only to illustrate the relationships between payoffs, profits, and expected values. They are not part of the model, but they do bring up a good question: What is the best—or easiest—way to price options in the model?

Black-Scholes Add-In One way to price the options is to expand the spreadsheet with additional columns to calculate expected payoffs as in the preceding section. Another way is to use the Black-Scholes calculator from Chapter 3, “Option Pricing Methods.” When you don’t need to extract a probability distribution, it is simpler to use the Black-Scholes formula. The advantage of using the Black-Scholes calculator is that you can copy it directly under the profit columns. It doesn’t require additional columns, and later when we look at multiple occurrences of option types, you don’t have to think about making changes to other parts of the spreadsheet. Because it is compact, it is easy to copy it twice to provide prices both at Time 0 and at the current date. Doing this with expected payoff columns would be more complicated. Figure 9.5 shows the Black-Scholes formula add-in. This is a view of Rows 97–152 in Columns T–Z, just below the Profit Calculator.

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The descriptions in the first column and the formulas in the second column are exactly the same as those in Chapter 3, with the exception of the deltas in Rows 151 and 152, which are explained later.

Figure 9.5 Black-Scholes add-in

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To create this section, copy the three-column Black-Scholes calculator from Chapter 3 into Columns T–V and define the input cells as: V99 = $B$3 / 365 V100 = $B$4 V101 = V$4 V102 = $B$5 V103 = $B$6 V104 = $B$7 Next, copy the formulas in Column V across through Column Y. That completes the Time 0 section. Then copy and paste to create the current date section, changing B to C in the cell references to match the display screen inputs for time Current Date. The input cells on the display screen under Time 0 (Column B) will feed the Time 0 section, and the input cells on the display screen under current date (Column C) will feed the current date section.

The Heading Formulas To complete the heading formulas, link the structured security’s number of shares and strike prices to the input cells on the display screen. This defines the cells in Rows 3 and 4 as: U3 = E3, V3 = F3, and so on, and V4 = F4, W4 = G4, and so on. Then set the stock prices in U5 = B4 and U6 = C4. These are also input items that you select. Next, set the option prices in Rows 5 and 6 equal to the values from the Black-Scholes add-in. These numbers are calculated based on the assumptions and scenarios you choose. The references are: V5 = V122 W5 = W122 X5 = X116 Y5 = Y116

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V6 = V149 W6 = W149 X6 = X143 Y6 = Y143 To finish, set the Deltas in Row 7 to the values from the BlackScholes add-in, and define the $Deltas in Row 8 as: V7 = V152 W7 = W152 X7 = X151 Y7 = Y151 V8 = V3 * V7 W8 = W3 * W7 X8 = X3 * X7 Y8 = Y3 * Y7

Delta Formulas The delta formula is explained in detail in the material on Greeks in Chapter 13. For now, enter the standard formulas for delta in Cells V151 and V152 as: V151 = EXP(–V130 * V126) * V140 V152 = EXP(–V130 * V126) * V148 Copy these formulas across through Row Y to complete the section.

Time Value and Total Premium Formulas The final step is to calculate option time value and total premium. In general, the total option value is the sum of time value and intrinsic value. The formulas below subtract the intrinsic value from the total value to obtain time value.

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V154 = –V3 * (V5 – (MAX(0,V4 – $B$4))) W154 = W3 * (W5 – (MAX(0,W4 – $B$4))) X154 = X3 * (X5 – (MAX(0,$B$4 – X4))) Y154 = –Y3 * (Y5 – (MAX(0,$B$4 – Y4))) Z154 = SUM(V154:Y154) V155 = –V3 * (V6 – (MAX(0,V4 – $C$4))) W155 = W3 * (W6 – (MAX(0,W4 – $C$4))) X155 = X3 * (X6 – (MAX(0,$C$4 – X4))) Y155 = –Y3 * (Y6 – (MAX(0,$C$4 – Y4))) Z155 = SUM(V155:Y155) The numbers in Z154 and Z155 are shown on the display page as Option Time Value in Cells B9 and C9, respectively. To keep track of gains and losses, we also need the total premium paid or received for the options. This is the number of options multiplied by the premium for one option, with a negative sign for long positions, as follows: V157 = –V3 * V5 W157 = W3 * W5 X157 = X3 * X5 Y157 = –Y3 * Y5 V158 = –V3 * V6 W158 = W3 * W6 X158 = X3 * X6 Y158 = –Y3 * Y6 The premiums in Row 157 are shown on the display page as Cash at Time 0, and the premiums in Row 158 are shown as Cash Now. Positive amounts indicate cash was paid for the option (long positions), and negative amounts mean cash was received for the option (short positions).

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10 Option Investment Profiles This chapter extends investment profiles to the four basic option variations: • • • •

Long call options Short call options (also referred to as written call options) Long put options Short put options (also referred to as written put options)

Option investment profiles combine standard option profit diagrams with a probability distribution. To provide a comparison benchmark, the graphs also include the stock profit line to highlight the difference between the derivative and its underlying security. This chapter also begins the discussion on leverage and capital allocation, where the intent may not be as clear with options as it is with stocks, and where metrics need to be defined in the context of specific objectives.

Long Call Option Investment Profile Figure 10.1 compares the stock-only profile to a long call option profile. Assume that the underlying stock and assumptions are the same as in the stock-only example in Figures 8.1 and 8.2: 100 shares of a stock currently trading for $100, a one-year option term, zero interest and dividend rates, a $100 strike price for the option, and implied volatility of 30%. The profile for the long call option is the darker shaded area. The stock-only profile is the straight-line lighter area. Having both profiles in the graph makes it easier to compare buying an option to what would happen if you bought the stock instead. 179

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Figure 10.1 Long call option profile

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You can see one of the advantages of buying a call option on the left side of the graph. If the stock plunges, the most you can lose is $11.92, the amount paid for the option. The breakeven point is $88.08. On the graph, this is the point where the two profit curves intersect. If the stock price is less than $88.08, the option performs better. If the stock price is above $88.08, the stock performs better. With the stock, you make a profit at any price over $100. But with the option, the stock price has to be above $111.92 to make money. On the graph, that point is where the option profit curve crosses the x-axis. The amount paid for the option, $11.92, is represented by the vertical distance between the two profit lines. This distance is the visual representation of the time value of the option.

Option Time Value When you buy a call option, you can think of the price as being composed of two parts: intrinsic value and time value. Intrinsic value is the difference between the stock price and the strike price. It is the amount by which the option is in-the-money. The time value is the remainder. As an example, let’s say you pay $15 for a call option with a $95 strike price, and the stock price is $100. The intrinsic value is $5, and the time value is $10. Time value is a measure of how much you will make on the option if the stock price stays constant at $100. In other words, time value isolates the effect of the passage of time versus changes in the stock price. In Figure 10.1, the stock price and the strike price are the same, so the entire value of the option is time value. One of the questions I like to ask with options is: Am I paying for time, or am I being paid for time—and how much? The answer, in this case, is: I am paying for time. The amount is $1,192.00. In other words, if I buy the call option contract for $1,192.00 and the stock price at expiration is the same as it is now ($100), I will have paid $1,192.00 for “time.”

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Capital at Risk There is an important distinction between stocks and options. With stocks, both the intent and the capital allocated to a trade are normally understood. If an investor or trader believes a stock is going up, they buy it; or if they think it is going down, they either avoid it or short it. The capital allocated to the position is somewhat defined by how much it costs. Of course, margin accounts allow for some leverage, but it is low compared to the leverage available with options. With options, the intent and the amount of capital may not be as clear. Therefore, the metrics associated with options are often more meaningful in the context of specific objectives. For instance, for longterm strategic investing, options may act as stock substitutes, insurance, or to generate income or change risk exposures for an existing position. In this context, it is natural to compare the behavior of the option to the behavior of the stock and portfolio capital allocated to the stock position. On the other hand, in the context of tactical trading, options can be used to create leverage or make short-term directional bets. In that context, it is better to measure option behavior as a stand-alone instrument and define risk and reward metrics differently. In either case, the investment profile gives you the framework to ask and answer a broad range of questions. The idea behind the investment profile is to accommodate both perspectives by presenting options as standalone instruments and in comparison to the underlying stock. What about the percentage gain or loss in this example? Let’s just look at the case where the stock price at expiration is the same as when the option was purchased, or $100. If you look at the option as speculation, then you would lose 100% of the money used to buy the option. If you look at the option as a stock substitute, and you put aside the $10,000 you would have used to buy the stock and kept it in cash, then you might look at the loss as 11.92% of the amount allocated to this position. If you use the option for leverage and, instead of buying one contract, you bought eight of them (approximately $10,000 ÷ $11.92) and kept no money in cash, you would definitely think of this as a 100% loss of premium. This is one example of how the metrics need to be interpreted in light of the intent and capital at risk.

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In Figure 10.1, the yield calculated in Cells B10 and C10 of –11.92% assume that $10,000 is the capital allocated to the trade, so $10,000 is used in the denominator of the fraction. Some people prefer to use a different definition of capital such as VaR or maximum loss. This is one of those metrics that is not straightforward and depends on the context of the investment or trade. Because the main focus of this book is the comparison of stock positions to a structured security that includes the underlying stock, capital at risk will normally be defined as the net amount invested in the structured security.

Fiduciary Calls and Protective Puts To make the distinction between stock substitution and speculation, sometimes the term fiduciary call is used. A fiduciary call refers to the combination of cash (or a bond) plus a long call option. A fiduciary call makes it clear that either cash or a bond is being held as capital on the position. A similar distinction applies to puts. The term protective put refers to the combination of a long stock position and a long put option. A protective put makes it clear that the put is being used as insurance against the associated long stock position rather than as a speculative bet on a decline in the price of the stock, which could be done without holding the stock. The investment profile of a fiduciary call and a protective put is identical. This is one example of put–call parity, a concept covered in more detail in Chapter 11, “Covered Calls, Condors, and SynAs.”

Annualized Average Theta Theta is the time decay of an option occurring in one day. It is possible to calculate theta by formula, or you can do it by changing the days until expiration from 365 to 364 in the model. In addition to daily theta, I like to track another number. I call it annualized average theta, and it is displayed in Cells B10 and C10. Annualized average theta is the time value of the option (in Cells B9 and C9) divided by the number of days left until expiration and then annualized. It is expressed as a percentage of the capital allocated

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to the position. In this case, with exactly one year left on the option term, it is already annualized; so average theta is –$1,192 ÷ $10,000 = –11.92%. This means we are paying for time at an average rate of 11.92% annually for holding this option. The negative sign indicates that we are paying for time. A positive sign indicates we are being paid.

Delta The delta of this option position of 100 shares is $55.96. As long as there is any time left before option expiration, the value of an option will not move dollar for dollar with the underlying security. If the underlying stock price increases by $1, the stock position increases in value by $100 for the 100 share position. Delta of $55.96 means that the value of the option position increases by $55.96, or 55.96% ($55.96 ÷ $100.00) of the stock increase. After all, there is still a year for the stock price to increase above the strike price. Understanding the immediate and lagged effects on changes in value is critical to understanding options. Delta changes with the time remaining in the option term, the level of the stock price, and volatility. The formula in the model takes all these factors into account. Also, you can use the model to generate delta under different scenarios. This is discussed in more detail in Chapter 13, “The Greeks.” Although delta has multiple interpretations, the one I tend to use most often is to think of delta as an equivalent number of shares of stock. This interpretation works only within a range around the current stock price. But assuming that the stock price stays within a narrow range, the behavior of 100 long call options is roughly equivalent to the behavior of 56 shares of stock. If the stock price increases by $1 from $100 to $101, you would make $56 by holding 56 shares of stock. You would also make about $56 by holding 100 long call options.

Crossover/Probability Building a structured security is a process of reshaping the profit curve. Normally, this creates a single crossover point. The crossover point is the stock price where, on one side of the price, the stock-only

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position performs better, and on the other side of the price, the structure performs better. In this case, the crossover point is the lowest stock price at which a stock-only position has a higher value than the structured security being modeled. In Figure 10.1, it is clear that the crossover point happens between $85 and $90. Because this is a simple structure, we can easily calculate the crossover exactly as $88.08, equal to the strike price of $100 minus the premium paid for the option of $11.92. In more complicated structures, it is convenient to have a general method of calculating the crossover price—and the probability that it will be exceeded. Figures 10.2a and 10.2b contain a sample format for determining the crossover point and the probability that the stock price will exceed the crossover. The columns also include the probability of a gain on the underlying security and the probability of a gain on the option. In terms of spreadsheet layout, the basic calculations of the model are performed in the Calc Engine and the Profit Calculator. The crossover/probability and other optional modules simply access and organize different aspects of the results. The first three columns of the Crossover/Probability module— the stock price in Column AJ, the stock-only profit in Column AK, and the structure profit in Column AL—are pulled in from the corresponding values in the Calc Engine and the Profit Calculator, as follows: AJ13 = O13 AK13 = U13 AL13 = Z13 The next four columns answer these questions: 1. What are the stock price points where the stock-only profit is higher? 2. What is the probability of those points occurring? 3. What is the probability of a gain for the stock-only position? 4. What is the probability of a gain for the structure?

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Figure 10.2a Calculating the crossover point and probability of gain

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Figure 10.2b Calculating the crossover point and probability of gain (Continued)

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Column AM compares the stock profit to the structure profit and shows the points where the stock outperforms. The lowest number in the column is shown in Cell AM95 and on the display page as the crossover. Column AN is blank when Column AM is blank, and equals the probability of the point from the Calc Engine when Column AM is not blank. The sum of the probabilities are shown in Cell AN95 and on the display page as the probability of crossover. In this case, it is 59.8%. Column AO shows the probabilities of the points where the structure has a gain. This number is also shown as one of the four summary metrics in the text block inside the graph. The probability that the stock price is equal to or above the crossover price is the sum of the probabilities for $90 and above. The probability that the stock will be profitable is 42.4% as shown in Cells AO95 and AO11. The probability that the structure will make a profit— that is, the probabilities of the stock price points above $111.92—is 30.67% as shown in Cells AP95 and AP11. The Excel code for Columns AM through AP for Row 13 is: AM13 = IF(AK13 >= AL13, AJ13, “ ”) AN13 = IF(AM13 = “ ”, “ ”, Q13) AO13 = IF(AK13 > 0, Q13, “ ”) AP13 = IF(AL13 > 0, Q13, “ ”) Copy these formulas down through Row 93, and insert the following formulas in Row 95: AM95 = MIN(AM13:AM93) AN95 = SUM(AN13:AN93) AO95 = SUM(AO13:AO93) AP95 = SUM(AP13:AP93) I should point out a technical issue here. As mentioned earlier, the model uses “risk-neutral” probabilities. Risk neutral probabilities are calculated assuming no upward drift in the stock price. Over the longer term, it is often reasonable to expect an upward drift based on pricing models such as the Capital Asset Pricing or Arbitrage models.

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Over the shorter term, you may also have a view, using technical or fundamental analysis, that the trend is up. In more advanced applications of the model (not covered in this book), it is possible to separate the risk-neutral probabilities calculated using the “risk-free rate” from probabilities using trend asssumptions. For instance, by assuming an expected return of 8% (but keeping the option pricing return assumption linked to the risk-free rate of zero), the probability of the option in this example producing a gain after one year is around 40% instead of 31%. With shorter option terms, the difference in the two methods narrows. And when the purpose is to compare a stock to a related structure, because both are being measured in the same way, the difference between the two is closer to the result using trend. As a practical matter, the main point is to be aware that probabilities associated with a single stock or a single structure may be understated over longer periods of time.

VaR 5.0% The VaR, in this case, is equal to $1,192, the premium paid for the option. Although it is correct to say that, at the 5% tail boundary, the loss is $1,192, the concept of VaR is not as meaningful for a long option, where the loss is capped. In fact, the level of VaR doesn’t affect the answer within a wide range. The maximum loss occurs in Row 37 of Figure 10.2. The cumulative probability from Figure 8.3 for Row 37 is roughly 58%. VaR at the 58% level is also $1,192. This is an attractive aspect of long options. Your downside loss is capped fairly early, and one of the reasons commentators such as Jim Cramer who otherwise don’t recommend options for most investors sometimes recommend deep in-the-money call options as a stock substitute. (The reason for deep in-the-money call options instead of at-the-money or out-of-themoney is their relatively low time value premiums—that is, you don’t have to pay that much for time.)

Include Stock? Y/N You might have noticed a new indicator in Column J. I included a switch that enables you to either include or exclude the stock component in Column E in the structure definition. If the entry in Cell J5 is

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N, the stock is not included and the structure is defined by the sum of Columns F–I. If the entry is Y, the stock component is included and the structure is defined by the sum of Columns E–I. For this chapter, the options are the only components of the structure, so the indicator is set to N. In the next chapter, covered calls are defined as a long stock position and a short call option position, so the indicator is set to Y in that case.

Short Call Option Consider what happens when you sell a call option contract. In this case, you receive the premium of $1,192. The most you can make is $1,192 which happens if the stock price is below $100 at expiration. Losses occur at stock prices above $111.92, and because the stock price could conceivably go to infinity, the losses are unbounded. Figure 10.3 shows the profit curve. If you sell a call option (short call option) and also have a position in the underlying stock, selling the option creates a covered call. If you don’t have a position in the underlying stock, it is an uncovered or naked call. Brokerage firms normally allow covered calls even for relatively inexperienced investors because the combined position is considered slightly more conservative than owning the stock by itself. On the other hand, to sell naked calls, you must apply for a higher level of options trading (for experienced option traders). This is because, with naked calls, there is the potential for large, even unlimited losses because there is no cap on how high a stock can go.

Option Time Value Because there is no intrinsic value of this option, the entire premium consists of time value. But instead of paying for time, as with the long call option, you are being paid for time. If the stock price at expiration is equal to its value today of $100, you will have earned $1,192 as the time value goes to zero over the term of the option.

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Figure 10.3 Short call option investment profile

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One of the most interesting aspects of options is the capability to generate yield by time decay. Regardless of how the stock price moves, time value goes to zero over the life of the option. Because this source of return is dependent only on the passage of time and not on the direction of the underlying security, it can be compared to the other time-related sources of investment return. Theta is to options what interest is to fixed income and dividends are to equities. And the potential return from theta is much higher than either interest or dividends. Going back to the example, assume that the portfolio allocation to the stock is $10,000. When you sell the option, you get $1,192, so the net investment is $8,808. If the option expires out of the money (the stock is below $100), the return can be calculated as $1,192 ÷ $8,808 = 13.5%. Returns from theta are higher with higher volatilities, and annualized returns are higher with shorter option terms. One- and twomonth options significantly increase the potential yields compared to annual options due to the rapid time decay during that time.

Delta Delta for a short option is opposite that of a long option, meaning that the short option value moves in the opposite direction of the stock price. The interpretation of delta is still the same and represents an equivalent number of shares of the stock, at least within a narrow range of stock prices.

Long Put Option Figure 10.4 illustrates the profile and metrics of a long put option.

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Figure 10.4 Long put option investment profile

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Long put options are often used as insurance. In concept, there is no difference between a homeowner’s policy that protects a house against fire and other damage, and a put option that protects a stock position against losses. Most people accept the idea that paying for homeowner’s insurance is prudent. However, very few investors use put options for portfolio insurance because of the cost. A typical homeowner’s policy costs less than 1% of the value of the house. Under these assumptions, the put option costs around 12% of the value of the stock. (One of the challenges for investors who want to use put options for insurance is finding ways to finance the put option at lower cost.) VaR is $1,192, the same as with a long call option. The difference is that the loss happens at higher stock values. As with the long call option, VaR looses most of its relevance because of the shape of the curve. Notice that the loss is the same at $100 and $240. So it is technically correct to say that VaR at the 5% level is $1,192, and it is also true that VaR at much higher levels is also $1,192. When thinking about risk, it is more useful in this situation to use the maximum loss.

Short Put Option Figure 10.5 illustrates the investment profile of a short put option. Short put options are an alternative way to get exposure to a security. A short put option produces gains at higher stock prices and losses at lower stock prices. Because the profits from a short put option move in the same direction as a stock position, delta is positive. By selling put options, you can increase your exposure to a stock. Short puts are useful instruments for experienced traders to adjust both risk and return metrics. If you are not familiar with short puts and would like to see some examples of short put options and their payoffs, there is an introductory video at Khan Academy. The link is www.khanacademy.org/ finance-economics/core-finance/v/put-writer-payoff-diagrams.

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Figure 10.5 Short put option investment profile

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Short put options are also helpful in financing partial protection as one leg of a put spread. You create a put spread by buying a put option at one strike price and selling a put option at a lower strike price. As an example, if you wanted to buy full protection on the stock in the example currently trading at $100, it would cost $1,192, or 11.92%. However, if you are satisfied with partial protection, such as covering only the first $10 in price drop, you could establish a put spread by buying the $100 strike put and selling the $90 strike put. The $90 strike put price is about $7. So the cost of insuring against a $10 drop in price is about $5 ($11.92 − $7). You have less protection, but the cost is less, too. To see the effect of this trade in the model, enter 100 shares and a $90 strike price in the short put column. The yield on a short put option can be interpreted in different ways, depending on how you are using it. If it is a purely speculative bet, you stand to make 100% if the stock price at expiration is above the strike price. The chances of making some level of profit is 59.8%, using the model’s risk neutral probabilities. In terms of yield, if the allocation to the position is $10,000, the potential return is a $1,192 gain on a net investment of $8,808, or 13.53%. Of course, the actual yield is determined by the stock price at expiration. This definition of yield is average annualized theta, based only on time decay of the option, which assumes a stock price at expiration of $100.

11 Covered Calls, Condors, and SynAs This chapter looks at two common structured securities, covered calls and iron condors. It also takes a more detailed look at synthetic annuities (SynAs) introduced in Chapter 7, “Investment Profiles and Synthetic Annuities.” I chose these three structures because they provide differing perspectives and some general points of reference. Covered calls are one of the simplest and most popular option strategies. They also provide a natural context in which to consider put–call parity. Iron condors represent a different but important point of view. Condors are well suited for people who want to take advantage of option time decay and don’t necessarily have a desire to invest in the underlying security. Synthetic annuities are included for a couple of reasons. The first is to illustrate how to modify the model to include more than one short call option. The second is to begin to think about managed strategies in which options positions are adjusted in response to stock price changes and market conditions. After this chapter, with the model completed, you can experiment with various design combinations by assembling the basic building blocks to create customized structures. The five basic building blocks of options-based structured securities are: • Stock (or other underlying security) • Long put options • Short put options • Short call options • Long call options

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Simple structures include only one or two elements. More complex structures can take on an almost endless variety of combinations because each of the four option types can appear in a structured security multiple times, with different strike prices or expirations. Each combination produces a distinct investment profile. The design process of finding the right mix for a particular situation depends on your investment risk tolerance and your trading goals. The purpose of this chapter is to begin to understand and quantify how adding options affects the behavior of the structures with respect to yield, market exposure, the probability of gain, and downside protection.

Covered Call Investment Profile A covered call is the combination of a long underlying security plus a short call option for every 100 shares owned. Selling call options has become a popular strategy because it tilts the investment profile in favor of current income in exchange for capital gains later. It is more comfortable for people who think that a bird in the hand is worth two in the bush, taking the smaller sure thing now in exchange for chances of a larger gain later. Figure 11.1 is the profile for a covered call. The lighter shaded area is the straight-line payoff of a stock-only investment. The darker shaded area that is sloped on the left and parallel to the x-axis on the right is the covered call profit curve. Compared to a stock-only investment, a covered call has a higher chance of making a profit because it generates a return even if the stock goes nowhere, or even declines somewhat. It also reduces the cost basis of the investment, reducing downside risk. Of course, there is a tradeoff: A covered call’s profit is capped. In this case, it is capped at $1,192. The text boxes inside the graph show that yield, as measured by average annualized theta, increases from 0% to 13.5%. Because the stock pays no dividend, the stock yield is 0% in the left box. In the right box, 13.5% is the time value of the option ($1,192.35, which declines to zero over the term of the option) divided by the net amount invested ($8,807.65 = $10,000 − $1,192.35).

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Figure 11.1 Covered call investment profile

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Market exposure, as measured by delta, is reduced from $100 to $44, meaning that the volatility of the position is less than half of the stock-only position. Of course, delta is not constant across stock prices. Keep in mind that delta changes as the stock price changes, so the market exposure changes as well. In general, the current value of delta is applicable for only a narrow range of prices close to the current price. The probability of a gain for the covered call is significantly higher than the stock-only position (59.8% compared to 45.7%), and VaR at the 5% level is reduced by almost $1,200. This combination of metrics illustrates that a covered call is a more conservative position due to a reduced cost basis, lower volatility, and a lower VaR.

Covered Calls and Short Puts You might have noticed that the covered call investment profile looks exactly like the short put investment profile from Figure 10.5. The two profiles have the same shape, the same option time value, the same VaR, the same crossover point, and the same probability of gain. So is it correct to say that a short put option is equivalent to a covered call position? Not exactly. Something is missing. A quantitative relationship between call and put options, referred to as “put–call parity” defines the exact connection between a short put and a covered call. Covered calls offer a natural context for put– call parity. This concept is important to understand from a theoretical point of view, and it has practical implications in option pricing. Let’s take a small diversion at this point before continuing with iron condors to address put–call parity.

Put–Call Parity Put–call parity is a frequently mentioned concept in quantitative finance. It describes the relationship between the price of a European call option and a European put option, where both options have

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the same strike price and expiration date. The general expression for put–call parity is: C(t) – P(t) = S(t) – K × B(t, T) where C(t) is the value of the call at time t, P(t) is the value of the put at time t, S(t) is the market price of the underlying asset, K is the strike price of both options, and B(t,T) is the present value of a zero-coupon bond that pays $1 at time T, the expiration date. When the interest rate is zero (as a practical matter, with shorter term options and with current low interest rates, this is a reasonable approximation), the term B(t,T) becomes 1.0, and the expression can be simplified to: Call − Put = Stock − K In other words, if you buy a “call option with strike price K” and sell a “put option with strike price K”, the transaction is equivalent to buying the underlying stock and borrowing an amount K in cash. Applying parity to our example gives this: Buying the call for $11.92 and selling the put for $11.92 = Buying the stock for $10,000 and borrowing $10,000 at 0% interest. In either case, you have spent $0, and the gain (loss) is equal to the stock gain (loss). The profit is equal to the stock profit because if the stock goes up, the call option pays the gain. If the stock goes down, the short put option has a loss. Although this relationship is true, it is not very useful. But when the expression is rearranged, it is easy to see when it might be. Following are three cases.

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“Covered Call” equals “Short Put Plus Bond” When the expression is rearranged as follows, you can see the exact relationship between a covered call and a short put: Stock − Call = K − Put Which can also be stated as: Stock + Short call = K + Short put The left side is a covered call. The right side is a short put plus K in cash. In the generalized case, when the interest rate is not zero, the right side becomes a short put plus a bond that pays K at the option expiration date. With low interest rates and shorter term options, the zero interest rate approximation is close. Earlier, we asked the question whether a covered call is equivalent to a short put. Now we know what was missing: The term K is from the short put side of the equation. To be equivalent, cash in the amount of K needs to be put aside and considered as part of the capital in the trade, along with the short put option. Maintaining the cash position is also related to the yield on the position discussed in the last chapter. In that discussion, the yield was calculated in two ways. The first way was based on the short put option by itself as a speculative trade. The second way was combined with cash as an investment. If you don’t maintain the cash balance along with the short put, the two positions are very different. However, if you use the asset allocation definition (the short put option plus cash) of potential yield, there is no difference between these two positions or the two investment profiles. So the equivalence is: A short put option plus cash is equivalent to a covered call position. Or, in exact terms with nonzero interest rates, cash is replaced with “a bond that pays the risk-free rate.” This form of put–call parity is a reminder of a practical relationship between call options and put options.

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Arbitrage Interpretation Arbitrage is an overriding principle in markets where synthetics exist. Arbitrage traders are always on the prowl for riskless profits. And put–call parity allows them to take advantage of mispricing by creating synthetic calls or synthetic puts. To create a synthetic call, rearrange the put–call parity expression as follows: Call = Stock − K + Put This means you have two ways of buying a call option. You can buy the call option trading on the exchange, or you can follow the recipe on the right side of the equation for a synthetic call: Buy the stock, borrow K in cash, and buy a put option. Either way, the gains and losses are identical. Imagine a call option that, for some reason, was “cheap” relative to the synthetic call. Then you would buy the real call option and sell the synthetic version. The result is a riskless profit. These mispricings won’t last long in the marketplace because they violate the law of one price. If the two positions are not priced correctly, then one version of the same thing is cheaper than another. Whenever option arbitragers see this, they buy the cheaper version and sell the more expensive version to make a riskless profit. In practice, their trades maintain the balance of pricing. This is one case of a more general pricing principle: no-arbitrage pricing. An upside to this story exists for investors using structured securities. Because the no-arbitrage pricing mechanism often overrides supply–demand pricing, sometimes either call or put options are relatively cheap for the amount of expected volatility, another very practical implication. For instance, when a one-sided event could push up a stock price significantly if it happens, but won’t affect the price as much on the downside if it doesn’t happen, then call options might be priced high to account for upside move that could happen. In effect, you could imagine that the upside volatility should be higher than the downside volatility and that the high price of the call option would not be reflected in high-priced put options. That is a reasonable argument for why the price of puts and calls should become disconnected, but

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in reality, no-arbitrage pricing normally overrides fundamental pricing. In those situations, you can sell put options for a higher price than justified. This plays out in specific cases and in turbulent markets in many ways, and is a direct result of put-call parity.

Fiduciary Call and Protective Put In the last chapter, fiduciary calls and protective puts were shown to have the same investment profile. This is another illustration of put–call parity arrived at by rearranging the parity expression as follows: Call + K = Stock + Put The practical application of this expression is that you can see the requirement to hold cash along with the call option in order to make the two positions equivalent.

An Accounting Illustration The following table compares a covered call to a “short put plus cash.” Assume that you have two accounts, both with $10,000. In account 1, you purchase 100 shares of XYZ at $100 and sell 100 call options expiring in one month for $3 per share. In account 2, you sell one naked short put contract for the same $3 and hold $10,000 in cash. The strike price on both options is $100. The following table shows how the accounts will appear at option expiration (Time 1) at three different stock prices ($90, $100, and $110): Time 0

$90

$100

$110

$300

$300

$300

$300

$10,000

$9,000

$10,000

$11,000

–$300

$0

$0

–$1,000

$10,000

$9,300

$10,300

$10,300

Account 1 Cash Stock value Option value Total account 1

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Time 0

$90

$100

$110

$10,300

$10,300

$10,300

$10,300

$0

$0

$0

$0

–$300

–$1,000

$0

$0

$10,000

$9,300

$10,300

$10,300

Account 2 Cash Stock value Option value Total account 2

Both account 1 and account 2 have the same value at each of the three stock prices in the example. In fact, this statement is generally true. These two accounts will have the same value at all stock prices. Because they act the same, they are, by definition, equivalent. For more information on put–call parity, see these resources: • www.theoptionsguide.com/understanding-put-call-parity.aspx • www.khanacademy.org/finance-economics/core-finance/v/ put-call-parity

Iron Condor Investment Profile Iron condors are popular options strategies. They involve selling out-of-the-money call and put options and then buying further outof-the-money call and put options. A condor has a shape resembling wings, hence the name. Figure 11.2 is an example of a condor. For the condor, the switch in Cell J5 is set to N. There is no stock component in this structure. Most traders using this strategy professionally do not think of themselves as investors. They are unabashed traders who know exactly what they want from this strategy: pure theta. As a rule, they do not believe in either technical or fundamental analysis, although some use these as entry and exit indicators. A condor trader banks on selling implied volatility and hoping that, on average, actual realized volatility will be lower. If the stock price stays within a certain trading range until or near expiration, the trader makes money. It is an interesting perspective, especially because it is consistent with the research and findings of the Efficient Market Hypothesis, indicating that technical and fundamental analysis are not useful in predicting stock prices.

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Figure 11.2 Iron condor investment profile

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In the figure, the condor consists of the following: • Ten contracts of a long put option with a strike of $80. This option was bought for $3.53 a share, for a total cost of $3,534.39. • Ten contracts of a short put option with strike $90. This option was sold for $7.01 per share, for a total cost of $7,012.88. • Ten contracts of a short call option with strike $110. This option was sold for $8.14 per share, for a total cost of $8,141.01. • Ten contracts of a long call option with strike $120. This option was bought for $5.44, for a total cost of $5,440.56. The net proceeds from the condor was $6,178.94. The maximum gross loss occurs at the tails, anything below $80 or above $120, and is fixed at $10,000. If the stock price at expiration is $70, the two call options expire worthless, the short put loses $20,000 ($90 − $70 × 1,000), and the long call pays $10,000. The difference is a $10,000 loss. But the proceeds were $6,178.94, so the maximum net loss is the difference, or $3,821.06. With a potential return of $6,178.94 and capital at risk of $3,821.06, the potential return is very high, at around 162%. There is a 41% chance of a gain and only a 29% chance of the maximum payoff of $6,178.94. Still, this is an attractive return with low correlation to the overall market. Because this is not an investment vehicle, the crossover or average theta formulas are not included here. One rule with condors is to keep the risk at manageable levels. In the example, a risk level was picked consistent with a $10,000 capital allocation. The capital at risk is close to the 5% VaR on the underlying stock. It is tempting to leverage these positions to much higher levels; just make sure you can absorb the maximum loss if you use leverage. For an excellent resource on Iron Condors, see Michael Benklifa’s Profiting with Iron Condors, FT Press.

Sources of Investment Return The iron condor is an example of a strategy targeting a particular source of investment return. In general, the sources of investment return are limited. You can earn a theoretical risk-free rate from short

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term Treasury securities. (Even these are not really risk-free because of currency devaluation and inflation.) Everything else involves risk. If you want more return in the fixed-income marketplace, you take credit risk. If you want more return in equities, you take equity risk. This is how the majority of institutional money and retail 401(k) capital is invested, with the intention of making money through interest payments on fixed income and capturing risk premiums available from diversified credit risk and diversified equity risk. Also referred to as beta investing, it is a classical long-term investor point of view, participating in the market through cyclical ups and downs. Then there is directional trading. Most traders make bets on the price of a security with long positions if they think it is going up and short positions if they think it is going down. Rather than rely on asset class returns over the long term, directional trading normally relies on technical analysis, fundamental analysis, or statistical arbitrage analysis to get an edge on the market. Sometimes this is called alpha investing, and the source of return it targets is capital gains. Those sources—risk-free returns, risk premiums (in the form of increased interest, dividends, and/or capital gains), and directional capital gains—account for almost all investment returns, at least within the traditional space. Options add a new source of investment return: theta, or time decay. Iron condors are an example of targeting this source of return. Iron condors target theta apart from risk premiums (beta) or directional trading (alpha). On a theoretical level, one question is whether it is possible to combine elements of different styles of investing to create new instruments or new strategies. That is, can you combine beta, alpha, and theta investing into a new structure? The challenge is to mix modern portfolio theory, with its emphasis on risk premium capture, with quantitative investing, with its emphasis on strict risk management in a way that takes advantage of the strength of both. The next section looks at a synthetic annuity, an example of an architecture that attempts to balance the two styles.

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Synthetic Annuity (SynA) Investment Profile A SynA is similar to a covered call position, with three distinctions. First, the short call options have staggered strike prices instead of one strike. Second, a portion of the proceeds from selling call options is used to buy one or more put options. Third, SynAs are designed to be managed, as opposed to many structured notes that are designed to provide a predefined payout regardless of changes in the price of the underlying security or market conditions. For example, if you establish a covered call at the beginning of the year and do nothing until option expiration date, it is an unmanaged security, where the profile is known in advance and accepted. If the market begins to drop rapidly, a covered call protects an investor only up to the amount of the call premium. Any price declines below that level affects the position dollar for dollar. This was a contentious point for many investors who bought structured notes before the 2008 crisis. They did not fully understand how these instruments would behave in extreme turbulence. Instead of owning a safe note, what they really owned was an instrument similar to a covered call, with a downside that was much greater than they realized. A SynA recognizes that risk is fluid and incorporates a set of management rules designed to maintain risk levels within prespecified limits. In this chapter, the SynA profile is the one that exists when it is first set up. In later chapters, we look at management tools and decision metrics for making real-time adjustments. The idea behind SynAs is not to create complex or unmanageable securities. In fact, it is just the opposite: It is to give more flexibility and choice, within a framework of quantitative discipline. The best way to think of SynAs is as alternatives to holding the underlying security alone. By adding an options overlay to the security, you have more choices in managing the position, not just buy– sell–hold. You can adjust risk exposures, buy priniciple protection, or reduce cost basis based on your risk appetite and market conditions.

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A SynA is a long-term alternative to holding a stock. The underlying stock is not intended to be sold unless a fundamental event would make you sell the stock otherwise. This means that options are rolled out over time instead of allowed to expire or execute. If an option is in-the-money on expiration date, it should be bought to close, and another option with a later expiration date should be sold in its place. Options provide the management framework and income-generating engine. They function as strategic elements of the position, not just as short-term trading instruments.

Modifying the Spreadsheet Because a SynA contains multiple short call options, it is necessary to modify the spreadsheet layout. Before discussing this specific adjustment, I would like to make some general comments about the spreadsheet. First, the spreadsheet was designed to be modified. The basic functionality is contained in just a few columns of the Calc Engine and Profit Calculator. The rest of the spreadsheet (the display page, the section on standard deviation markers and VaR, and the section that calculates the crossover point) just pulls this information together in different ways. Hopefully, as you get familiar with the layout, you will be comfortable modifying the spreadsheet to highlight the information that interests you most. At this point, the layout of the spreadsheet is Columns A through J

Display Screen

Column K

Blank

Columns L through R

Calc Engine

Column S

Blank

Columns T through Z

Profit Calculator and Black-Scholes Add-In

Column AA

Blank

Columns AB through AH

Optional: StdDev / VaR Markers

Column AI

Blank

Columns AJ through AP

Optional: Crossover and Probability

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As you work with the spreadsheet, keep in mind that the model’s basic purpose is to compare two random variables. The two random variables are the Stock-Only Profit and the Structure Profit. The values of these random variables are shown in the Profit Calculation module (Columns U and Z, respectively). The probabilities of these values, which are the same for both random variables, are shown in the Calc Engine (Column Q). These three items are the only required source data for the chart and most of the spreadsheet calculations. The standard deviation markers and related VaR measures, and various probability summaries are all derived from the random variables. In the next section, we look at how to expand the Profit Calculator section to include more option columns. That means the column references will change for the Profit Calculator and the sections that follow it. That is okay, as long as the two primary output columns— the Stock-Only profit and the Structure profit—are still referenced properly. When adding columns, just make sure that the Structure profit column, which is the sum of the stock and options columns, spans the correct range when you are finished. Excel will automatically adjust column references, but be careful when you copy and paste close to one of the end columns to make sure all the option columns are included in the summation for the Structure profit.

Expanding the Spreadsheet for Multiple Options SynAs normally include short call options with different strike prices. But the model has only one column for short call options, so we need to add more columns to the Profit Calculator section. In general, you can add as many columns as you like by copying and pasting whichever of the option types you need more of to build the structure you want to model. Figure 11.3 is an example of an expanded Profit Calculator with three short call options and one long put option. In this configuration, there are two additional columns. For display here, the columns for the short put and long call options are hidden, but not deleted. In my default spreadsheet, there are three columns for each option type. This is usually enough to handle most setups, although you can add as many columns as you want.

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Figure 11.3 Profit Calculator with three short call options

One nice feature of the spreadsheet that we have been using up to this point is that the input was all done on the display page. The display page has four columns to input option data, one for each option type. With only one of each option type, the number and strike prices can be entered on the display page and fed into the Profit Calculator. But now, with multiple occurrences of option types, it is necessary to scroll over and enter the number and strike prices of the options directly into the heading of the Profit Calculator. The display page can be left blank where this information was previously entered. Or, if you like, you can change the display page to reflect the total number and average strike prices, summarizing what is entered in the Profit Calculator. Because Rows 7 and 8 of the display page option section reference Rows 157 and 158 from the Black-Scholes add-in, it doesn’t affect any calculations.

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After the adjustments, the new column references should be: Columns A through J

Display Screen

Column K

Blank

Columns L through R

Calc Engine

Column S

Blank

Columns T through AB

Expanded Profit Calculator

Column AC

Blank

Columns AD through AJ

Optional: StdDev / VaR Markers

Column AK

Blank

Columns AL through AR

Optional: Crossover and Probability

Figure 11.4 shows the synthetic annuity described in Figure 11.3. In this case, the SynA has three short call options (with different strike prices) and one long put.

SynA Versus Stock: The Metrics Because the SynA at Time 0 is similar to a covered call position, most of the comments about covered calls apply here as well. You can see the tradeoffs in the graph. Most strategies involving covered calls have a large gap on the right side of the graph between the two profit curves. The gap represents the potential gains the SynA gives up. The difference between a SynA and a covered call is that if the stock price moves up past a certain amount, the position is adjusted so that the profile changes to capture more of the upside move. This is illustrated later in the section “Delta Adjustment.”

Yield The yield is 10.55%. This measure of yield is defined as the time value of the options (Cells B9 and C9) divided by the net investment (Cells J7 and J8). It is the amount of return based solely on option time decay, assuming that the stock price at expiration is the same as the price at Time 0. The yield is less than the yield on a covered call mainly because some of the premium collected from selling the call options is used to buy a put option. With the put option, the time decay works against you. Instead of getting paid for time, you are paying for time.

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Figure 11.4 SynA investment profile

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Delta Delta is $118.58, meaning that if the stock price goes up $1, the value of the structure increases by $118.58. This is compared to the stock-only position, which increases in value by $300 for a $1 price increase. On the other hand, if the stock price goes down by $1, the structure loses only about $118.58, whereas the stock position loses $300. In general, if the delta for the structure is less than the stock delta, the structure is less volatile.

Probability of Gain The results here are similar to those in the covered call. The probability of gain increases. The graph shows that the increase in probability is represented by the shift of the profit curve up and to the left for the SynA. The SynA crosses the x-axis further to the left. The crossover can be read from the Profit Calculator, or you can divide the net investment of $27,267 by 300 shares to get $90.89 as the breakeven stock price. This means that the probabilities associated with the points between $100 and $90.89 are added to the probability of a stock gain, moving it up from 46% to 60%.

VaR 5% The SynA VaR is $7,622, compared to a stock VaR of $12,750. This is due to the net cost basis reduction from the option proceeds and the purchase of the put option. You can see the effect of the put option in the graph. Starting at the strike price of the put ($75), the graph flattens a little for any further price declines. The effect of the put is that only 200 of the 300 shares continue to lose money below $75. One of the objectives of a SynA is to provide some principle protection as part of the setup and then use a set of rules to finance more principle protection over time so that VaR gradually decreases to zero.

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Delta and Theta Adjustments I have included chapters from the book Profiting with Synthetic Annuities on practical trading issues as supplemental material in this book (Chapters 14 and 15). Those chapters go into more detail, but briefly, delta and theta are monitored and adjusted by making additional option transactions when needed. The first thing to know is how different options affect the direction of delta and theta. • Buying options, whether a call or put, decreases theta. • Selling options, whether a call or put, increases theta. • Buying a call or selling a put option increases delta. • Buying a put or selling a call option decreases delta. To give you an idea of how this works, let’s say that everything goes as planned, and after nine months, the stock price is $105 and volatility has decreased to 25% (it is not uncommon for volatility to decrease in an uptrending market). Figure 11.5 shows how the SynA looks. With the stock price at $105, the current profit on the stock position is $1,500 (Cell E9). The profit is the difference between the cash paid to purchase the stock, $30,000 (Cell E7), and the cash it would bring today, $31,500 (Cell E8). The display page formulas for Column E, Rows 7 and 8, are shown here: E7 = –E3 × E5, where E5 = $B4 E8 = –E3 × E6, where E6 = $C4 The minus sign indicates a cash outlay. These formulas can be copied over to Column I to include the option position profits. The formula for Column J is the sum of Columns E–I if the Include Stock switch is set to Y, or Columns F–I if the switch is set to N. The SynA profit after nine months is $2,674.30, more than $1,000 higher than the stock position. Even though two of the three call options were in-the-money (the $95 strike and the $100 strike) and did not participate in the move from $100 to $105, time decay of the options outweighed this effect. Going forward, however, any further price increases will be limited. Currently, delta is $99.

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Figure 11.5 SynA investment profile after 9 Months

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On the other hand, theta is stronger. At Time 0, average theta was 10.5%. Now, if the position were held for the last three months, the current amount of time decay would produce average theta of 14.30%. This is the current time value divided by the net investment. Option time decay in general increases as the time until expiration decreases. You will see this pattern clearly in Chapter 13, when we look at the Greeks. The probability of a gain on both the stock and SynA has increased. This makes sense because the stock price has moved up and the probability distribution has narrowed as the option term shortened. The probability of a gain on the SynA increased to 90%. Related to the change in the probability distribution is the change in VaR, where the stock VaR decreased to $4,500 and the SynA VaR was only $1,122, or roughly a quarter of the stock VaR, representing the more conservative nature of the SynA.

Delta Adjustment If you hold the SynA at this point, you should feel comfortable. You have a bigger gain than if you had bought the stock, you are earning a theoretical yield of more than 14%, and the chances of a loss are minimal. But two situations might affect your attitude: a big move up or down in the stock price. If the stock price moves up enough, you might feel left out and think that you should have just bought the stock—without selling any options—because the short call options will have kept you out of most of the upward move. On the other hand, if the stock price moves down significantly, you might experience losses higher than you wanted. You might begin to question whether you should have sold the stock when it was up instead of taking a longer-term view. Of course, the results are still better than if you simply bought the stock and held it because the SynA losses are smaller, and you do have some downside protection. In terms of risk management, SynAs have rules, including setting a maximum loss and doing something to reduce cost basis when that loss is exceeded. For example, if you want to limit your loss to 10% of principle, the rule requires reducing cost basis by selling more call

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options when the 10% loss is reached. And the put options are already in place to help limit further losses. In terms of upside profit management, SynAs also have rules for price spikes. Missing out on upside moves is something anyone using a covered call strategy has to think about. So how does a SynA work when prices go up? Let’s say that a product announcement or a good earnings report immediately pushes the stock price up $10, to $115. Now the SynA looks like Figure 11.6. At this point, you might begin to feel left out. Even though the SynA gain increased to $3,680, the stock gain is now better, at $4,500. Delta, the indicator of how much the SynA value increases with the next $1 in stock price increase, is low, at $38.50. In other words, if the stock price goes to $116, the stock position will increase in value by $300, whereas the SynA will go up only about 15% of that. Also, now that the short calls have moved further in-the-money, the yield goes down to 4.54%. If this stock keeps going up, you will not get much out of it, in terms of either delta-related capital gains or theta-related yield gains. So have you missed the boat? Not necessarily. One of the nice things about options is the capability to make retroactive adjustments. Let’s say you think that the fundamental news that drove the price up $10 is significant to the long-term value of the company, and you think there might be further upside. Then one tactic is to sell put options into this price strength. Another tactic is to buy-to-close (buy back) the most in-the-money short call, the $95 strike short call. The reason for buying this option is to remove its drag on any further price increases. At this point, the $95 strike has a delta of almost –$100, meaning that it is moving dollar for dollar against the position and had almost no theta. Let’s assume you decided to sell a put. In particular, what effect would selling a $115 short put have? First, you would get $5.69 for the put under the new option term and assumptions, a three-month term, and 25% volatility. This would make the potential profit about the same for the stock and SynA. Potential profit refers to the profit you would make if the stock price stays the same until the option expiration. The transaction would also increase both delta and theta.

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Figure 11.6 SynA investment profile in Up Market

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Figure 11.7 is the profile after selling the put option. Several items that changed are highlighted, including the crossover, which moved from $112 to $120, meaning that the stock performed better at prices above $120. The probability that the stock would outperform went down from 58% to 37%. In the short put column, the premium of $569.17 is included. The premium is all time value, which increased the total time value in the Now column and increased average theta from 4.54% to 13.29%. SynA delta also increased from $38.52 to $86.05. The put added $47.53 to delta (Cell G10). All these changes were in the right direction, giving more exposure to the stock price, more theta, and more potential profit. As with almost any option transaction, there are tradeoffs. In this case, some of the downside protection was sacrificed. This is evident in the change in the shape of the profit curve. Before this, the left tail was flatter. Now by selling the put, the tail changed shape so that it became very similar to the stock loss line. If the stock were to fall below $75, the original configuration would have been better. Technical note: When you enter new options transactions such as this one that occurred after Time 0, be careful to set Cell G5 equal to Cell G6. Cell G5 normally uses the information in the Time 0 input Column B to price the option. That is, it assumes that there are 365 days left, the stock price is $100, and so on. But in this example, the put option is being sold with only 90 days left and a stock price of $115. So you need to override what it naturally does to make sure the price in Cell G5 is equal to $5.69, to reflect the current information. In the chapters on SynA metrics and the covered SynA, I explain how to create a template to track performance data over longer periods. Still, selling the put (after the fact) is an effective way to recapture some of the gain initially given up by selling the call options. It also illustrates some of the power of the method to generate high levels of theta. In fact, by selling 300 put options, you can push theta over 30% and still maintain delta below $200. This rate of return given the level of risk is hard to achieve without options.

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Figure 11.7 SynA investment profile with Delta Adjustment

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This technique of pushing against a price move is tempting, and it quickly leverages theta, but whenever you do this, consider using put spreads instead of naked puts. Sometimes a price spike can be attributed to anticipation of something not widely known. If it fails to materialize, the price can reverse quickly. In terms of emotional indifference, pushing against a price move by selling options is effective. With only minor adjustments, you can maintain indifference in light of high theta. You might not care if the stock price moves up or down by $5 because you are comfortable with the outcome. If it goes up, you capture your original target gains (capital gains plus theta) plus more for the put option. If it goes down, it centers the price better in terms of the call option strikes, which is good for theta. The newly sold put option might still be out-of-themoney, creating even more theta. And the put option doesn’t begin to lose money until it falls more than the premium. Another consideration often comes into play when a stock moves to the upside. If your portfolio is correlated to the broader market, and the upside move in this stock was because of market movements, then the other positions in your portfolio that might not have calls written against them will gain as well. Often this is the case with money in 401(k) and other qualified plans, as well as real estate and other assets that do well when the market is in an uptrend.

Adding a Customized Utility Function In Chapter 7, “Investment Profiles,” an example of an investment profile reflecting a utility function was presented. A utility function transforms the actual “dollar” gain or loss into an “emotional” measure of gain or loss. This transformation is based on the findings of behavioral finance that indicate most investors dislike large losses more than they value large gains. If you would like to add a customized utility function to the model, you can do it by inserting a section immediately after the Profit Calculator. The format for this section includes columns to import the Stock-Only profit and the Structure profit, columns to define the utility function, and the results of the transformation.

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Figure 11.8 shows a sample layout.

Figure 11.8 Utility Function Template

The utility function in this example uses a simple linear rule to measure the effect of large losses. It is 0.01 times the distance from the current stock price as the multiplier. You can use whatever rule or formula you like, or you can just type in the numbers corresponding to a pattern you would like to duplicate. To be consistent between the utility of losses for stock-only and the structures, make sure that the utility is the same for the same dollar level of loss. If you use a formula, it should match. If you create the stock-only utilities in some other way, you can be consistent by defining the structure utilities by lookup, as illustrated in the example. The Black-Scholes formula, which is reproduced in this book without the utility function, is consistent with a risk-averse investor, even though the probabilities are described as risk-neutral. Overriding

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the actual gains and losses with a customized utility function takes the calculation beyond the Black-Scholes boundaries; however, over shorter periods of time, understanding how an investor might react to different levels of gains and losses is often more important than staying within technical boundaries. For a more detailed discussion on risk-neutral pricing, see McDonald.1 If you use a utility function, make sure all spreadsheet references to the two output columns in the Profit Calculator (the Stock-Only and Structure profit columns) are changed to the two corresponding columns from the Utility Function module. In other words, wherever there is a reference to the stock profit in Column U, change it to reference the stock profit utility, currently shown in Column AE. Similarly, change any reference to the structure profit to the structure profit utility. The effect is to replace the actual dollar amount of gains and losses to the “perceived” value of those gains and losses. If you know in advance that you want to use a utility function, it is easier to include this module immediately after the Profit Calculator (in Columns AB to AH) and then complete the optional sections by referring to the output from this module. In this configuration, you can include a switch to turn this module on or off. In the off mode, the utility function is defined as 1.0 in all rows, so the stock-only and structure utilities are the same as the profits. In the on mode, you can define the utility factors any way you want and they will be reflected in the chart and other calculations automatically.

Endnotes 1. McDonald, Robert L. Derivatives Markets, 2nd Ed. 2006, AddisonWesley, Pearson Education.

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12 Understanding Price Changes Option prices are in constant movement. So are the values of structured securities that contain options. This chapter looks at the components of option value and the factors that drive changes in those values. The chapter begins with a hypothetical investment, explains the role of implied volatility in constructing an investment profile, and then tracks option prices in terms of intrinsic value, time value, and the elements of extrinsic value. The Greeks are covered in more detail in the next chapter.

Investing in XYZ Imagine you are watching CNBC’s Mad Money. Jim Cramer just interviewed the CEO of Company XYZ. You already know the company and like what you heard in the interview. You think now is a good time to buy. You have a choice of buying the stock or buying an option on the stock. The option you are considering has a one-year term and a strike price of $95. The current stock price is $100. The company doesn’t pay a dividend. Assuming a zero interest rate and 30% volatility, the model results in Figure 12.1. The option price is $14.29 (Cell I5). The model uses the BlackScholes formula and the pricing assumptions on the display screen. Then assume that you look at a real-time quote from your trading platform, and the quote is $13.50. It is cheap compared to the BlackScholes value, so you look more closely at the pricing assumptions.

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Figure 12.1 Investment profile using assumed volatility

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The option term, the stock price, the dividend rate, and the strike price of the option are all known. The only two pricing variables that could be different are the risk-free rate and volatility. For purposes of this example, assume that the risk-free rate is zero and not a source of difference between the two prices. That leaves only volatility.

Using Implied Volatility to Estimate Future Stock Price Distributions When you price an option using the Black-Scholes formula, you have to specify the volatility assumption. It might be an estimate of volatility based on historical averages or some other estimate that reflects your outlook. The estimate is a pricing assumption. In this case, an assumption of 30% volatility translates into an option price of $14.29. The graph on the display page also reflects the 30% assumption. Whatever volatility you enter in Cell C8 is used to construct the probability distribution and to price the option. The question now is whether you should adjust your estimate of volatility to reflect the option price of $13.50. That depends on your point of view. Some traders put a lot of effort into developing a volatility estimate. If they have the conviction that 30% is the right number, they might use it for analysis, regardless of the difference in price. They might even decide to buy the option for $13.50 because it is cheap relative to their estimated price of $14.29. However, when I build structured securities, I am more interested in market information than I am in figuring out whether a particular option is cheap or expensive. That is why, at least as a starting point, I want the investment profile to reflect volatility implied in the option price. So instead of estimating volatility, the investment profile uses implied volatility. Implied volatility is the volatility currently priced into exchange-traded options. Using implied volatility is one way of listening to the market as it tells you what is currently being priced and what that means about possible future stock prices. Of course, you can adjust the volatility to reflect a particular view, but by using implied volatility as a starting point, you will be more consistent with market costs to hedge the position.

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How do you know the implied volatility? If you have access to a trading platform or real-time market data that includes implied volatility, it is provided for you. You can also calculate it yourself by trial and error. (There is no easy way to do it by reversing the BlackScholes formula.) One way to do this is with Goal Seek, a standard function in Excel. Goal Seek uses an iterative trial-and-error process that zeros in on the right answer. To do this, go to the Black-Scholes add-in or other option pricing page. Click Tools in the Excel menu and select Goal Seek. A box like the one shown in Figure 12.2 appears.

Figure 12.2 Calculating implied volatility

In the Set Cell field, enter the cell location for the option price. In the To Value field, enter the option price you are looking for; in the By Changing Cell field, enter the cell where volatility is entered. After clicking OK, you see that the answer from Goal Seek is 27.905%, which is the volatility that produces an option price of $13.50. This is the implied volatility that should be used to construct the investment profile so that the probability distribution is consistent with actual option pricing on the exchange. Figure 12.3 shows the new starting point.

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Figure 12.3 Investment profile at Time 0

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In this implied volatility version, yield is slightly less negative because the option costs less. Delta, the probability of a gain, and VaR are roughly the same as before. Even though the results are similar to those using 30% assumed volatility, this is a more logical starting point for our analysis.

Intrinsic Value, Time Value, and Extrinsic Value Several terms, including intrinsic value, time value, in-the-money, time decay, time value, time value premium, and extrinsic value, describe components of option value or explain changes in option values. The following definitions are from the Chicago Board Options Exchange website (www.CBOE.com): • Intrinsic value: The value of an option if it were to expire immediately, with the underlying stock at its current price; the amount by which an option is in-the-money. For call options, this is the difference between the stock price and the striking price, if that difference is a positive number, or zero otherwise. For put options, it is the difference between the strike price and the stock price, if that difference is positive, and zero otherwise. • In-the-money: A term describing any option that has intrinsic value. A call option is in-the-money if the underlying security is higher than the strike price of the call. A put option is in-themoney if the security is below the strike price. • Time decay: A term used to describe how the theoretical value of an option “erodes” or reduces with the passage of time. Time decay is especially quantified by theta. • Time value: The portion of the option premium that is attributable to the amount of time remaining until the expiration of the option contract. Time value is whatever value the option has in addition to its intrinsic value. • Time value premium: The amount by which an option’s total premium exceeds its intrinsic value. Although not defined on the CBOE website, extrinsic value is a term generally used to indicate sources of change in value that are not “predictable.”

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For our example (dollars are per share): • • • • •

The option’s intrinsic value is $5.00. The option is “in-the-money.” The option will experience time decay. The option’s time value is $8.50. The option’s time value premium is $8.50.

Because the option contract in the model covers 100 shares, the model displays these per share values times 100.

Example of Two Outcomes Now let’s run the clock forward six months and consider two different outcomes in Figures 12.4 and 12.5. In the first scenario, the option price on the exchange is $10.98; in the second, the option price is $10.63. The two option prices are close, but the situations and analysis are different. The following summarizes the stock prices, the option prices, and implied volatilities with six months remaining on the option term: Figure 12.4 Stock price Option price Implied volatility

Figure 12.5

$105.00

$90.00

$10.98

$10.63

15%

50%

The first scenario is consistent with a calm, up-trending market, where prices increase and volatility decreases to reflect investor optimism. To visualize this market condition, look at the investment profile in Figure 12.4—particularly the center point and the relative spreads of the probability distributions. Compared to Figure 12.3 six months earlier, Figure 12.4 has a narrower profile (reflecting the lower volatility), and the center point of the distribution has moved to the right (reflecting the higher stock price).

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Figure 12.4 Investment profile after six months in calm market

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In terms of metrics, the yield is less negative because there is less time value; delta is higher because the option is deeper in-the-money. The more an option is in-the-money, the higher the delta. This makes sense because the option begins to approach a dollar-for-dollar move with the stock as the stock price moves above the call’s strike price. Now look at the second case, where the stock price has moved down and volatility has moved up. These conditions are consistent with a more turbulent market. Even though the option price is close to the first scenario, the profiles look nothing alike. Neither do the metrics. Now the option is almost all time value because the stock price moved from above to below the strike price. At this point, the option has no intrinsic value and is $5 out-of-the-money. Here is a summary of the two scenarios: Figure 12.4

Figure 12.5

$105.00

$90.00

$10.98

$10.63

15%

50%

Yield

–1.96%

–21.3%

Delta

$84.05

$50.95

35%

23%

$1,250

$5,250

Stock price Option price Implied volatility

Probability of gain VaR 5%

The question now is, how do you analyze these two possible outcomes in terms of intrinsic value and time value? Here is a comparison of intrinsic and time value for the two scenarios: Scenario 1: Time 0

Time .5

Change

$100.00

$105.00

+$5.00

$13.50

$10.98

–$2.52

Intrinsic value

$5.00

$10.00

+$5.00

Time value

$8.50

$0.98

–$7.52

Stock price Option price

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Figure 12.5 Investment profile after six months in turbulent market

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Scenario 2:

Stock price Option price

Time 0

Time .5

Change

$100.00

$90.00

+$5.00

$13.50

$10.63

–$2.87

Intrinsic value

$5.00

$0.00

–$5.00

Time value

$8.50

$10.98

+$2.48

These numbers might be accurate, but how useful are they in explaining what happened? For instance, does knowing that the intrinsic value went up by $5 in the first case and down by $5 in the second case really account for the $15 stock price difference ($105 vs. $90)? And what about time value? In the first case, it declined by $7.52; in the second case, it increased by $2.48. Can time value really increase? That seems to violate the basic notion that time value should decay. Some experts make a distinction between time value and another metric, extrinsic value, explaining that time decay is a predictable process and should not be confused with or combined with the effects of other pricing variables, such as volatility. For example, Michael Thomsett writes: Time value by itself is quite predictable and, if it could be isolated, would be easily predicted over the course of time. Simply put, time value tends to change very little with many months to go, but as expiration nears, the rate of decline in time value accelerates and ends up at zero on the day of expiration... Time/volatility value is often described as a single version of “time value premium.” If these two elements are separated, option analysis is much more logical.1 And: The portion attributed to volatility might be accurately named “extrinsic value.” This is the portion of an option’s OTM premium beyond pure time value. Extrinsic value can be tracked and estimated based on a comparison between option premium trends and stock volatility.

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Looking at value in this way, time value increasing by $2.48 doesn’t make sense.

Attribution: Explaining Why the Option Price Changed It helps to make a distinction here between the components of option value “at a point in time” and the attribution of changes in option value occurring “between two points in time.” There is nothing wrong with using the terms intrinsic value and time value to talk about the components of an option “at a point in time.” But those terms don’t tell the whole story when looking at changes in option values across time. Because option prices are determined by six pricing variables, a change in any of those variables affects the price. Of the six, two are fixed after a particular option is chosen: the strike price and the dividend rate. In addition, although the risk-free interest rate can change, it is not a significant contributor to changes in option values, particularly now, with very low market interest rates in effect. So let’s ignore it. That leaves three pricing variables: time, stock price, and volatility. The analysis of changes in option prices should include the effects of these three variables. Look at Figure 12.6. It lays out a procedure for isolating and measuring the effects of the variables. It is the BlackScholes formula add-in, modified by copying the formula into three additional columns. The pricing variables in Column C are the conditions at Time 0 (one-year term, $100 stock price, and 27.905% volatility). The last column shows the conditions after six months (six months remaining, $105 stock price, and 15% volatility).

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Figure 12.6 Option price changes by source: calm market

The two intermediate columns help to isolate the effects of changes. In Column D, the only variable that changes from Column C is time. By comparing the option price in Column D—which assumes both the stock price and the volatility are the same as Time 0 conditions—to Column C, you see the effect of “time alone.” In this case, with the other variables held constant, the change in the option price due to time is the difference in the option price at Time 0 ($13.50) and the option price at Time .5 ($10.42), or a decrease of $3.08. Similarly, the effect of the change in the stock price is the difference between Column E and Column D, or an increase of $3.43. Volatility accounts for a decrease of $2.86.

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• Point in time summary: Rows 22 and 23 show the straightforward breakdown of intrinsic and time value across the columns. These numbers are correct; they indicate how much of an option value is in-the-money and how much is not in-themoney. That is all they are intended for. Not that these numbers are not useful—in fact, the model uses time value quite a bit, to give you an idea at “a point in time” how much average yield is possible, assuming that stock prices and volatility remain constant. It is one of the dashboard displays. • Sources of change analysis: Rows 25–30 show the real story of what happened to the option price and why it changed. The arrows indicate how each component is related to the stepthrough pricing. A natural question at this point is whether the two sections can be related by formula. Can you derive Rows 25–30 on the basis of Rows 22–23? That depends. If all you have is the first and last columns, the answer is no. If you have the four columns, it can be done, but it is more confusing than it is worth. To do it, you have to track the excess of the stock price drop over the reduction in intrinsic value. For example, in the second scenario, the drop in stock price is $10. Because intrinsic value started at $5, it can go only to $0. So there is an additional drop in price of $5 below intrinsic value.

The Probability Distribution It is also possible to visualize the components of change. By running the model iteratively, you can step through the changes in the probability distribution attributable to a particular pricing variable change. For example, if you want to see the effect of time, run the model and change only the time remaining on the option term; keep the stock price and volatility constant. See Figure 12.7. This view enables you to see not only in hindsight why something happened, but also in advance what to expect. For example, when this option was purchased at Time 0, this view will forecast the outcome if nothing changes other than time. Notice that the Gain (Loss) in Cell J9 is a loss of $308.09. Also notice that the probability of a gain in the text boxes dropped from 43.6% to 30.0%. In addition, the rate of time decay accelerated from –8.50% to –10.84%.

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Figure 12.7 Change in time value: constant price and volatility

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You can continue this process, stepping further out in time. Figure 12.8 is a snapshot with only one month left on the option term. (If you put together a series of snapshots, it creates what I call time lapse photography, a kind of options movie you can create with macros to step through a predefined scenario.)

The Second Scenario: Turbulent Market Figure 12.9 illustrates the sources of price change for the second scenario, in which the stock price drops to $90 and the volatility increases to 50%. When the stock price falls significantly below the strike price, the option quickly becomes “disconnected” from the stock, and delta falls quickly as well. In this example, the stock price was only $5 below the strike price. If the stock price were $80, delta would be $37; at $70, delta would be $25. Time decay in this scenario is the same as before, a decrease of $3.08. This is a reasonable answer. It moves in the right direction and is the same for both examples. As Thomsett pointed out, time value is predictable. Here is a summary of the changes in value: Figure 12.4

Figure 12.5

Time decay

–$3.08

–$3.08

Stock price

+$3.43

–$5.38

Implied volatility

–$2.86

+$5.59

Total

–$2.52

–$2.87

Time decay is the same, but the other two factors are very different, moving in opposite directions. The net effect is that the factors offset each other to a large extent, ending with similar option values.

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Figure 12.8 Change in value: looking forward 11 months

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Figure 12.9 Option price changes by source: turbulent market

The Gain (Loss) Analysis Perspective Why would anyone buy a long call option if they expected the stock price to stay the same? The trade loses money as the time value decays to nothing. You buy a call option if you think the stock is going up. And when you have an opinion about how much you think it is going up, it makes sense to add another step to the analysis. Instead of comparing the total change from beginning of the period to the end of the period, first calculate an expected value at the end of the period, based on what you know about the predictable elements such as time decay and your assumptions about price and volatility movements. Then you can compare what you “expected” to happen with what actually does happen. In many contexts in finance, this is a more

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meaningful way of analyzing results. It encourages a certain discipline with regard to your investment thesis. When buying and selling stocks, the primary consideration is price direction. But when buying and selling options, you need to expand the narrative to include value changes related to price and volatility. In terms of format changes to Figure 12.9, the first column would be Beginning of Period, the second Expected End of Period, and the last column Actual End of Period. The intermediate columns reconcile the difference between expected and actual by changing price and volatility in steps. In the expected column, you can incorporate your views on the correlation.

Market Direction/Volatility Correlation Should you expect higher volatilities when markets are falling and lower volatilities when markets are rising? Not always, but these factors have well-established correlations, so as a benchmark, yes, you should. This often works against long call option holders and in favor of long call option sellers. It is not an absolute rule, but as you think through your expected scenario, it is, on average, the right assumption.

Options Versus Structured Securities In this chapter, we looked at the changes in value of one option. But what about structured securities in general? You can use the same steps to analyze the sources of change for any structured security, whether it is an option, a combination of options, or stock plus options. The only difference is that you use the model instead of the Black-Scholes formula. Using the model, you can walk through the same sequence of steps discussed in this chapter. In the next chapter, we look at this with respect to the Greeks.

Endnote 1. Thomsett, Michael. The Options Trading Body of Knowledge. FT Press 2009.

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13 The Greeks The Greeks measure the sensitivity of derivative prices to changes in pricing variables such as time, the price of the underlying security, interest rates, and volatility. The name comes from the Greek letters used to represent them. Normally, the Greeks refer to options, although they can apply to any derivative. The concept also extends to any structured security that contains derivatives. Delta and theta, discussed throughout the book, are two of the option Greeks. In this chapter, these and other Greeks are presented more formally. Also, beginning with this material, we shift focus from attributing changes in value to understanding how to manage those changes. To illustrate the difference in contexts, think about the term investment vehicle as a car metaphor. The earlier chapters on the elements of structured securities can be viewed as designing the car’s performance and safety features (engine power, the suspension, and braking distance, for example). Then in Chapter 12, “Understanding Price Changes,” we looked at the factors influencing option values, primarily over the longer term. The discussion on the changes in value due to pricing assumptions was similar to planning a trip, estimating how long it would take and what might happen along the way. Now we focus on the shorter term. In the next three chapters, we look at how to drive the car. This is where the Greeks are useful. To take advantage of the increased flexibility of structured securities, it is important to have a good dashboard that tells you how fast you are going, how much gas is in the tank, and has a “check engine” light. The Greeks are part of that dashboard, providing real-time information to make tactical adjustments in risk–return profiles. The

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shorter-term nature of the Greeks make them valuable tools for measuring and managing portfolio positions.

The Option Greeks The option Greeks measure changes in option price for a specified change in one of the option pricing variables. For example, delta is the change in option price that results from a $1 increase in the stock price. Theta is the change in option price that results from time decay—specifically, the decrease in option time value that occurs in one day. When Greeks measure changes with respect to one of the pricing variables directly, they are known as first-order Greeks. There are also second-order Greeks, which measure changes with respect to another Greek. Gamma is an example of a second-order Greek. Gamma is the change in delta that results from a $1 increase in the stock price. The following are the most common Greeks: • Delta: First order, related to stock price • Theta: First order, related to time until expiration • Rho: First order, related to risk-free rate • Vega: First order, related to volatility • Gamma: Second order, related to delta Because option values change whenever one of the pricing variables changes, and the pricing variables can all be changing at once, how do you measure the effects of just one variable? The assumption is this: When measuring the effect of one variable on option price, it is assumed that the other variables are constant. In reality, it is difficult to strictly isolate one of the pricing variables from the others. Pricing variables are in constant change. Time is always moving. Stock prices move during trading hours and after hours. Volatility is, well, volatile. But to make these calculations, some assumption has to be made about what is going on with the other variables. As you consider the Greeks, keep in mind the assumption behind them:

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Each Greek measures the change in option price with respect to a small change in one variable, assuming the other variables are constant. That is why the Greeks are somewhat theoretical. Still, over small time intervals, they provide practical information for managing options positions. They are indispensable to market makers and other market participants who need to adjust risk exposures rapidly. By measuring the component risks separately, these investors can rebalance a portfolio of options positions to achieve a desired exposure. For example, delta hedging refers to the practice of adjusting the number of shares of stock held in the portfolio in order to offset the change in value of the options positions. As a reminder of a point made earlier in the book, when I use the term stock, I am really referring to any underlying security (including ETFs) that provide exposures to any number of assets and asset classes, including individual stocks, equity indexes, fixed income, interest rates, real estate, currencies, commodities, and other assets. So rather than being precise by using the term “underlying security,” I use stock to indicate the most common underlying security. However, I don’t want to give the impression that structured securities, or the Greeks, are restricted to equities. Although I use the term more loosely to describe changes in an option with respect to changes in stock price, the Greeks technically describe changes in any derivative with respect to its underlying security. The same principles apply, for example, to a fixed-income portfolio or a commodity portfolio.

Calculating Greeks: Formulas, Models, and Platforms There are two common ways to calculate option Greeks. The first is to use a formula, if one exists. An example is Black-Scholes, which can be solved for the Greeks and expressed as closed-form solutions. The second way to calculate option Greeks is to use a model or simulation tool. Because Black-Scholes prices European options

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under a fairly restrictive assumption set (including constant volatility over the option term), more accurate models, such as the binomial model, are often used to calculate Greeks. But with these models, there might not be a closed-form solution. In this case, you can obtain the Greeks by changing pricing variables and observing the change in the option price. As a practical matter, most portfolio management is done with trading platforms. The platforms calculate the Greeks for you, although depending on the platform, it might use the Black-Scholes formula as an approximation to more exact American option pricing. Option trading platforms are provided by most asset-management firms and from specialty brokers such as TradeStation, optionMONSTER, thinkorswim, Interactive Brokers, and optionsXpress. The platforms are powerful and constantly improving. One recently introduced a probability distribution similar to the one developed in this book. Chapters 14, “Managing Positions,” and 15, “More on Synthetic Annuities,” introduce the TradeStation platform and use it to illustrate a synthetic annuity.

Greeks As Mathematical Derivatives A mathematical derivative has a different meaning than a financial derivative. An option is an example of a financial derivative. This simply means the value of the option is derived from the value of another instrument. In math, a derivative has a very specific meaning. A mathematical derivative measures the instantaneous rate of change of one variable with respect to another variable. The Greeks are mathematical derivatives. Specifically, they are partial derivatives.

Partial Derivatives Say you have a function Y that is related to X, expressed generally as Y = f(X). A specific example might be: Y = X2 The derivative of Y with respect to X, denoted as dY/dX, measures the instantaneous rate of change in Y, given a specific value of X. For

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many formulas, the derivative can be solved and expressed as another formula. In this case, the derivative is: dY/dX = 2X If X = 0, the rate of change in Y at that specific point is 2X = 0. If X = 1, the rate of change in Y is 2. This illustrates a property of derivatives. The rate of change depends on the specific value of X. The Greeks are a little more complicated. That is because the value of an option depends on six pricing variables. Of the six variables, only four can change after a specific option is chosen. For example, if you buy an option with a strike price of $100 and the underlying stock does not pay a dividend, those two pricing variables are fixed and will not change during the option term (the company could announce a dividend, but let’s assume that doesn’t happen). That leaves four pricing variables that can change. The value of the option can and will change as (1) time (t) passes, (2) with changes in stock price (S), (3) as interest rates (r) change and (4) as volatility (X) changes. If the value of the option is V, then: V = f(t, S, r, X) When a function such as V depends on more than one variable, and you want to calculate a derivative, you have to be specific about what derivative means. For instance, if you want to know how V changes with respect to S, you need to specify exactly what is going on with t, r, and X. To make the solution manageable, the assumption is that the variables are constant. To make it clear that a derivative is being calculated under this assumption, the term partial derivative is used. It reminds you that the derivative is being calculated with respect to one variable—assuming that the other variables are constant. Delta, for instance, measures the rate of change of the option value V with respect to changes in the stock price S, with all other pricing variables constant. This assumption leads to restrictions on the interpretation of the derivative, but within narrow ranges (of time, stock price, and volatility), is a useful approximation. Under the Black-Scholes pricing assumptions, it is possible to derive formulas for the Greeks.

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Delta Delta is defined as the change in option value (V) that occurs when the stock price (S) increases by $1. Delta is the partial derivative of V with respect to S, written as follows: )=

yV yS

The solution for a call option is: e–qY+(d1) where q is the dividend yield, t is the time to maturity, and d1 =

ln(S / K) + (r – q + X2 /2)Y X™Y

This formula for delta is used in the model and contained in the Black-Scholes add-in at the bottom of the Profit Calculator. Call and put deltas are calculated in Rows 151 and 152, respectively (refer to Figure 9.5 in Chapter 9) and are also displayed in Row 10, Columns E–J of the display screen. If you look at the formulas for delta in the model, you will see that the formulas are not as complex as the previous expression. The reason is that the term d1 was saved as an intermediate step in the BlackScholes calculation. See, for example, Row 136 under the column for a short or long call. With d1 available, delta is simplified to the first line of the delta expression above.

Example Consider the call option used frequently throughout the book, a one-year option with a strike price of $100 on an underlying stock currently trading for $100 with 30% volatility and zero interest and dividends. The price of the option is $11.92. Delta, in this case, is 0.5596, or 55.96%, in Row 136 of the Profit Calculator. By definition, this is the delta of one share. This means that the option value is changing at the rate of 0.5596 when the stock

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price changes by $1. But remember, we have to be specific about the value of S and about the values of the other pricing variables. Here is almost the exact language: The option value is changing at the rate of 0.5596 as the stock price changes from $100 to $101, with the other pricing variables being held constant at t = 1 year, r = 0%, and X = 30%. I say almost because the instantaneous rate of change applies to only an infinitely small range around $100, not to the entire range between $100 and $101. Technically, the rate of change increases at a different rate as the stock price moves to $101.01. By the time the stock price reaches $101, delta is changing at a new, higher rate. But the approximation is close enough for most purposes. On the model display page, delta is scaled for the size of the option position. In Figure 13.1, delta shown in Cell I10 is equal to $55.96, or 100 times the per-share value.

Figure 13.1 Call option delta

In practical terms, how should you interpret this? The model compares two alternatives: a stock-only investment and a specified structured security. Here the structured security is one long call option contract covering 100 shares. If you buy 100 shares of stock and the stock price increases by $1, the position value increases in value by $100 (Cell E10). On the other hand, if you buy an option contract and the stock price increases by $1, delta tells you that the option position should increase in value by about $55.96. You can see how much the value of the option actually increases by changing the stock price in the model from $100 to $101, as in Figure 13.2.

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Figure 13.2 Call option delta at $101

The option price changes from $11.92 to $12.49. This creates a position gain of $56.62, as shown in Cells I9 and J9. This is close to the delta estimate of $55.96. The difference results from the fact that delta increases as the stock price increases. At a stock price of $101, delta is $57.27, as shown in Cells I10 and J10. It makes sense that the actual change in option price is somewhere between the delta rate of $55.96 that applies at a stock price of $100 and the delta rate of $57.27 that applies at a stock price of $101. The only time $55.96 is an exact number is when the stock price is exactly $100.

Deltas Are Additive In the model, delta for the structure in Cell J10 is the sum of the deltas in Columns E–I when the indicator in J5 is set to Y, or the sum of the deltas in Columns F–I when the indicator is set to N. This is because the delta of a combination of stocks and options is equal to the sum of the individual deltas. To avoid confusion, keep in mind that delta can be expressed in different ways. For the model, the deltas in Row 10 are expressed as dollar amounts and reflect the size of the stock or option position. In this case the position is 100 shares, so delta is $55.96, or 100 times the per-share delta of $0.5596. If the position were 200 shares, delta would be $111.92. For 200 shares, the display page would show a stock-only delta of $200 and a call option delta of $111.92, meaning the following: The option position (two contracts, or 200 shares) increases in value by $111.92 when the stock position (200 shares) increases in value by $1 per share, or $200 for the total position.

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This is also a 55.96% change ($111.92 ÷ $200.00 = 0.5596). For one option, or one option position, it is just as easy to use either the percentage or the dollar amount. When there is a complicated position involving several options positions and an underlying security, it is better to know how the structure as a whole is moving instead of tracking each individual piece. Here is summary of delta expressions: • For one share, delta can be expressed as a number (0.5596), a percentage (55.96%), or a dollar amount ($0.5596). • For a position consisting of more than one share, delta is normally expressed as a dollar amount. For a 100-share position, delta is shown as $55.96. You can also express delta as a percentage by dividing the dollar delta of the structure by the dollar delta of the stock, or 55.96% = $55.96 ÷ $100. • In the model, delta in Row 10 will always be expressed as a dollar amount scaled to the position size. One of the reasons for building the model was to allow you to look at any combination of stocks and options. Using the model, you can increase the stock price by $1 and see what happens to the value of the options and the structure as a whole. If you are looking at a single option position, you can use a platform or websites such as www.hoadley.net to produce tables and graphs illustrating how the Greeks behave under different conditions. When the position is more complicated, you can use the model to generate these tables and graphs. As illustrated in the next section, you can run whatever scenario you like, including multiple steps to create a delta table.

Delta Tables A delta table shows you delta values across a hypothetical scenario. Figure 13.3 is an example of a delta table. The position being modeled is a synthetic annuity (SynA) composed of 300 shares of stock trading at $105, multiple short call options, and a single long put option. Trying to get a good feel for this many moving parts is difficult with a formula alone. You can use the model to do this by playing

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around with different scenarios. As a first step, you can produce a delta table such as the one in Figure 13.3.

Figure 13.3 Delta table

The bold highlighted row is the current situation: 22 days until expiration, a stock price of $105, and implied volatility of 37%. The value of the stock-only position is $31,500, and the value of the SynA is $30,381. The current delta of the stock-only position is $300 (by definition), and the current delta of SynA is $119. The question is, how does the delta of this particular SynA behave in the short term with a sudden change in stock price? To answer this question, the days until expiration and volatility are held constant, and the stock and SynA values are determined by increasing and decreasing the stock price. For example, in Row 6, the stock price is assumed to jump to $110. At $110, the stock-only value goes from $31,500 to $33,000 (a $5 increase in price times 300 shares). At the same time, the SynA value goes up only about $500 ($30,882 − $31,500). Notice also the pattern of increases. Column F shows how much the SynA value changes with each $1 increase in the stock price. Between $105 and $106, the SynA value increases by $115. But between $109 and $110, it goes up by only $85. This is an important property of delta for positions with short calls. As the stock price increases, the value of the position goes up by smaller amounts. The opposite is true when the stock price goes down. As the stock price falls from $105 to $104, the value of the SynA decreases by

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$123. But from $101 to $100, the value goes down by $145. There is a significant damping effect compared to the stock-only position (which goes down by $300, regardless of stock price level), but the damping effect is smaller with further price drops. The general rule for delta, whether for calls or puts, is: The more in-the-money an option is, the higher the delta. And the more out-of-the-money an option is, the lower the delta. The pattern of delta changes depends on the specific components of the structured security. One of the important aspects of planning for and monitoring delta is to understand the pattern of change. An easy way to do that is to create the delta table. To create a table, run the model once for each row. You can step through it manually, or you can set up a macro to run through the stock prices. In terms of formulas, Cell F11 ($119) is delta from the model. The other values of Column F are the differences in consecutive values of Column E. For example, F10 = E10 − E11. Column G is Column F divided by the stock-only delta of $300. The SynA delta goes from 28% to 48% of the stock-only delta, even within this fairly narrow range of stock prices.

Theta Theta is defined as the negative partial derivative of option value V with respect to time. V=

yV yY

Theta can be calculated with the following formula: –e–qY

S‰(d1)X

–rKe –rY +(d2) + qSe –qY +(d1) 2™Y In this expression, the Greek symbol in the numerator of the first term represents the normal density function, and the Greek symbol in the second and third terms represents the normal cumulative density function.

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Theta is like delta, in the sense that it is nonexact and constantly changing. And this expression applies only to a single option. Instead of going through the calculation in detail, it might be more instructive to walk though an example using the model and then discuss some practical issues.

Theta Tables You can create a theta table in the same way as a delta table. In this case, keep the stock price and volatility constant, and step through the remaining days until expiration. Figure 13.4 shows a theta table for the same SynA.

Figure 13.4 Theta table

There is a slight disconnect between the formal partial derivative definition given earlier and the more commonly used definition of theta as the change in value over one day. It is the same issue raised for delta, the distinction between the “instantaneous” nature of the partial derivative and the “over a period of one day” practical definition. The model conforms better to the one-day definition because it measures exactly that.

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In this case, the value of the SynA increases by $14.99 over a oneday period. For reference, the instantaneous value of theta is close to $14 because it technically measures the change in value of a single instant of time at 24 hours remaining. Both the magnitude and pattern of change are important. Notice how theta changes over the 22 days until expiration. On the first day, it is $14.99. Close to expiration, it is over $40. The wide variation in theta over relatively small periods is the reason the model includes a modified version of theta, or average annualized theta. Average annualized theta is shown on the display page in Cells B10 and C10. This measure averages the effects of shorter-term thetas over the remaining option term. For example, the value of the SynA in Figure 13.4 at expiration is $30,991, compared to $30,381, with 22 days left in the option term. The difference, $610, is due solely to the effect of time because the other variables were held constant.

Calculating Yield As Average Annualized Theta With 22 days left, the option time value is $610. This amount will decay to zero over the 22-day period. If stock price and volatility remain constant, the table reveals the pattern of decay. However, regardless of the actual values of stock price and volatility over the period, the time value must decay to zero. We might not know the pattern of the decay, but we can still calculate the average daily decay. Average daily time decay = $610 ÷ 22 days = $27.7272 On an annualized basis, this is: Annualized average time decay = $27.2772 × 365 = $10,120 Expressed as a percentage of the position value, this is: Average annualized theta = $10,120 ÷ $30,381 = 33.3% In the table, theta expressed in these terms varied from 18.0% on Day 22 to 50.8% on the next-to-last day. It makes sense that the average value falls close to the middle of this range.

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Look at this again with regard to Figure 13.1. In the figure: B9 = $1,192.35 C9 = $1,148.97 The formulas for these two cells are: B9 = Z156 C9 = Z157 Z156 and Z157 are developed as: V156 = –V$3 × (V$5 − (MAX(0,V$4 − $B$4))) W156 = W$3 × (W$5 − (MAX(0,W$4 − $B$4))) X156 = X$3 × (X$5 − (MAX(0,$B$4 − X$4))) Y156 = –Y$3 × (Y$5 − (MAX(0,$B$4 − Y$4))) Z156 = SUM(V156:Y156) V157 = –V$3 × (V6 − (MAX(0,V$4 − $C$4))) W157 = W$3 × (W6 − (MAX(0,W$4 − $C$4))) X157 = X$3 × (X6 − (MAX(0,$C$4 − X$4))) Y157 = –Y$3 × (Y6 − (MAX(0,$C$4 − Y$4))) Z157 = SUM(V157:Y157) These formulas are the time value portion of total option premium for each option type. The total time value is shown in Cells B9 and C9 of the display page, so you can track the amount of time value for any structured position. Time value is converted into annualized average theta in the same way as earlier, as follows: Average daily time decay = $1,192.35 ÷ 365 days = $3.2667 On an annualized basis, this is: Annualized average time decay = $3.2667 × 365 = $1,192.35

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Expressed as a percentage of the position value, this is: Average annualized theta = $1,192.35 ÷ 10,000 = 11.92% B9 = IF(B9 = 0,0, – (B9 × 365 / B3) / $E$7) C9 = IF(C9 = 0,0, – (C9 × 365 / C3) / $E$7) When the stock price increases to $101, the time value changes to $1,148.97 because part of the value is now intrinsic. As a consequence, the average annualized theta drops to 11.49% from 11.92%.

The Importance of Theta In the previous example, theta was relatively large because of the high level of implied volatility (37%). A more typical volatility level of 15% to 20% will produce lower levels of average annualized theta. Still, in my opinion, theta represents the most exciting source of investment returns available to investors today. Theta is to options what interest is to bonds and dividends are to stocks. That is, theta represents a “time only” source of income that is not dependent on being right about whether a stock moves up or down. Stocks, bonds, and options all have two sources of yield. One is dependent on price direction; the other depends only on the passage of time. With equities, dividends represent the time-only source of yield, whereas capital gains depend on stock price direction. With fixed income, interest or coupon payments are the time-only source of yield, and bond price changes depend on the direction of interest rates. The same is true of options. There is both a directional component and a time-only component of yield. By overlaying short options on a stock or bond security, you create a hybrid vehicle, adding the option time component to the dividend or interest yields. When designing structured securities, you can build in target levels of yield. These levels depend on how volatile the overall market is at the time and the combination of term and number of options.

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Vega To construct a vega table, keep stock price and time constant, and step through volatility. In Figure 13.5, volatility is adjusted up and down to illustrate the effects on SynA value.

Figure 13.5 Vega table

Over longer periods, the effects of volatility are much more pronounced. It is not uncommon to see volatility spike over 40 during turbulent periods or to settle below 20 during calm periods. Even in normal market conditions, changes in implied volatility for a single security create interesting trading opportunities. Several options brokerages and commentators follow these changes in implied volatility as indicators. Joe Cusick of optionsXpress, for example, includes a section called “Implied Volatility Mover” in his daily newsletter: Commentary by Joe Cusick — December 10, 2012, http://marketing.optionsxpress.com/index.php/email/ Implied volatility in the options on Diamond Foods (DMND) is up, as shares fall Monday. ...Volume is a brisk 1.2 million shares and roughly 16,000 options traded. 11,000 puts and 5,000 calls so far and 30-day implied volatility in DMND options is up 20 percent to 69.

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In terms of general risk management, it is important to stress test portfolios in anticipation of volatility changes. One of the most important aspects of management is to set risk tolerances that you are comfortable with and understand in advance what tools you have to manage that risk. The next two chapters look at some of the metrics and design considerations to think about in building positions and portfolios.

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Introduction to Chapters 14, “Tracking Performance,” and 15, “Covered Synthetic Annuities” The final two chapters are supplemental material reprinted with permission from Pearson Education. These chapters originally appeared as Chapters 3 and 4 of Profiting with Synthetic Annuities: Options Strategies to Increase Yield and Control Portfolio Risk. Profiting with Synthetic Annuities (PwSA) is about adding options to portfolios—not as trading instruments—but as integral components of investment positions. The goal of synthetic annuities is to create a hybrid architecture that balances the long-term investor perspective of traditional portfolios with the risk discipline of quantitative-based strategies. Because synthetic annuities are designed and managed using quantitative models, I wanted to follow up in this book with an explanation of how the models are constructed. These final two chapters extend the conversation from the tools of quantitative finance to applied portfolio management. While this book defines the quantitative tools, PwSA is more of a practical guide for investors who are ready to begin building and managing structured securities. My hope is that by seeing a practical application of the quantitative finance material in this book, the models and metrics will be more meaningful as you think about structured securities, synthetic annuities—or any other options strategy. Chapter 14, “Tracking Performance,” introduces a trading platform (TradeStation) and a generalized template for tracking performance. Chapter 15, “Covered Synthetic Annuities,” shows you how to build a conservative SynA and illustrates it with Deere & Company 265

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(NYSE: DE). A covered SynA is a form of SynA that uses only covered calls and protective puts. By nature, it is less volatile and less risky than its underlying security. Option strategies such as the covered synthetic annuity that are restricted to covered calls and protective puts are well-suited to the retail investor market exactly because they are conservative. As a way to generate extra yield and create better risk profiles, these strategies are popular with many investors. For example, BlackRock’s Michael Fredericks, head of retail asset allocation for the multi-asset group, talked about his group’s use of covered calls and protective puts in a CNBC interview.1 Options strategies have been a growing part of institutional allocations for many years, but it was interesting to see the level of enthusiasm from Fredericks and the panel with respect to the retail market, where options are often criticized as being too risky and too complex. Fredericks: “We’re going to be more creative in where we find income. We’ve been selling covered calls...” Fredericks said they prefer to sell 5% to 10% out-of-the-money calls, where the combination of the options premium and dividends on the underlying stocks are yielding around 9.5%. Compared to credit markets (which are subject to losses in rising interest rate environments) high-quality equities with options-enhanced yields are very attractive. Fredericks: “You get upside and very competitive levels of yield. I don’t know where else you’re going to find that.” In terms of risk management, he said that they are also taking advantage of low-volatility levels to buy protective puts on the S&P 500 index. BlackRock’s approach is consistent with the covered form of SynA described in Chapter 15. As these strategies become more mainstream, there is, at the same time, the need for increased communication and education. At one level, investors and traders who manage their own portfolios will be looking for advice on how to use the information embedded in

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267

options prices and the options themselves to gain a trading edge, or create a comfortable investment style suited to them personally. At a much broader level, institutional trustees and individual investors, who cannot by law or simply don’t want to manage portfolios, will still be influenced by industry trends. They will need to decide when and whether to participate in customized solutions or offerings from investment firms or “smart ETF” providers. The issue for them will be to understand enough to make smart choices. As an example of new fund offerings, a few years ago the Wisdom Funds launched a family of funds based on the investment theme of dividends and the importance of dividends to total returns over the long term. Rather than weighting an index by stock market capitalization, they use a weighting system related to dividends. Going forward, it is logical to imagine that many investment management firms and ETF providers will launch new funds based on an options theme. As discussed in this book, options create a new source of return related to time decay and measured by theta. In the next two chapters, theta is translated into projected payback periods. Accelerating payback periods while reducing volatility are part of the design goals of synthetic annuities. Because of the potential advantages, it is easy to justify the rationale behind an options theme. It is harder to identify exactly what that means. Options themes are very broad and include an incredible array of possibilities. They can be conservative by nature, such as the covered varieties, or extremely aggressive, leveraging capital and market exposure. They can be applied to an entire portfolio or only to specific parts of it. Some option strategies will be right for some, and probably many, investors. It is just a matter of matching strategy style with investor preference—something that has not and will not change.

1

Michael Frederick interview: CNBC Half-Time Report, December, 2012 video link: http://www.cnbc.com/id/100335213/Go_RiskOn_in_2013_ BlackRockrsquos_Fredericks

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14 Tracking Performance This chapter introduces the administrative tools and graphic displays used in the book to illustrate SynAs. The material also presents the basic metrics for describing and tracking performance. If you trade options and use cost basis or net principle in your trading decisions, you are probably already familiar with the ideas behind most of this material. If you are new to structured securities, please keep in mind this chapter provides only a brief overview of option Greeks. The examples in the chapter on setup of SynAs (Chapter 15, “Covered Synthetic Annuities”) should provide a more meaningful context and interpretation of the Greeks and the rationale behind trading decisions. The first section presents a sample template that I use to track cost basis, position gain/loss, and payback periods. The template simply offers an example of how to summarize the information to set up and monitor performance. For example, it shows the current cost basis and how to roll it forward to the next period. Because a SynA is adjusted periodically based on its relationship to cost basis, it is necessary to track cost basis—at least during the first few months. The template also shows the calculation of position gain/loss and the projected payback period. The position gain/loss is compared to maximum loss to indicate when a trade needs to be made, and the projected payback period measures the rate at which the SynA is creating virtual dividends.

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The second section covers a feature of the TradeStation® platform, called a theoretical position. The theoretical position helps to describe a SynA by combining information for a security and options on the security. Using this feature, you can follow SynA metrics such as delta, gamma, and theta to structure and manage positions.

Note The term cost basis as used in this book refers to net principle or net capital invested. It is used for trading decisions and is not intended to be a tax definition of cost basis.

Tracking Template The tracking template consolidates the information needed to make trading decisions. The most important decision is when and how to adjust cost basis. It is not necessary to use this particular template—in fact, many times I don’t use any template. As long as you know when you have reached your maximum loss, any method works. If the underlying security is not volatile, you could keep a running total of options transactions on a notepad next to the computer. I generally have an idea of the price at which I should start looking more closely at gain/loss. If the price is comfortably above that amount, I just download transaction records periodically to update the cost basis. For example, if you buy at $25 and don’t reach your maximum loss unless the price drops below $22, then you don’t need real-time cost basis information unless the price gets close to $22. The template format in Figure 14.1 attempts to standardize the process, probably at the cost of making it seem overcomplicated. The important point about whatever method you use is that you have easy access to the information you need to make trading decisions. When the market becomes volatile, having an easy-to-read display showing current cost basis and how close the position is to a trading trigger is helpful.

CHAPTER 14 • TRACKING PERFORMANCE

Figure 14.1 Deere & Company tracking template

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Cost Basis Cost basis is the net amount invested in a security. For a stock that doesn’t pay a dividend, cost basis is simply the amount paid for the stock. For a stock or equity index that does pay a dividend, cost basis is reduced each time you receive a dividend. Similarly, for a fixed-income security, cost basis is reduced by interest or coupon payments as they are received. For a SynA, the calculation also involves tracking cash flows from options transactions. The cost basis of the underlying security is adjusted down by the amount received from selling options; the cost basis is adjusted up by the amount paid for options. The terms net options credit or net options premium is the difference between the amount received from selling options and the amount paid for options. I also refer to the net options credit as a virtual dividend. Cost basis is rolled forward from the beginning of a period to the end of the period, as follows: Beginning-of-period cost basis Minus actual dividends Minus virtual dividends (net options credit) Equals end-of-period cost basis

Trade Trigger The first rule of SynA management is to not lose more than you are comfortable losing without doing something about it. The gain or loss on the position is the difference between the current value of the position and the cost basis. Comparing the loss, if it exists, to the maximum loss indicates when you need to make a trade. For example, if your maximum loss is $2,000, the current value of the position is $20,000, and the cost basis is $22,500, the template would indicate that you are $500 over the maximum loss. Therefore, you need to make a trade to reduce the cost basis by at least $500.

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Projected Payback Period The payback period is the time it takes to recover invested capital, or cost basis, through interest payments, dividends, or optionsrelated cash flows. For example, an average S&P 500 stock paying a 2.5% percent dividend has a payback period of 40 years. One of the objectives of the SynA is to steadily reduce cost basis over time. With a SynA, the payback period can often be reduced significantly. The ability to accelerate the payback period has important implications for yield and for financing insurance protection. The payback period in the template is shown in months for both the underlying security and the corresponding SynA.

Example of Tracking Template The template in Figure 14.1 for Deere & Company (DE) has entries for cost basis, the current gain/loss, the trade trigger, and projected payback months. • Cost Basis: The four columns under Cost Basis show the roll forward from the beginning of the period (BOP) to the end of the period (EOP), based on actual dividends and the net credit from options transactions. In this case, the stock has just been purchased for $21,300. The net credit from setting up the DE SynA ($292.92) is subtracted from the cost basis to calculate the EOP cost basis, or $21,007. The row labeled Stock tracks the performance of the stock position without regard to options transactions. It is a pro-forma view of how a buy-and-hold position would have performed, and therefore is updated only with actual dividends.

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• Trade Trigger: The display should make it clear when you have reached a trigger point. In this case, the cost basis is $21,007 and the position value is $21,300, resulting in a current gain of $293. Because there is a gain, nothing needs to be done. If there were a loss, it would be compared to the max loss of $2,000 and the excess, if any, would be displayed in the Sell Call $ column. • Payback Months: For the pro-forma Stock row, payback months are estimated as the cost basis divided by the monthly dividend, where the monthly dividend is the latest declared quarterly dividend divided by three. For the SynA, payback months are calculated by dividing the cost basis by the sum of the monthly dividend and the estimated monthly theta. (See the section “Calculation of Payback Period” in Chapter 15 for more information.)

TradeStation Platform The examples in this book use a particular screen view from the TradeStation® platform.1 As in most trading platforms, various display screens give you access to standard and customized ways of organizing information. In TradeStation, these screens are referred to as workspaces. One of these workspaces, the Options Analysis Workspace, is useful in building and tracking synthetic annuities.

Options Analysis Workspace Figure 14.2 is an example of a TradeStation Options Analysis Workspace for Deere & Company (DE).

1

Certain screenshots, including Options Analysis Workspace and Theoretical Positions, were created with TradeStation. ©TradeStation Technologies, Inc. All rights reserved.

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Figure 14.2 TradeStation options analysis for DE on October 12, 2011

275

Source: TradeStation Technologies, Inc.

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The workspace is divided into three sections. The top section shows the current price of DE, along with other real-time information such as price and market-implied volatility. The large middle section contains call and put option quotes, market-implied volatilities, and selected Greeks—in this case, delta, gamma, and theta. The bottom section combines a user-defined set of stock and options. In TradeStation, this section is referred to as a theoretical position.

Theoretical Position A theoretical position is simply a consolidated view of a SynA; that is, it shows the underlying security and the options on the security that make up a SynA. The theoretical position is used in two ways. First, in setting up a SynA, it enables you to quickly see if the position meets your design criteria before executing the trades. Second, after you execute the trades to establish a SynA, the theoretical position helps you track and manage it. The theoretical position updates automatically in the platform, so it is a real-time representation of the SynA going forward. Figure 14.3 is a closer look at the information in the theoretical position. Most of the SynA examples presented in the book use this display format. This particular example is a representation of the Deere & Company SynA, where each of the five rows describes a particular component of the SynA. For example, the bottom row is the stock component. Reading across the bottom row is the stock symbol (DE), the number of shares (or Qty, which is 300 here), the price ($71.00), the profit and loss (Gross P&L of $21.00), the Spread Quote (or bid and ask), the Maximum Gain (unlimited), and the maximum loss ($21,300 or the amount invested). The position delta is $300, or simply the number of shares. Gamma and theta are both zero (by definition).

Figure 14.3 Deere & Company theoretical position on October 12, 2011

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Source: TradeStation Technologies, Inc.

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The row immediately above the stock component describes one of the options, labeled DE 111022C70. The symbol format is read as follows: • DE is the stock symbol. • 111022 is the option expiration date, with year first (2011, Oct 22). • C means this is a call option, and 70 is the strike price. Reading across, Quantity (Qty) is –1, meaning short 1 contract, or 100 shares. Also shown is Price ($2.50), Max Gain (Loss) ($250 = $2.50 × 100 shares), and the option Greeks: Delta$ (–$61.02), Gamma$ (–$8.13), and Theta$ ($10.08). Continuing to read up in the theoretical position, the next two rows show the details of a $72.5 strike call and a $67.5 strike put. The top line adds the individual component rows. This consolidated row describes the SynA, including the three Greeks on the right side of the screen view, as follows: • Position Delta$: The SynA has a delta of $178.46. This means that, for each $1 change in the stock price, the value of the SynA changes by $178.46.68, creating a damping effect on volatility, compared to a stock position with a delta of $300. • Position Gamma$: Gamma åis –$10.33. Gamma is a measure of how much position delta changes as the stock price increases by $1. In this case, if Deere goes up to $1 from $71.00 to $72.00, Delta$ will decrease by approximately $10.33, from $178.46 to $168.13. In general, the smaller the gamma, the more stable the position. The SynAs covered in the next chapter—the covered SynAs—are fairly stable by nature, so gamma tends to be small and plays a minor role.

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• Position Theta$: Theta is $12.82, meaning that the options are decaying in value by $12.82 per day. Because the options are a net short position, this time decay adds to the position value. Multiplying daily theta by 30 gives an estimate of monthly theta. $12.82 × 30 is $384.60, or the monthly “virtual dividend.”

Note You might see delta, gamma, and theta presented in any one of three ways: as a number, a percentage, or a dollar amount. The context usually determines which is used. When speaking in general terms, referring to a number or percentage is easier. For example, saying that delta is 0.40 (or, equivalently, 40%) means that the position moves by 40¢ for each dollar move in the underlying security. Without knowing how many shares are in the position, stating the Greeks in dollar amounts is not possible. However, when speaking specifically about a position such as the preceding one, in which you know the number of shares, referring to a dollar amount is usually more descriptive. In the examples that describe a specific SynA, the number of shares is known, so the Greeks are shown in dollar amounts. When talking about delta ranges or delta adjustments without referencing a specific SynA, either decimal numbers (for example, a delta target range of 0.30 to 0.70) or their equivalent percentages (as with a delta target range of 30% to 70%) are used.

Although I do not use other trading platforms, I understand that many are capable of providing the same basic information as Trade Station. You might want to verify that you can create theoretical positions in a format similar to the one shown, or at least have an idea of what would be required to view it easily. If your platform does not allow you to summarize Greeks easily, you can use another method, based on specific dollar moves in the underlying security, as an alternative to delta-based adjustments.

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Transaction Records After setting up a SynA, I download the transactions to keep a record of the net premium received. Normally, any trading platform enables you to download transaction information directly into an Excel spreadsheet or an Access database. Figure 14.4 shows the Excel file for the Deere options transactions: The net credit from the options transactions is – $70.03

Paid for the long $67.5 strike put

+ $113.97

Received for the short $72.5 strike call

+ $248.98

Received for the short $70 strike call

= $292.92

Net credit

Note The column labeled Amount is the option price × quantity – commission, where the commission is per contract. For example, the premium received for the $70 strike call is $250 and the commission on the trade is $1.02, for a net credit of $248.98.

In addition to the downloaded information, I add two more columns—intrinsic and time value—where intrinsic value measures how much the option is in-the-money (or ITM). Time value is the remainder of the premium. How often you download and update information depends on how volatile the stock price is. If a particular position has not been close to the maximum loss, I usually wait until the end of the month or quarter, download the transaction information, and update the cost basis at that point.

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Figure 14.4 Deere & Company options transactions

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I also want to point out that cost basis tracking is usually not required over the long term. Hopefully within a reasonable period after setting up a SynA, the cost basis will be reduced enough so that full protection of the cost basis can be financed through the normal process of rolling out options over time. The idea here is that as the cost basis of the position is steadily reduced over time, the normal amount spent on put options will be enough to cover the reduced cost basis in total. At that point, it is not necessary to continue to track cost basis.

Putting It All Together: Synthetic Annuity Overview Combining all the information into a single view provides an easyto-use overview of a synthetic annuity. Figure 14.5 shows the combination of the theoretical position, the tracking summary, and the transaction records. In this figure, you can see the complete picture. The top section, the theoretical position, shows the individual pieces and the consolidated view of the SynA. You can see, for example, how much smoothing is being accomplished and the rate at which virtual dividends are being generated. Delta measures smoothing. With a position delta of $178.46, the SynA moves up or down about 59% ($178.46 ÷ $300.00) of the stock move. Theta, or the rate of virtual dividends, is $12.82 per day. The middle section, or tracking template, updates cost basis information from period to period and highlights the need to trade when the maximum loss is exceeded. The bottom section documents the actual trades and shows how information flows into the tracking template. Again, it is not necessary to use this or any particular method. The only critical element in management knowing when you have reached your risk budget and knowing the general level of delta so that the SynA does not become too detached from the underlying security.

Figure 14.5 Deere & Company SynA Overview

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Source of screenshot: TradeStation Technologies, Inc.

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15 Covered Synthetic Annuities A covered synthetic annuity (CSynA) is a synthetic annuity that uses only covered call options and protective put options. A CSynA is a more conservative investment than its underlying security because covered calls and protective puts work to make the potential losses from a CSynA less than those of the underlying security. To illustrate how a CSynA functions, the chapter begins by comparing a CSynA to a covered call strategy and addressing the main objections to using covered calls. Then, a CSynA is constructed using Deere & Company stock as the underlying security. The CSynA transforms the Deere stock position into a related security that is less volatile and produces higher levels of current income. Of course, nothing is free in investing; the tradeoff here is that you give up some upside. The performance metric section presents a more detailed explanation of risk and return expectations, based on the number of short options, ranging from a contingent position (zero options) to a fully covered position. This chapter ends with a description of an algorithmic CSynA, or standardized version of a CSynA, that you can implement and manage using a simple set of rules.

Note For new options traders, most brokers include covered calls and protective puts as part of Level 1 options approval, which is normally available even to those without options experience.

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Covered Synthetic Annuity (CSynA) A CSynA is a form of “managed” covered call strategy. Covered call strategies are popular tools for enhancing returns and providing limited amounts of risk protection. Still, many investors don’t like to use covered calls because you give up the upside but still have the downside. That’s true, to a large degree. Imagine that you buy a stock for $100 and sell an at-the-money (ATM) call for $3. If the stock goes to $150, you get $3. If it drops to $50, you suffer the loss—except for $3. The problem is that options can quickly become “disconnected” from the security price as the security price declines, and can quickly become “too connected” as the security price rises. Figure 15.1 shows a one-year call option with a strike price of $45 on a stock trading at $45. The ATM option, in the center column, has a delta of 0.5727, or about 57%, meaning that the option moves about 57¢ for each dollar move in the stock price. The other columns illustrate the effect on delta of a quick price change in the underlying security. In the first column, where the security price has dropped to $30, delta falls to 0.1212, or 12%. At that point, the option price changes by only 12¢ for each dollar of change in the stock price. In general, the lower the delta, the more “disconnected” the option is and the less the option protects against further declines for a covered call position. On the other end of the table, where the stock price has risen to $60, the option delta rises to 87%, meaning that the option is very connected to the stock price, moving 87¢ for each dollar of stock price change. For a covered call position, the short option eliminates most of the gains from any further increases in stock price. With shorter-term options, the effect of volatility and time is much stronger. Option deltas can approach 0 on the downside or 1 on the upside after only a few points of price movement for one-month options.

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Figure 15.1 Call option deltas at various stock prices

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How do option deltas translate into covered call position deltas? The delta of a covered call position is the delta of the stock (or underlying security) minus the delta of the option. By definition, stock delta is 1.0, so as the stock price declines and the option delta approaches zero, the covered call delta approaches 1.0. Conversely, as the stock price increases, the option delta approaches 1.0 and the covered call delta approaches zero. In other words, during price declines, the option fades away and the covered call begins to act like the stock by itself, with a delta of 1.0. During price increases, the option delta approaches 1.0, so the covered call delta approaches zero. This is just a restatement of the common objection to covered calls: You give up the upside and still have the downside. Theta is affected as well. In general, theta is highest when the stock price is close to the strike price of the option. That is, ATM options produce the highest levels of theta. As the stock price moves away from the strike price, theta goes down regardless of the direction the price moves. As the covered call delta moves below about 0.2 or above about 0.8, the rate of theta is relatively low and, therefore, time decay is minimal. Because of delta and theta effects, a covered call strategy is much more effective as long as the underlying security price stays close to the option strike price, where the balance between delta and theta works to your advantage. The balance between delta (the market exposure) and theta (the rate of time decay or virtual dividend payments) is fundamental to options-based strategies. To address balance, a CSynA differs in three ways from a covered call position. The first difference is that the call option strike prices are normally staggered. Staggering the strike prices of the short call options makes delta more stable across a wider range of prices. The second difference is that part of the cash received from selling the call options

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is used to purchase put options. Reinvesting part of the call option premium in long put options creates stronger downside protection. Third, the CSynA includes management rules that help to maintain and balance the relationship between delta and theta over the longer term.

Example: Deere & Company I have been looking at Deere recently and wondering whether it is a good time to buy. As I write this in October 2011, the right answer in this case (as with everything else) probably depends on Europe. But with a P/E below 12 (a discount to its historical multiple) and a yield higher than the 10-year Treasury, and considering that Deere is a beneficiary of the long-term agricultural secular trend, buying doesn’t sound like a bad idea. After all, with the world’s population expanding and farmers needing to increase production to satisfy an appetite for more and better food, Deere seems to be in the right place for the long term.

Building a CSynA: The Steps Setting up a CSynA involves these steps: 1. Select the underlying security, normally a stock or other security that you would like to own long term. 2. If you don’t expect the price to go up right away, sell one or more covered call options on the security. 3. Buy out-of-the-money put options, using part of the money from the call options. After reviewing Deere, everything looks reasonable. Here are the steps I went through to build the DE CSynA.

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Figure 15.2 Deere & Co option quotes

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Step 1: Buy the Underlying Security I bought 300 shares of DE at $71, for a total position value of $21,300. Of course, the number of shares is determined by your portfolio size and how diversified you want to be. For this example, I am assuming a total portfolio size of around $300,000, so that this position represents about 7% of the total. The results can be scaled for larger portfolios. Step 2: Unless You Expect the Price to Go Up Right Away, Sell One or More Covered Call Options Deere has already reported earnings, and the results of the crop report are out. Of course, I hope it goes up from here, but I don’t know of any short-term catalysts. As with most other stocks, it will probably move with the macro picture. If I did have a view of direction, I would probably give it some room to run. Instead of selling calls immediately, I could either put in limit sell orders for options at higher strike prices or simply wait to see how far momentum might take it. In setting up a covered SynA, no requirement states that you must begin to extract theta right away. To give you an idea of the option prices available as I began to set up the CSynA, Figure 15.2 shows near-the-money options prices at the market close on October 12, 2011. In this case, I sold two option contracts, a $70 strike call for $248.98 and a $72.5 strike call for $113.97, for a total of $362.95. Because a CSynA can use only covered calls, the limit is three option contracts for this position. Because I think Deere is slightly undervalued, I will wait to sell the third option. Normally at setup and at monthly roll-forwards, if I have no opinion about stock price direction, I sell the maximum number of call options.

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Step 3: Use 20% to 30% of the “Time Value” of the Call Option Premiums to Purchase One or More Put Options The call premiums totaled $362.95, but some of that is intrinsic value. The $70 strike call option is ITM (in-the-money), so it needs to be split between intrinsic value and time value. Of the $248.98 premium received, $100 is intrinsic value ($71 current price – $70 strike price). Therefore, the time value of the $70 strike option is $148.98. Because the $72.5 strike call is OTM (out-of-the-money), the entire premium, $113.97, is time value. The total time value is $262.95. I bought a put option with a strike of $67.5 for $70.03, or about 27% of the time value. Subtracting the price of the put from the proceeds from the two calls leaves $292.92 as a net credit. That’s it—finished. Figure 15.3 shows the overview. This is the same overview from Chapter 7, indicating the following: • Delta of $178 (or 59% of the stock-only delta) • Daily theta of $12.82 (or payback rate of 49 months) • $292.92 net transaction credit (reflected as a reduction in cost basis) The observations on the CSynA Greeks are repeated here: • Position Delta$: The CSynA has a delta of $178.46. This means that, for each $1 change in the stock price, the value of the CSynA will change by $178.46. This creates a damping effect on volatility, compared to a stock position with a delta of $300.

Figure 15.3 Deere & Company CSynA at setup on October 12, 2011

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Source of screenshot: TradeStation Technologies, Inc.

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• Position Gamma$: Gamma is –$10.33. Gamma tells you how quickly delta changes. Specifically, gamma is a measure of how much position delta changes if the stock price increases by $1. For example, if Deere stock price goes up by $1 from $71 to $72, Delta will decrease by $10.33, from $178.46 to $168.13. In other words, instead of capturing about $178 gain as the stock price moved from $71 to $72, gamma tells you that you will capture only about $168 if the stock price goes up by another dollar. This is consistent with the earlier result of diminishing returns during stock price increases for covered positions. The opposite is true for declining stock prices. If Deere stock moves from $71 to $70, gamma tells you that the position delta will increase by $10.33, from $178.46 to $188.79. That is not as bad as the stock itself, which would fall by $300, but the cushioning effect of the position begins to deteriorate as the price of the stock drops. Again, you get the same result as the covered position. In general, the closer gamma is to zero, the more stable the position delta is. Most CSynAs are fairly stable by nature, so gamma tends to be small and plays a minor role.

Note Keep in mind that delta and gamma are not exact. They are instaneaous approximations and generally apply within fairly narrow price ranges. It is not correct to think that they will describe the downside risk if the stock price falls by $10 or $15. The further the price is from the current value, the worse the approximation.

• Position Theta$: Theta is $12.82, meaning that the options are decaying in value by $12.82 per day. Because you are short the options here, time decay is a credit; that is, it adds to the position value. Multiplying daily theta by 30 gives an estimate of

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monthly theta. $12.82 × 30 is $384.60, or the monthly “virtual dividend,” shown in the column labeled Monthly Theta.

Calculation of Payback Period Deere pays a quarterly dividend of 41¢ per share. How long will it take to pay off an investment in Deere from the dividend alone? The amount invested in the position is $21,300, and the annual dividend rate is $1.64 per share. On 300 shares, the annual dividend is $492, or $41 per month. Payback months on the stock-only position is calculated by dividing the current cost basis of $21,300 by the monthly dividend of $41, which equals 520. That is the number of months it will take to recover the cost basis from the dividend alone. Now look at what happens to the payback period for the CSynA. With the CSynA, there are two sources of cash flow. First, because the CSynA includes the stock position, there is the actual dividend of $41 per month. Second, selling two call options and buying one put option produced a net credit of $296 (of which $196 is time value). This is not a large reduction in cost basis immediately, but when the rate of cost basis reduction (as measured by theta) is projected into the future, the number of months required to pay back the investment is cut by a factor of 10. Let’s walk through the calculation. From the theoretical position, Position Theta$ is $12.82 per day. Multiplying the daily rate by 30 gives an estimate for monthly theta of $384.60. Adding the monthly dividend of $41 produces a total monthly “dividend” of $425.60 for the CSynA. Dividing the current cost basis of $21,004 ($21,300 reduced by the net option credit) by $425.60 gives a payback period of 49 months. Am I really saying that by making a couple of options transactions, you can cut the payback period by a factor of 10? Yes, potentially.

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But you have to be careful when interpreting these numbers. To get a little more comfortable, take a look at an alternative calculation of theta and payback period.

A Different Calculation of Payback Period The longer-term power of theta is evident in the payback period. But what does this mean? Is it reasonable to think that you can recover your entire cost basis in 49 months? Maybe something about the calculation of theta is confusing the issue. To get an intuitive feel for the payback period, let’s approach the calculation of theta in a simpler way. Let’s look at the options and what the CSynA will be worth at option expiration. Figure 15.4 breaks the option premium into intrinsic value and time value, and shows how the CSynA will look at expiration.

Figure 15.4 Time value and payback period

Both the put option and the $72.5 strike call option are OTM, so they are composed of only time value, as shown in the table. The $70 strike call option is ITM. The ITM, or intrinsic value, is $1, based on the current stock price of $71. At any time before expiration, the options have both intrinsic value and time value. But just before options expiration, the options have no time left, thus there is no time value. The price of the call options at expiration is simply the amount by which the stock price exceeds the strike price. The column labeled Expiration Date Value shows the value the options will have at expiration if the stock price is

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the same as it is today. The assumption that the stock price remains the same at expiration is the best way to isolate the effects of the passage of time on option prices. Of course, in reality, the stock price will be whatever it is. But say that the stock price is $72 at expiration. The $70 strike call will be worth $2. But we could say that $1 of it was because of stock price movement and $1 was due to the passage of time. So you would get to the same place anyway. Assuming that price stays constant is the easiest way to measure the time decay or theta. The options transactions took place on October 12, 2011. The options expire on October 22, 2011, or ten days later. That means the time value of the options will go to zero over the next ten days. At that point, the options will have the value shown in the column labeled Expiration Date Value. Under the strategy, the options are used to modify the behavior of the combined stock option position. The underlying security shares are long-term investments, so you do not want to let any of the options be exercised. Before the market close on the 22nd, the options that are ITM are to be repurchased and rolled over to the next month. If the stock price is $71 at expiration, the options will be repurchased at their current intrinsic value of $100. The net gain over the next ten days, in this scenario, will be $196, equal to the time value. That is, the gain will be equal to the premium received for the options minus the amount to repurchase the ITM options. So, you will make $196 over the next ten days due to the passage of time. If you make $196 over ten days, the monthly amount is about three times that, or $588 a month. Dividing the cost basis of $21,004 by $588 is 36 months, even better than the 49 months using the platform (or more exact) definition of theta. It seems that the back-ofthe-envelope calculation of payback period confirms the speed at which theta is working.

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Still, can this be right? To get paid back this quickly on an investment, there must be more to the story. There is. The assumptions themselves are almost contradictory. You are getting paid high premiums for the call options because the market is assuming a certain level of volatilty. At the same time, the assumption is that the stock price will not move. Actually, the assumption was that the stock price at expiration would be the same as today. (Technically, that is not an assumption that it doesn’t move in the meantime—just that it finishes at the same place.) The chance also exists that you will need to adjust the options positions prior to expiration. Realistically, the more a stock price moves around, the more likely, under the strategy, that an adjustment will be made. If adjustments are made to keep delta within a target range, for instance, those adjustments, depending on how they are made, could reduce the returns and lengthen the payback period. Volatility is the most important driver of options pricing. Right now, volatility is relatively high. At lower levels of volatility, the call option premiums would be less, which would lengthen the payback period. However, if the stock remains within a certain range and the pricing of options stays at elevated levels so that you could roll out the options month after month at current pricing, it would take only about four years to fully recover the cost basis. This is maybe not likely, but it’s possible. To avoid too much optimism, I normally think of the payback period as a reminder of what is possible, not an assumption about returns. It is simply an indication of the power of theta.

Note It can be argued that I reduced the cost basis too much by including both the intrinsic value and the time value of net options credit. I agree, but the projection is dominated by the rate of theta rather than an extra $100 in cost basis reduction, especially because

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the $100 is not projected into the future—only the time value is projected. I think the extra complexity of an exact calculation of cost basis reduction for one month is not worth the effort and does not significantly distort the overall result. However, for multiperiod projections, it is important to include only the time value of options in cost basis reduction. In other words, including total net credit does not anticipate the repurchase of option intrinsic value before expiration.

The Number of Options and the Strike Prices Notice that I didn’t sell three short call options. Selling the maximum number of call options is not necessary. In fact, this example offers a choice to sell zero, one, two, or three short calls. The strike prices are also flexible. Deciding how many options to sell and the strike prices of those options depends on what you want to achieve in terms of market exposure (delta) and income generation (theta) and the tradeoffs between them. In general, the more options you sell, the lower the delta and the higher the theta. If you sell less than the maximum number of options to set up the CSynA, you can always sell the others later, if you want to. For example, you might decide to make future option sales contingent on a higher or lower stock price. Or if the stock trends upward and you think it might keep going, you are not required to sell the other options. To get a feel for the level of tradeoffs for Deere, consider Figure 15.5. The figure makes it clear that delta and theta work together to simultaneously decrease market exposure and increase current income. To maximize theta, sell the maximum number of options. To maximize delta, don’t sell any options.

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Figure 15.5 Delta and theta by number of short options

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Stock-Only or Contingent SynA Of course, if you don’t sell any options, you just have 300 shares of stock. You might wonder how a stock position can be a CSynA. This scenario points out that selling options is not necessary to create a CSynA. The only rule required for a CSynA is that you must establish a maximum loss amount and then, when you reach that amount, you must do something about it. Because the important aspect of the CSynA is an intent to do something if it becomes necessary, I think of this form of CSynA as a “contingent” CSynA. (Of course, if you never sell any options, you will also not create any theta, so it is assumed that you will sell options at some point in the future.) Sometimes, you might enter a position because you think there is a near-term catalyst. In such a case, you might not want to sell any call options or buy any put options at the time you enter the trade. If you are right about the catalyst, you can get better pricing on the options after a moveup. You do nothing unless the price falls enough to exceed the max loss.

One to Three Short Options: The Tradeoff Between Delta and Theta When you sell even one call option, you begin to reduce delta and create theta. By looking at the individual options available and the theoretical position before you execute the trades, you can balance, within ranges, the level of delta you prefer and how much theta you want to generate. This balance between delta (the market exposure) and theta (the rate of time decay or virtual dividend payments) is fundamental to options-based strategies. In selling call options, you give up potential upside, but you get theta in return. Sometimes, you might prefer to sell only OTM options to maintain most of the upside, with the understanding that OTM options have smaller deltas and, therefore, less protection on the downside.

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The more theta you decide to generate, the more upside you will be giving up, but as a positive, the smaller the volatility will be. Earlier, Figure 15.3 showed the stock-only delta as $300 (by definition, the delta of a stock position is the number of shares). By selling options, you can decrease market exposure and the related volatility of the CSynA. Selling one option reduced delta to $223, selling two short options reduced it to $176, and selling three short options reduced it to $150, cutting delta in half. Theta moves in the opposite direction—that is, as you sell more options, delta goes down and theta goes up. In any particular situation, you might be more interested in reducing volatility or increasing yield. Think of stocks such as Southern Company, Verizon, or Kinder Morgan, all of which pay high dividends. They have also been remarkably stable, even in historically volatile periods. If you are not concerned about volatility and are satisfied with the yield, you can look at options as simply a yield-enhancement mechanism. For instance, you could sell one or more OTM or farOTM options to boost the yield. In low-interest-rate environments, certain high-yielding stocks have “natural” put protection because, as the price falls, the yield rises. This is especially true for companies that have disciplined cash-management practices such as Master Limited Partnerships or those that are regulated by public commissions such as utilities. The only drawbacks to pursuing yield enhancement CSynA’s is that the option premiums are very low due to the stable nature of prices. The possibility also exists that, in a recovering economy, prices on relatively defensive securities can decline simply due to an outflow of funds as the “risk on” trade returns. For securities with higher volatility, you might want to sell more call options as a volatility-reduction mechanism. Even though improving yield might not be your main objective, you get that as well. No one right answer exists. Each situation presents its own opportunities, depending on your outlook on risk, your belief in the long-term

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prospects of the company, and the degree to which you want to emphasize volatility reduction and yield over capital gains. You can also reflect your view of the importance of dividends, both actual and virtual, to investment returns.

How Important Are Dividends to Total Return? Over the past 80 years, dividends have contributed an almost equal share of total returns as capital gains to the S&P 500. One factor in CSynA design was to recognize the importance of the role dividends have played in total portfolio returns. As Figure 15.3 pointed out, CSynAs let you adjust the yield source. The more options you sell, the more the expectation moves away from capital gains and toward theta (dividends). Whether a CSynA works in a particular situation depends not only on how the underlying security acts, but also on your expectations about where return will come from and how patient you are in waiting for it. For this reason, the more you think that the return of a security will come from capital gains (that is, the more you think the price will move up), the fewer options you should sell. The more you want to emphasize dividends over capital gains, the more options you should sell. Of course, the balance can change as events unfold, but in the CSynA structure, you have the flexibility to make adjustments.

Longer-Term Delta Targets One of the objectives of the CSynA is to dampen the effect of market volatility. By selecting the number and strike prices of the call options you sell, you can achieve a higher or lower delta. As a general rule, the more options you sell, the lower the delta and the lower the volatiity. Also, the delta reduction depends not only on the number of options sold, but also on the strike prices of those options. As a

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general rule, for any particular option, the more ITM, the higher the option delta and, therefore, the lower the CSynA delta. For instance, in Figure 15.1, the delta of an ATM option was 57%. As the stock price increased, making the option more ITM, delta increased to 70% for a $5 ITM option, 80% for a $10 ITM option, and 87% for a $15 ITM option. In most market conditions, I usually like the delta target percentage to be between 0.4 and 0.6—that is, I like to have about 40% to 60% of the stock price movement flow through to the CSynA. I think of this range as a strategic target that applies in average conditions to average securities. In turbulent market conditions, or when tactical trading opportunities exist, I adjust delta down or up to fit my outlook. The average longer-term strategic target can be described in terms of a standardized CSynA.

The Standard CSynA In practice, I normally set up CSynAs in a straightforward way. If I have 300 shares of the underlying, I simply sell an ITM call, an ATM call, and an OTM call. I use about 25% of the time value of the proceeds to purchase an OTM put option. On options expiration day, I repurchase any ITM options and sell the next month’s options in the same formation: one ITM, one ATM, and one OTM. If the underlying is not that volatile, I look at it once a week or about every 500 Dow points (whichever comes first) and make tactical adjustments if delta is below 0.25 or above 0.75. That’s it. This middle-of-the-road management style is consistent with a belief in the EMH (efficient market hypothesis—in other words, it is hard to beat a liquid market in heavily traded securities). It generally has a delta of about 0.50 and theta of around 20% to 30% annually, depending on the level of implied volatility.

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In general, the CSynA is flexible—it can be used in different ways in different situations. But what if you are more interested in a simple application of the basic principles of a CSynA? In that case, you can use a simplified form of the strategy, called a standard CSynA. The standard CSynA is easy to set up and manage. It can be applied in an almost algorithmic fashion. After you select the underlying security and setting a risk tolerance, you have little to think about. As with the typical setup discussed previously, it uses a full covered position, has a delta target of around 50%, and makes delta adjustments at 25% and 75%. It is easy because it narrows the set of possibilities. Sometimes, to narrow a set of possibilities, you first need to understand what the possibilities are. That way, in deciding to exclude a decision point, at least you know what you are excluding—and why. If you think about simply selling a call option, you need to specify a few parameters, such as the underlying security, the strike price, and the expiration date. For more complicated strategies, such as iron condors, the number of parameters goes up because you must specify information for all four legs. And that is just for the setup. Trying to parameterize the management rules would become unmanageable as you attempted to articulate where and how to make adjustments, terminate the trade, or initiate the next setup. Still, simplicity is important, if it exists. I admit that I was a little relieved when I went through the exercise of imagining how someone might enter a CSynA trade into a drop-down menu on a trading platform. In fact, only three inputs are required: the risk allocation or budget and the high and low delta adjustment points. And if you are willing to accept “average” values for those inputs, the process is as simple as clicking a CSynA box. To illustrate what I mean, it is helpful to start with a more generalized parameter set and narrow it so that the assumptions and investment “beliefs” built into a standard CSynA are obvious. The following is a list of some of the general parameters that might apply to a standard CSynA. A discussion of each follows.

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Standard SynA Parameters —

Fundamental/technical valuation high

—

Fundamental/technical valuation low

—

Minimum value

E

Momentum (M), reverting (R), or micro-efficient (E)

—

Price-related delta (custom SynA)

7%

Max drawdown % (risk budget)

100% Covered % (100% = full covered position) Yes

Spread evenly

—

Exclude ITM

—

Exclude ATM

—

Exclude OTM

75%

Upper delta adjustment (0.75 for reverting; 0.65 for momentum) Lower delta adjustment (0.25 for reverting; 0.40 for momentum)

25%

Fundamental/Technical Valuation High and Low If you have views on valuation levels, you can express them in the way you set strike prices and time the options trades. For example, you might buy a security for $50 because you think it is really worth $60. In that case, it makes sense to use the contingent CSynA setup and wait until the price gets close to your view of the real value. The first two parameters refer to the buy and sell points for securities based on perceived value. The standard version of a CSynA bucks convention and does not use predefined valuation ranges. It does not have opinions about the real value of securities. It assumes that the market price is the correct value, at least in the sense that fundamental or technical analysis is not helpful in determining a better value.

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More on this is discussed shortly when covering the micro-efficient parameter. As with all the parameters, these assumptions apply only to the standard version. Nothing prevents you from using these parameters as guidelines for setting up trades for other CSynAs.

Minimum Value Particularly for hard assets such as commodities and real estate, many people like to set minimum values, or prices at which they want to own more of the asset. Or they might think that certain companies are so good at managing invested capital that there are predefined entry points. If you think that oil is a bargain at $65, or that you should buy gold at $1,000, or that Berkshire Hathaway stock is a must-buy at $55, you can modify the normal CSynA operation to change the profile at these minimum levels. The idea of intrinsic value of an asset might be compelling, but the standard CSynA doesn’t recognize this, either.

Momentum, Reverting, or Micro-Efficient Specifying this parameter helps to determine how and when to make adjustments. A momentum investor looks for price strength and might increase exposure as the price increases. A mean-reverting investor might interpret price strength differently if he or she believes that price strength means the exposure should be decreased. As you might have guessed, the standard CSynA doesn’t use this information. The standard CSynA is consistent with a micro-efficient view of security prices. In other words, it makes the assumption that the market price is hard to beat—not necessarily that the market price is right or correct, just that beating the market is hard with either

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fundamental or technical analysis. In fact, this is probably one of the most researched areas of investment theory. Paul Samuelson, among others, put forth an interesting corollary to this. Samuelson believes that although the market exhibits incredible micro-efficiency (meaning, at the individual security level), it often exhibits macro-inefficiency (meaning, at the asset class level). The standard version assumes micro-efficiency—that is, it agrees with the efficient market hypothesis in its weak (technical analysis cannot give you an advantage) and semistrong (fundamental analysis cannot give you an advantage) forms. Given all the publicly known information, the standard version doesn’t try to make predictions about where the price is going because it believes the market price is where it should be.

Price-Related Delta In some cases, you might want to customize delta exposures based on the security price level. Within bounds, it is consistent with the automatic delta adjustments built into the SynA structure. If you prefer to exaggerate the effect, you can easily do so. From a theoretical standpoint, it is a customized blend of views on the previous parameters. The standard version doesn’t use this parameter.

Maximum Drawdown (Setting a Risk Tolerance) The maximum drawdown or risk tolerance is the amount you are willing to lose before taking some kind of action. Risk tolerance is also sometimes referred to as risk allocation or risk budget. When the loss on the position reaches this amount, the strategy dictates that you sell enough call options to lower cost basis so that the loss is back within the 7% limit. The standard SynA, and all versions of any SynA, uses this parameter. It is required for the quantitative risk-management discipline.

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Covered Percentage This parameter and the next four parameters describe how many options are sold and which ones to exclude, if any. The standard version is 100%, meaning that it sells the maximum number of covered options. If there are 300 shares of the underlying security, it sells 300 short options. If you have 1,000 shares, it sells 1,000 options. In nonstandard setups, this parameter enables you to specify that you want to sell only 50% of the maximum covered options. For example, someone who simply wants to enhance the return from a high-yielding stock could decide to sell only 200 options on a 400share position and to sell only OTM options. In that case, this person would exclude the sale of ITM and ATM options. This type of setup and management produces less theta but maintains more upside and decreases the chances that the stock could be called away prior to expiration.

Upper and Lower Delta Adjustments These parameters determine when a delta adjustment is made. The wider the range, the less frequently the adjustments are made. The purpose of the adjustments is to not lose contact with the underlying. The standard version uses a lower bound of 25% and an upper bound of 75%, although you can change this parameter.

Summary of the Standarized CSynA The standardized CSynA simplifies the setup and management of a CSynA. It requires only three parameters: 1. Maximum drawdown on cost basis 2. Upper delta adjustment level 3. Lower delta adjustment level

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If average values for these parameters (for example, 7% on drawdown, 75% on upper, and 25% on lower) are used, nothing else is required.

The BuyWrite Index By specifying a particular set of parameter values, a standard CSynA can be evaluated in the same way as the CBOE BuyWrite Index (BXM). The BXM is a simple covered call index on the S&P 500, where it is assumed that each month the index is purchased and an ATM option or slightly OTM option is sold against the index. Even though it is simple, it has outperformed the S&P 500 in terms of both the return achieved and the risk taken to get that return. The CBOE website includes a summary of the risk and return metrics of the BXM and reports from Callan and Ibbotson. See www.cboe.com/micro/bxm/ for a complete description of the BXM. Although I haven’t done it yet, I think it would be interesting to run parallel models of the S&P 500, the BXM, and the standard CSynA at various settings of the parameters to see how they stack up. With regard to investment theory, the standard CSynA assumes micro efficiency—that is, it eliminates the need to make directional price bets. Chapter 1, “Introduction,” discussed a typical CSynA. Generally, when I am deciding which call options to sell and how many of them, I don’t have a view on the direction of the security price. I typically sell the maximum number in a level pattern of ITM, ATM, and OTM options. For example, if I buy 100 shares of the underlying security, I sell one ATM call option. If I buy 200 shares, I sell the closest ITM call option and the closest OTM call option. For 300 shares: 1 ITM, 1 ATM, and 1 OTM; for 400 shares: 1 ITM, 2 ATM, 1 OTM, and so on, up to 1,000 shares with 3 ITM, 4 ATM, and 3 OTM. This practice is consistent with a view that the market is efficient on a micro basis and that the price of a security reflects available

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information. At times, I think I know more than the market, but when I don’t, I focus more on theta than on trying to pick a price direction. In Deere’s case, I did have a slight belief that the stock was undervalued, so I sold two options instead of three. In terms of performance, the standardized CSynA at setup has the following characteristics: • Volatility, or delta, of about 50% of the underlying • A payback period that is five to ten times less than the average dividend-paying stock • Significantly lower drawdown expectations Payback periods and yield are influenced by the implied volatility of the options. For the S&P 500 Index, a rule of thumb is to divide the VIX by 10 to get an estimate of monthly yield. For individual securities, volatilities are normally higher than the index, with some of the highest implied volatilities on volatility itself. The higher the implied volatility of the option, the more potential there is to accelerate payback periods.

Supplemental Material The CBOE S&P 500 BuyWrite Index The CBOE S&P 500 BuyWrite Index (BXM) is a benchmark index designed to track the performance of a hypothetical buy–write strategy on the S&P 500 Index. Announced in April 2002, the BXM Index was developed by the CBOE in cooperation with Standard & Poor’s. To help in the development of the BXM Index, the CBOE commissioned Professor Robert Whaley to compile and analyze relevant data from June 1988 through December 2001. Data on daily BXM prices now is available from June 30, 1986, to the present time (see the next section). The BXM is a passive total return index based on (1) buying an S&P 500 stock index portfolio and (2) “writing” (or

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selling) the near-term S&P 500 Index (SPXSM) “covered” call option, generally on the third Friday of each month. The SPX call written has about one month remaining to expiration, with an exercise price just above the prevailing index level (that is, it is slightly out of the money). The SPX call is held until expiration and cash settled, at which time, a new one-month, near-the-money call is written. Visit the BXM FAQ for more information about the construction of the index.

BXM Study by Callan Associates In 2006, Callan Associates, an investment services consulting firm, published a new study on the CBOE S&P 500 BuyWrite Index, with an analysis of performance from June 1988 through August 2006. The study builds upon the earlier studies done by Professor Robert Whaley (now at Vanderbilt University) and by Ibbotson Associates. The new Callan Associates study had several key findings, including these: 1. BXM generated superior risk-adjusted returns over the last 18 years, generating a return comparable to that of the S&P 500 with approximately two-thirds of the risk. (The compound annual return of the BXM was 11.77%, compared to 11.67% for the S&P 500, and BXM returns were generated with a standard deviation of 9.29%, two-thirds of the 13.89% volatility of the S&P 500.) 2. The risk-adjusted performance, as measured by the monthly Stutzer Index over the 18-year period, was 0.20 for the BXM versus 0.15 for the S&P 500. A comparison using the monthly Sharpe Ratio yielded similar results (0.22 versus 0.16, respectively), confirming the relative efficiency of the BXM over the 219-month study period.

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3. The BXM underperformed the S&P 500 during most rising equity markets and consistently outperformed the S&P 500 in all periods of declining equity markets, demonstrating the return cushion provided by income from writing the calls. 4. The BXM generates a return pattern different from that of the S&P 500, offering a source of potential diversification. The addition of the BXM to a diversified investor portfolio would have generated significant improvement in risk-adjusted performance over the past 18 years.

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Index

A accuracy, comparing alternatives versus, 56-57 additive, delta as, 254-255 “Adjustments for Anticipated Days of Higher Volatility” (McDonald), 59 alternative comparison with visual option pricing method, 56-57 annualized average theta, 183-184 annuities. See SynAs (synthetic annuities) Apple (turbulent markets example), 138-143 AQR Capital Management, 2 arbitrage in put-call parity, 203-204 Asness, Cliff, 2 assigning probabilities, 107-110 assumed drift, 156 assumptions in Black-Scholes formula, 48-49, 100-102 effect of changes, 93 visualizing, 94-95 average annualized theta in short put option investment profile, 196 yield as, 259-261 Average Value at Risk. See CVaR (Conditional Value-at-Risk) axes in charts, 159-160

B backward equation, 102-104 behavioral finance adjusting investment profiles for, 125-128 concentrated stock example, 133-135

Benklifa, Michael, 207 Bernstein, Peter, 8 beta investing, 208 binomial option pricing model, 49-51, 54 Black, Fischer, 43-46, 55 Black-Scholes formula, 8, 43-48, 120 assign probabilities to investment profiles, 120-122 assumptions, 48-49, 100-102 full functionality in option pricing spreadsheet, 77, 79 discount factor, 84-85 effect of assumption changes, 93 Excel code, 90-93 put option pricing, 88-89 stock price median, 85-88 Stock Return, 94 Stock Return Mean, 79-82, 93 Stock Return Standard Deviation, 82-84, 94 visualizing assumptions, 94-95 history of, 44-46 notation for, 46-48 in Profit Calculator, 173-175 relationship with binomial model, 50-51 visual method, compared, 53 breakeven point, stocks versus options, 181 Brown, Aaron, 69 BXM (BuyWrite Index), 310-313

C Calc Engine, 104-107 assigning probabilities, 107-110 normal and lognormal distributions, visualizing, 112-114 315

316

INDEX

stock-only investment profile, 151-157 stock price range, setting, 110-112 Callan Associates (BXM study), 312-313 calm markets, turbulent markets versus, in option pricing outcomes, 233-238 Capital Asset Pricing Model (CAPM), 44 capital at risk in long call option investment profile, 182-183 Capital Ideas Evolving (Bernstein), 8 Capital Ideas (Bernstein), 8 CAPM (Capital Asset Pricing Model), 44 CBOE (Chicago Board Options Exchange), 43, 46, 232 charts primary and secondary axes, 159-160 standard deviation markers, 160-162 in stock-only investment profile, creating, 159-162 visualizing Excel code, 14-17 Chicago Board Options Exchange (CBOE), 43, 46, 232 Clarke, Roger, 4 comparing alternatives with visual option pricing method, 56-57 compound interest, 32-33 concentrated stocks in investment profiles, 128-138 conditional expectations, 70-73 Conditional Value-at-Risk. See CVar (Conditional Value-at-Risk) condor investment profile. See iron condor investment profile contingent CSynAs, 301 continuous compounding, 34, 37 continuous random variables, 25-26 correlations, VaR for multiple stocks, 68 cost basis, 270-273 cost basis reduction, time value in, 299 covered call investment profile, 198-200 covered calls, 285 CSynAs versus, 288-289 definition, 190 disadvantages, 286

short puts versus, 200, 202 SynAs versus, 118, 209, 213 covered percentage parameter (CSynA), 309 covered synthetic annuities. See CSynAs (covered synthetic annuities) Cox-Ross-Rubinstein model. See binomial option pricing model crossover in long call option investment profile, 184-189 stock-only investment profile, 148, 155-156 CSynAs (covered synthetic annuities), 266, 285 building, 289-295 contingent CSynAs, 301 covered calls versus, 288-289 Deere & Company example, 289-304 delta versus theta, 301-303 dividends, 303 explained, 286-289 long-term delta targets, 303-304 option sales and strike prices, 299 payback periods, 295-296, 296-299 standard CSynAs. See standard CSynAs stock-only, 301 cumulative probabilities, 24-25, 65-66 Cusick, Joe, 262 customizing utility functions, 223-225 CVaR (Conditional Value-at-Risk), 61, 69-75 conditional expectations, 70-73 with fat tails, 74-75 in stock-only investment profile, calculations, 151-154 VaR (Value-at-Risk) compared, 74

D Deere & Company example (CSynAs), 289-304 delta, 248, 252-257 as additive, 254-255 in covered call strategy, 286-288 in CSynAs, 292, 303-304 delta tables, 255-257

INDEX in long call option investment profile, 184 as number, percentage, or dollar amount, 279 price-related delta in standard CSynAs, 308 in Profit Calculator, 176 in short call option investment profile, 192 for SynAs, 215-223 in theoretical positions (TradeStation), 278 theta versus, in CSynAs, 301-303 delta tables, 255-257 Derivative Markets (McDonald), 8 derivatives, the Greeks as, 250-251 directional trading, 208 discount factor, 84-85 discrete normal distribution, creating, 28-30 discrete random variables, 25-26 diversification, VaR (Value-at-Risk) for multiple stocks, 68 for stocks and options, 68-69 dividend paying stocks in BlackScholes formula, 46 dividends in CSynAs, 303 in structured securities, 3-4 drift, 39-40, 81, 94

E Efficient Market Hypothesis, 17-18, 205 Einhorn, David, 69 e (number), 33-34, 37 Excel code. See also Calc Engine; option pricing spreadsheet; Profit Calculator assigning probabilities, 107-110 backward equation, 102-104 for Black-Scholes formula, 48 Calc Engine, 104-107 changes in pricing variables, 238-240 delta, calculating, 252-254 discount factor, 84-85 EXP(X) function, 34 forward equation, 99-100

317

full Black-Scholes functionality, 90-93 Goal Seek, 230 initial development of, 10-14 modifications to, 210-211 for Monte Carlo methods, 52 NORMSDIST function, 27, 30 probabilities within standard deviations, 62-66 put option pricing, 88-89 Stock Return Mean, 79-82 Stock Return Standard Deviation, 82-84 SynA delta adjustments, 216 visualizing, 14-17 expanding Profit Calculator for multiple options, 211-213 Expected Shortfall. See CVaR (Conditional Value-at-Risk) Expected Tail Loss. See CVaR (Conditional Value-at-Risk) expected value in Profit Calculator options section, 170-173 in random variables, 23-24 EXP(X) function, 34 extrinsic value definition, 232 effect on option pricing, 237-238

F Fama, Eugene, 46 fat tails, CVaR with, 74-75 fiduciary calls, 183, 204 financial crises, resources for information, 57-59 first-order Greeks, 248 formulas, calculating the Greeks, 249-251 Forsyth, Peter, 1 forward equation, 99-102 Fredericks, Michael, 266 fundamental/technical valuation high and low parameters (CSynA), 306-307

318

INDEX

G–H gain (loss) analysis, changes in pricing variables, 244-245 gamma, 248 CSynAs, 294 as number, percentage, or dollar amount, 279 in theoretical positions (TradeStation), 278 Geometric Browian motion, 100 Goal Seek, 230 the Greeks calculating, 249-251 definition, 247 delta, 252-257 explained, 248-249 as mathematical derivatives, 250-251 as number, percentage, or dollar amount, 279 theta, 257-263 heading formulas in Profit Calculator, 175-176 heading sections in stock-only investment profile, 148 history of Black-Scholes formula, 44-46 Hoadley.net website, 43

I “Implications for Asset Returns in the Implied Volatility Skew” (Doran), 58 implied volatility investment profiles, 120-122 modeling stock behavior, 146 price estimations using, 229-232 included stock in long call option investment profile, 189-190 insurance, put options as, 194 integrals, sums versus, 53 intent in long call option investment profile, 182-183 interest compounding, 32-33 in-the-money, definition, 232 intrinsic value definition, 181, 232 effect on option pricing, 235-238

“An Introduction to Computational Finance Without Agonizing Pain” (Forsyth), 1 investment profiles, 116, 119-120 adjusting for behavioral finance, 125-128 assigning probabilities, 120-122 company XYZ example, 227-238 calm versus turbulent market outcomes, 233-238 implied volatility calculations, 229-232 concentrated stock example, 128-138 option investment profiles. See option investment profiles payoff curves versus, 116-117 probability distributions, 116, 120 reshaping with options, 123-124 stock-only. See stock-only investment profile turbulent markets example, 138-143 utility functions, customizing, 223-225 iron condor investment profile, 205-208 “Is the Recent Financial Crisis Really a ‘Once-in-a-Century’ Event?” (Zhou), 58 IV. See implied volatility

J–K–L Jorion, Philippe, 69 Khan Academy, 194 Level 1 options approval, 285 leverage in long call option investment profile, 182-183 Lipner, Seth, 5 lognormal distribution assigning probabilities, 107-110 Calc Engine, 104-107 definition, 98-99 relationship with normal distribution, 104 resources for information, 98 visualizing, 112-114

INDEX long call option investment profile, 179-190 annualized average theta, 183-184 capital at risk, 182-183 crossover/probability, 184-189 delta, 184 fiduciary calls and protective puts, 183 included stock, 189-190 profit formula, 167 time value, 181 VaR 5.0%, 189 long put gain (loss) random variable, 168-169 long put option investment profile, 167, 192-194 lower delta adjustments parameter (CSynA), 309 low volatility in structured securities, 3-4

M market direction/volatility correlation, 245 mathematical derivatives, the Greeks as, 250-251 MathWorks/MatLab website, 43 maximum drawdown parameter (CSynA), 308 mean in random variables, 23-24 Stock Return Mean, 79-82, 93 volatility reducing, 36-37 median in stock-only investment profile, 156-157 stock price median, 85-88 Merton, Robert, 43 history of Black-Scholes formula, 45-46 micro-efficient parameter (CSynA), 307-308 Miller, Merton, 46 minimum value parameter (CSynA), 307 mistake in option pricing spreadsheet, correcting, 36-41 models, calculating the Greeks, 249-250

319

momentum parameter (CSynA), 307-308 Monte Carlo method, 40, 51-52 visual method, compared, 54-55 multiple options, expanding Profit Calculator, 211-213 multiple stock VaR (Value-at-Risk), 68

N naked calls, definition, 190 net options credit, 272 net options premium, 272 net payoff, 116 normal distribution, 26-27 assigning probabilities, 107-110 Calc Engine, 104-107 converting standard normal distribution to, 28-30 relationship with lognormal distribution, 98-99, 104 resources for information, 41 visualizing, 112-114 normal distributions, 17-18 NORMSDIST function, 27, 30 assigning probabilities, 108-109 probabilities within standard deviations, 62

O one-to-one relationships in probabilities, 65 option Greeks. See the Greeks option investment profiles building blocks of, 197-198 covered calls, 198-200 iron condors, 205-208 long call options, 179-190 annualized average theta, 183-184 capital at risk, 182-183 crossover/probability, 184-189 delta, 184 fiduciary calls and protective puts, 183 included stock, 189-190 time value, 181 VaR 5.0%, 189 long put options, 192-194

320

INDEX

put-call parity, 200-205 short call options, 190-192 short put options, 194-196 SynAs, 209-223 option payoff random variable calculating, 35 relationship with other random variables, 77-79 option payoffs, option profits versus, 167, 173 option prices changes in pricing variables, 238-245 gain (loss) analysis, 244-245 market direction/volatility correlation, 245 probability distribution, 240-242 for structured securities, 245 in turbulent markets, 242 company XYZ example, 227-238 calm versus turbulent market outcomes, 233-238 implied volatility calculations, 229-232 the Greeks. See the Greeks terminology, 232 option pricing methods binomial model, 49-51 Black-Scholes formula, 43-48 assumptions, 48-49 history of, 44-46 notation for, 46-48 relationship with binomial model, 50-51 Monte Carlo method, 51-52 resources for information, 43, 57-59 visual method, 52-57 binomial method, compared, 54 Black-Scholes formula, compared, 53 Monte Carlo method, compared, 54-55 PDEs (partial differential equations) and, 55-56 purpose of, 56-57 option pricing spreadsheet. See also Excel code correcting mistake in, 36-41 correct version of, 37-40 incorrect version of, 28-29

option payoff random variable, calculating, 35 random variables in, 27 standard normal distribution, creating discrete version of, 28-30 stock price random variable, calculating, 32-35 stock return random variable, creating, 31-32 weighted option payoff, calculating, 35-36 option profit curves, stock profit curves versus, 181 option profits, option payoffs versus, 167, 173 options profit formulas, 167 reshaping investment profiles, 123-124 option sales in CSynAs, 299 Options Analysis Workspace (TradeStation), 274-276 The Options Institute website, 43 options section of Profit Calculator, 167-178 Black-Scholes formula, 173-175 delta formulas, 176 expected values, 170-173 heading formulas, 175-176 long put gain (loss) random variable, 168-169 short put gain (loss) random variable, 169 time value and total premium formulas, 176-177

P partial derivatives, the Greeks as, 250-251 partial differential equations (PDEs), visual option pricing method and, 55-56 payback period CSynACs, 295-299 in tracking template, 273, 274 payoff curves, investment profiles versus, 116-117

INDEX PDEs (partial differential equations), visual option pricing method and, 55-56 performance, tracking SynA overview, 282-283 tracking template, 270-274 TradeStation, 274-282 platforms, calculating the Greeks, 249-250 precision. See accuracy price-related delta parameter (CSynA), 308 pricing options. See option prices pricing variables, changes in, 238-245. See also the Greeks gain (loss) analysis, 244-245 market direction/volatility correlation, 245 probability distribution, 240-242 for structured securities, 245 in turbulent markets, 242 primary axis in charts, 159-160 probabilities assigning, 107-110 in discrete normal distribution, 28-30 in investment profiles, 116, 120-122 in long call option investment profile, 184-189 random variables and cumulative probabilities, 24-25 explained, 22-23 mean (expected value), 23-24 in option pricing spreadsheet, 27 “risk-neutral” probabilities, 156 in stock-only investment profiles, 149, 155-156 within standard deviations, 62-66 probability distribution, changes in pricing variables, 240-242 Profit Calculator expanding for multiple options, 211-213 options section, 167-178 in stock-only investment profile, 157-159 profit curves, 116 for covered calls, 198-199 stocks versus options, 181 for SynAs, 215 profit formulas for options, 167

321

Profiting with Iron Condors (Benklifa), 207 Profiting with Synthetic Annuities, 115, 265 projected payback period in tracking template, 273, 274 protective puts, 183, 204, 285 put-call parity, 183, 200-205 arbitrage interpretation, 203-204 covered calls versus short puts, 202 fiduciary calls and protective puts, 204 put option pricing, 88-89 put options, as insurance, 194 put spreads, 196

Q–R quantitative finance demand for skills in, 5-6 direction of, 6-7 random variables, 120 cumulative probabilities, 24-25 definition, 21 discrete versus continuous, 25-26 explained, 22-23 long put gain (loss), 168-169 mean (expected value), 23-24 normal distribution, 26-27 option payoffs calculating, 35 relationship with option profits, 173 in option pricing spreadsheet, 27 relationships between, 77-79 resources for information, 41 short put gain (loss), 169 stock gain (loss), 158-159 stock price calculating, 32-35 volatility reduces mean, 36-37 stock return, 31-32, 94 stock return mean, 79-82, 93 stock return standard deviation, 82-84, 94 resources for information financial crises, 57-59 iron condors, 207 lognormal distribution, 98 normal distribution, 41

322

INDEX

option pricing methods, 43 put-call parity, 205 random variables, 41 reverting parameter (CSynA), 307-308 rho, 248 risk management, 18-19 “risk-neutral” probabilities, 156, 188 risk of loss. See CVaR (Conditional Value-at-Risk); VaR (Value-at-Risk) risk tolerance, CSynAs, 308

S Samuelson, Paul, 18, 45, 308 Scholes, Myron, 43, 45-46 secondary axis in charts, 159-160 second-order Greeks, 248 securities, structured. See structured securities model short call option investment profile, 167, 190-192 Short Put Gain (Loss) random variable, 169 short put option investment profile, 167, 194-196 short puts, covered calls versus, 200, 202 sources of investment return in iron condors, 207-208 spreadsheet. See Excel code; option pricing spreadsheet standard CSynAs, 304-311 BXM (BuyWrite Index), 310-313 covered percentage, 309 fundamental/technical valuation, 306-307 lower delta adjustments, 309 maximum drawdown, 308 minimum value, 307 momentum, reverting, microefficient parameter, 307-308 parameters, 306 price-related delta, 308 upper delta adjustments, 309 standard deviation markers in charts, 160-162 standard deviations probabilities within, 62-66 Stock Return Standard Deviation, 82-84, 94 VaR 5.0% calculations, 154-155

standard normal distribution, 26-27 creating discrete version of, 28-30 relationship with other random variables, 77-79 stochastic differential equations, forward equation and, 100-102 stochastic math, 120 stock and option VaR (Value-at-Risk), 68-69 stock gain (loss) random variable, 158-159 stock-only CSynAs, 301 stock-only investment profile, 146-149 Calc Engine, 151-157 chart, creating, 159-162 crossover, 148 heading sections, 148 probabilities, 149 Profit Calculator, 157-159 stock price range, 149 SynAs versus, 131-132 testing the spreadsheet, 162-166 VaR 5.0%, 149 stock price calculating, 32-35 drift, 39-40 normal distributions, 17-18 relationship with other random variables, 77-79 volatility reduces mean, 36-37 stock price median, 85-88 stock price range setting, 110-112 in stock-only investment profile, 149 stock profit curves, option profit curves versus, 181 stock return, 94 creating, 31-32 relationship with other random variables, 77-79 stock return mean, 79-82, 93 stock return standard deviation, 82-84, 94 stocks concentrated in investment profiles, 128-138 in long call option investment profile, including, 189-190 as underlying securities, 249 strike prices in CSynAs, 299

INDEX structured securities, 116 criticism of, 4-5 growth in, 2-3 low volatility and high dividends, 3-4 structured securities model changes in pricing variables, 245 option investment profiles. See option investment profiles Profit Calculator, options section, 167-178 purpose of, 145-146 stock-only investment profile. See stock only-investment profile sums, integrals versus, 53 SynAs (synthetic annuities), 117-119, 209-223. See also CSynAs (covered synthetic annuities) aggressive approach, 119 concentrated stock example, 128-138 covered calls versus, 118, 209, 213 covered SynAs, 266 creating, 118-119 delta and theta adjustments, 215-223 Profit Calculator, expanding for multiple options, 211-213 profit curve, 215 spreadsheet modifications, 210-211 stock-only positions versus, 131-132 tracking performance, 282-283 in turbulent markets, 138-143 utility curve, applying, 135-138 VaR 5%, 215 yield, 213 synthetic call, creating, 203

T tail (of distribution) cumulative probabilities, 65-66 fat tails, CVaR (Conditional Value-at-Risk) with, 74-75 Taleb, Nassim, 69 testing stock-only investment profile spreadsheet, 162-166 theoretical positions, 270, 276-279 theta, 248, 257-263 annualized average theta, 183-184, 196 in covered call strategy, 288 CSynAs, 294

323

definition, 183 delta versus, in CSynAs, 301-303 importance of, 261 as number, percentage, or dollar amount, 279 for SynAs, adjustments, 216-223 in theoretical positions (TradeStation), 279 theta tables, 258-259 vega tables, 262-263 yield as average annualized theta, 259-261 theta tables, 258-259 Thomsett, Michael, 6, 237 time decay, definition, 232 time value definition, 181, 232 effect on option pricing, 235-238 in long call option investment profile, 181 in short call option investment profile, 190-192 time value formulas in Profit Calculator, 176-177 time value premium, definition, 232 total premium formulas in Profit Calculator, 176-177 tracking performance SynA overview, 282-283 tracking template, 270-274 TradeStation, 274-282 tracking template, 270-274 cost basis, 272 example of, 273-274 projected payback period, 273 trade triggers, 272 TradeStation, 274-282 Options Analysis Workspace, 274-276 theoretical positions, 276-279 transaction records, 280-282 trade triggers in tracking template, 272, 274 transaction records, 280-282 trend assumptions, 189 Treynor, Jack, 44 turbulent markets calm markets versus in option pricing outcomes, 233-238 changes in pricing variables, 242 synthetic annuities in, 138-143

324

INDEX

U–V uncovered calls, definition, 190 underlying securities, stocks as, 249 upper delta adjustments parameter (CSynA), 309 utility curve, 125-128 applying to SynA profile, 135-136 concentrated stock example, 133-135 utility functions, customizing, 223-225 Value-at-Risk. See VaR (Value-at-Risk) VaR 5.0% calculations with standard deviations, 154-155 in long call option investment profile, 189 in stock-only investment profile, 149, 151-154 on SynAs, 215 VaR (Value-at-Risk), 66-68 criticism of, 69 CVaR (Conditional Value-at-Risk) compared, 74 definition, 66-67 formula approach, 67-68, 71 for multiple stocks, 68 for stocks and options, 68-69 variables pricing variables, changes in, 238-245 random variables. See random variables vega, 248 vega tables, 262-263 virtual dividends, 129 visualizing assumptions, 94-95 Excel code, 14-17 normal and lognormal distributions, 112-114

visual option pricing method, 52-57 binomial option pricing model, compared, 54 Black-Scholes formula, compared, 53 Monte Carlo method, compared, 54-55 PDEs (partial differential equations) and, 55-56 purpose of, 56-57 volatility implied volatility. See implied volatility market direction/volatility correlation, 245 reduction of mean, 36-37 stock price median, 85-88 in stock return random variable, 31 stock return standard deviation, 82-84, 94 in structured securities, 3-4 synthetic annuities in turbulent markets, 138-143

W–Z warrants in history of Black-Scholes formula, 44 websites. See resources for information weighted option payoff, calculating, 35-36 weighted outcomes in random variables, 24 Weiner process, 100 Whaley, Robert, 312 Wilmott, Paul, 6 written call options. See short call options written put options. See short put options yield as average annualized theta, 259-261 on SynAs, 213