ASM SOA Exam IFM: Study Manual [1 ed.] 9781647565022

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a,s,m SOAExamIFM Study Manual

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1st Edition Abraham Weishaus,Ph.D., F.S.A.,C.F.A.,M.A.A.A. NO RETURN IF OPENED

a,s,m Act11aria/ShuiyMaterials

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©Copyright 2018 by Actuarial study Materials {A.S.M.), PO Box 69, Greenland, NH 03840. All rights reserved. Reproduction in whole or in part without express written permission from the publisher is strictly prohibited. 1'1~

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Contents 1

Introduction to Derivatives Exerci.es Solutions

2

Projec!Analysis 2.1 NPV. 2.2

2.3 10 11 14

3

Monie Carlo Simulation Exerci.es Solutions

19

23 39

4

Efficient Markets Hypothesis Exercises Solutions

(EMH)

5

Mean-Variance Portfolio Theory Exercises Solutions

59 6S

6

Capital Asset Pricing Model (CAPM) 6.1 Required return 6.2 CAPM Exerci.es Solutions

77 77

53 56 57

72

"'

81 85

89 89 90 92 98 8

Exercises Solutions

103 103 lOl 106 107

Capila!Struclure

109

Exercises

112

Behavioral

Finance and Mullifactor

Models

8.1 8.2

9

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iv

CONTENTS

Solutions

115

10 The Effect of Taxes on Capital Structure Exercises Solutions

119 121

11 Other Factors Affecting Optimal Debt-Equity Ratio 11.1 Bankruptcy and Financial Distress 11.2 Agency costs and benefits 11.3 Asymmetric information Exercises Solutions

127 127 129 130 131 135

12 Equity Financing 12.1 Sources of capital for private companies 12.2 Allocation of company value among investors 12.3 Going public the !PO 12.4 IPOpuzzles Exercises Solutions

139 139 139 141 142 143 145

13 Debi Financing 13.1 CorporateDebt 13.2 Other Debt Exercises Solutions

147 147

14 Forwards

151

123

148 149 150 151 152 155 155 156 156 157 161

14.3.2 Compar 14.4 Syntheticf Exercises Solutions

165 165 166 167 170

15 Variations on the Forward Concept 15.1 Prepaid forwards

15.2 Futures Exercises Solutions

173 173 175

16 Options 16.1 Calloptions

16.2 Put options Exercises Solutions

176 181 185 185 187

17 Option Strategies 17.1 Options with Underlying As.sets 17.2 Synthetic Forwards lFMStudyManuaJ-i"edilion Copy'W't02!118ASM

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CONTENTS

17.3 Bear Spreads, Bull Spreads, and Coll 17.3.1 Spreads: buyinganoptionan 17.3.2 Collars: buying one option 17.4 Straddles, Exercises Solutions

188 188 192 193 196 206

18 Put-Call Parity 18.1 Stock put-call parity. 18.2 Synthetic stocks and Treasuries 18.3 Synthetic options 18.4 Currency options Exercises Solutions

211 212 213 214 215 217 224

19 Comparing Options 19.1 Bounds for Option Pri 19.2 Earlyexerci 19.3 Timetoexp 19.4 Differents 19.4.1 Tore 19.4.2 Op Exercises Solutions

2:l3 233 235 238 242 242

20 Binomial Trees-Stock, One Period 20.1 Risk-neutral pricing 20.2 Replicating portfolio 20.3 Volatility Exercises Solutions

259 259 261

21 Binomial Trees-General 21.1 Multi-period binomial trees 21.2 American options 21.3 Currency options 21.4 Futures 21.5 Other assets Exercises Solutions

287 287

250 251 253

265 268 276

288 289

291 294 295

30l

22 Binomial Trees: Understanding Early Exercise of Options Exercises Solutions

319

23 Modeling Stock Prices with the Lognormal Distribution 23.1 The normal and lognormal distributions 23.1.1 The normal distribution 23.1.2 The lognormal distribution 23.1.3 Jensen's inequality 23.2 The lognormal distribution as a model for stock prices. 23.2.1 Stocks without dividends

32.3 323 323 324 325 32h 32h

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320 321

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CONTENTS

328 329 330 336 338 24 The Black-Scholes 24.1 Black-Scholes 24.2 Black-Scholes 24.3 Black-Scholes currency Solutions

Formula Formula for common stock options . formula for currency options formula for options on futures

345 34h 349 350 351 357

25 The Black-Scholes 25.1 Greeks

Formula: Greeks

367

36' 36' 373 374 374 379 381 386 387 387 389 389 391 391 398

25.1.7 G 25.2 Elasticity 25.2.1 El 25.2.2 R 25.2.3 El 25.3 Whatwil Exercises Solutions 26 Della Hedging 26.1 Overnight profit on a delta-hedged portfolio 26.2 The delta-gamma-theta approximation. 26.3 Hedging multiple Greeks 26.4 What will I be tested on7 Exercises Solutions

41)7

27 Asian, Barrier, and Compound 27.1 Asian optio 27.2 Barrieropti 27.3 Maximaan 27.4 Compound 27.4.1 Co Exercises Solutions

433 433 437 440 442 443 444 451

407 411 413 415 416 424

Options

459 459 459 461 463

28 Gap, Exchange, and Other Options 28.1 Gap options 28.1.1 Definition of gap options 28.1.2 Pricing gap options using Black-Scholes 28.1.3 Delta hedging gap options. lFMStudyManuaJ-i"edilion Copynght02018ASM

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vii

CONTENTS

464 466 466 467 469

"" 29 Supplementary Questions-Derivatives Solutions

487 495

30 Real Options 30.1 Decision trees 30.2 Black-Scholes

505 505

507

"" 509 511

31 Actuarial Applications of Options 31.1 Variable annuities 31.2 Mortgage guar 31.3 Other insuranc 31.4 Static hedging 31.5 Dynamic hedgi 31.6 Hedging c Exercises Solutions

515 517 517 518 519 519 520

Practice Exams

525

515

522

1

PracliceF.xaml

527

2

PracliceF.xam2

537

3

PracliceF.xam3

547

4

PracliceF.xam4

557

5

PracliceF.xamS

567

6

PracliceF.xam6

575

7

PracliceF.xam7

583

8

PracliceF.xam8

591

9

PracliceF.xam9

599

10 PracliceF.xaml0

607

11 PracliceF.xamll

617

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viii

CONTENTS

Appendices A Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions

627 629 629 635 642 650 657 665 673 681 689 697 707

for the Practice Exams for Practice Exam 1 for Practice Exam 2 for Practice Exam 3 for Practice Exam 4 for Practice Exam 5 for Practice Exam 6 for Practice Exam 7 for Practice Exam 8 for Practice Exam 9 for Practice Exam 10 for Practice Exam 11

8 Solutions lo Old Exams 8.1 Solutions to SOA Exam MFE, Spring 2007 8.2 Solutions toCAS Exam 3, Spring 2007 8.3 Solutions toCAS Exam 3, Fall 2(J(J7 8.4 Solutions to Exam MFE/3F, Spring 2009 8.5 Solutions to Advanced Derivatives Sample Questions

717

C Lessons Corresponding

743

lo Questions on Released and Practice Exams

D Standard Normal Distribution

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717 721 724 728 731

Function Table

747

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Preface Welcome to the !FM exam1 This course combines Corporate Finance and Oerivahve Instruments. topics:

You will study the following

1. Investment risk and project analysis 2. Efficient markets hypothesis 3. Mean-variance portfolio theory 4. Capital as.set pricing model (CAPM) 5. Capital structure 6. Debt and equity financing 7. Forwards and futures 8. Call and put options 9. Strategies combining call and put ophons 10. Principles relating prices of calls and puts, and relating prices of op hons to each other and bounding them. These principles do not develop exact prices for options, but are general and easy to derive. 11. Valuing options using binomial trees. 12. The lognormal mOOel for stocks. 13. Valuing options using analytic methOO.s (Black-Scholes). 14. Definitions of exohc options, and pricing methOOs forexohc options where available. 15. Real options 16. Actuarial applications of option pricing The JFM exam is a 3--hourexam with 30 questions. The following table gives the syllabus topics and their weights, and the lessons discussing each topic Topic

Weight

Mean-Variance Portfolio Theory Asset Pricing MOOels Market Efficiency and Behavioral Finance Investment Risk and Project Analysis Capital Structure Introductory Derivatives-Forwards and Futures General Properhes of Options Binomial Pricing Models Black-Scholes Option Pricing MOOel Option Greeks and Risk Management

10-15% 5-10% 5-10% 10-15% 10% 5-10% 10-15% 10% 10-15% 5-10%

Lessons

.... 5

4,8 2-3,30 9-13 1,14-15 16--18 19-22 23-24,27 25,26,31

The syllabus includes parts of two textbooks, and two study notes. Download the syllabus from IFMStudyManual-i"edilion CopynghtC2!118ASM

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PREFACE

https: / /www.soa.org/Files/Edu/2&18/2&18-exam-ifm-syllabi.

pdf,

and refer to the last page of the syllabus for links to the two study notes. One textbook and one study note deal with Corporate Finance. The other textbook and study note deal with Derivative Instruments. If you use the Derirn/ives Markets textbook, refer to the following website for errata: http:/

/derivatives.

kellogg .northwestern.

edu/errata3e

.html

This manual This manual gives you complete cm·erage of all syllabus topics. The corporate finance topics, for the most part, come first. However, the first lesson is based on the Derirntires Markets textbook. This material is basic, and in parhcular includes the definition of shorting an asset, a concept that appears in the corporate finance part of the course, so it comes first. Also, one corporate finance topic, real options, was placed after most of the derivatives material, since, after all, a real ophon is an option, and one of the formulas for valuing it, Black-Scholes, is cm•Eeredin the derivahves part of the course. The two textbooks deal with interest rates differently. Corpamle Fi11a11ce uses annual effective interest rates, whereas Derivatives Markets uses continuously compounded interest rates. Exams will be clear as to the meaning of the interest rate they provide. But in this manual, if you're not told othenvise, assume that an interest rate discu~d in a lesson in the corporate finance part is annual effechve and an interest rate in the derivatives part is continuously compounded. To help you check how much you are learning, there are quizzes within most lessons; these are straightfonvard exercises which you should work out as you get to them. Solutions to quizzes are at the end of the lesson, after the solutions to the exercises. The exercises at the end of each lesson are designed to be exam-like (although sometimes they are a bit long for an exam), requiring only a calculator and a normal distribuhon calculator (like a spreadsheet program) to solve. Working these out will help you learn the concepts. Note the following valuable features at the end of the manual: • Eleven practice exams. • Solutions to relevant questions from all released exams: the Spring 2007 and Spring 2009 Exam MFE and the Spring 2007 and Fall 2007 CAS Exams 3 .. • Solutions to all sample queshons on derivahves. The solutions provided by the SOA are longer and often include additional commentary and educational material. My solutions get to the point and are meant to indicate the method you'd use on an exam. The SOA provides sample questions for Corporate Finance as well. See the syllabus for the links to the ~mple questions. References to these sample questions are provided at the ends of exercise sets of lessons related to those ~mple questions. • A cross-reference indicating the lesson covering every relevant queshon from the released exams and the ~mp le questions, and a similar cross-reference for all questions in the eleven prachce exams. • An index. A note on notation: the McDonald textbook uses N(x) to indicate the cumulative standard normal distribuhon at x. Many other textbooks use (x)for the same concept, and in fact J use (x)in the Exam LTAM, MAS-I, and STAM manuals. However, for this course, J follow McDonald and financial economics tradition and use N(x). IFMStudyManual-i"edilion CopynghtC2!118ASM

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PREFACE

The normal distribution table Formulas in this course use the normal distribution. Most students will be taking this exam at a Pro metric site. Prometric provides a standard normal distribution calculator. See http://www.

prometric.

com/SOA/l1FE3F_calculator

.htm

to .ee how it works. The calculator provides values of the cumulahve normal distribution function and itsinve15et0Sdecimal places. When working problems in this manual, you may use Excel's NORJ1SDISTand NORJ1SINVfunctions to perform normal distribution calculations in a manner similar to the way you will do them on an exam. A normal distribution table is provided in Appendix D. You can u,e this in a pinch if you don't have access to a calculator or program that can calculate normal distribuhon values. At the exam, you will have access to a formula sheet. sheet. https:

/ See

-table.

pdf

for /www.sea.org/Files/Edu/2&18/exam-ifm-cbt this Jt sheet. provides standard normal the function, the lognormal the funchon, density of moments lognormal distribution, and formulas for the the greeks 25). If you are taking the exam at a paper-and-pencil site, you will be given a formula sheet and a cumulative normal distribution table. Currently, the url for this table is https:

/ /www.soa.org/Files/Edu/2&18/exam-ifm-pp-ta.ble.

pdf

The formula sheet has rules for the use of the table, indicahng that you should not interpolate in the table; use rounded values. Read those rules for more details. Since most students will be taking the exam under CBT, this manual uses the more exact methOO of calculahng normal distribution values, namely 5-place precision.

Errata Unfortunately, there are likely to be more errors than usual in the Corporate Finance material which was newly added to this manual. Please report all errors to the author. You may ,end them to the publisher at [email protected] or directly to me at [email protected]. Plea,e identify the manual and edition the error is in. This is the 1' 1 edition of the Exam JFM manual. An errata list will be posted at http://errata.aceyourexams.net. Check this errata list frequently.

Acknowledgements I would like to thank the SOA and CAS for allowing me to use questions from their old exams. The creato15 of TI,X, J!.\TI,X, and its multitude of packages all de.erw thanks for making possible the profe~ional typesetting of this mathematical material. I'd like to thank Yeng Miller-Chang for poinhng out some errors ina draft of this manual. J wish to acknowledge all students who sent in errata or recommendations for improvements for the former MFE manual, whether many or few. Special thanks to Jeff Raven for his mathematical insights. A partial list of other students who pointed out errors in the former MFE manual: Mark Adams, Jonathan Akerman, Anas Al-Hammouri, German Altgelt, William Torres Amesty, Ryan Amman, Tophe Anderson, Michael Aronowitz, Dan Baker, Daniel Balzekas, Etai Barach, Elizabeth Barclay, Bethany Barger, Andrea Tsai Barker, Michael Baznik, Andrew Bender, Samuel Benidt, Sheryl Berman, Jean-Philippe Bernier, Eli Bochner, Batya Bogopulsky, Marc Bow.er, Howard Brandon, Nicholas Braniecki, David Bre~ler, Arishdes Briceno, Ken Burton, Wu Cao, Yang Cao, Mitchell Carr, Michael Carvell, Jeff Cecil, Garland Chan, Grace Chang, Jacob Chapman, Aaron Chase, Baiyu Chen, Esther Cheng, Jacky Chew, lFMStudyManuaJ-i"edilion Copynght02!118ASM

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PREFACE

Axiom Choi, Fuiying Chong, Alice Chu, Pauline Chung, Steven Clark, Brian Cohen, TOO.dCorley, [kmglas Cox, Jessica Crumrine, Brandon Cruz, Sue Curbs, Thomas Daly, Cameron Daniel, Jackie Daniels, Lindsey Daniels, Hannah David, Andrew de Leon, Christopher Derasmo, Magdalena Diedrich, Patrick Digan, Elizabeth Dinehart, David Dobor, Trevor J:kJnahue, Bryan Donkersgoed, Danielle Drinkmann, Andries Du Preez, Dominic Duke, Jacob Efron, Ian Elliott, Matthew Englebert, Yochewd Ephrathi, Matthew Estes, Jeremy Evans, Joe Famularo, Shoaib Farooq, Amarya Feinberg, Yucheng Feng, Daniel Fernandez, Erin Flickinger, Nechama Florans, Patrick Ford, Josh Fox, Brian Frey, Grace Gao, Joseph Garlinsky, Georges Ghannam, Aryeh Gielchinsky, Yoram Gilboa, Michael Goldstein, Maria Marin Gonzalez, Arthur Goodell, Joseph Gracyalny, Carl Grafmuller, Travis Grasse], Shaun Griffin, Jemmy Gu, Jenny Guan, Alison Guest, David Haarmann, Aaron Hahn, Brian Hall, Scott Handley, Marshall Li Hanhua, Tommy He, Flynn Heiss, William Henderson, Abby Hirshkowitz, Jennifer Ho, Quang Hoang, Symphone Home, Susan Home, Stephanie Horst, Brandon Howard, Henry Huang, Scott Humphries, Colin Hwang, Doyun Hwang, Matthew lseler, Gillian Jackson, Phil Jang, Derek Jansen, Matthew Jewczyn, Yong Jiang, Steven Jones, Randy Kabeya, Hyson Kang, Derrick Kaufman, Andrew Kim, Frank Kirby, Stew Koh, Parker Koppelman, &nily Kozlowski, Daniel Kramer, Reuvain Krasner, Neal Kroells, Anna Krylova, Takehiro Kumazawa, Duke Kung, Asher Kurland, Jacqueline Kusnitz, Chrishna Lai, Chris Lally, Alex Larionov, Gary Larson, Pascal Lataille, An Le, Dominic Lee, Joseph Seung Lee, York Lee, Aw Yong Chor Leong, Jason Leong, Jimmy Lim, Charles Lindberg, Joyce Liu, Allison Louie, David Lovit, Sheryn Low, Yitzy Lowy, Yuanzhe Ma, Megan MacEwen, Wilson Mack, Wilson Mak, Christina Malleo, Chris Manhave, Cipriano Mascote, Gloria Mascote, Jason Mastrogiacomo, Russell Mawk, Kevin McBeth, Stuart McCroden, Jake McDougle, Johnny Mc Hone, Philip McKeman, Julie Meadows, Max Mehta, Xiangchen Meng, Marston Miller, Joseph Molinar, Paul Mollema, Mitchell Momanyi, Jose Monteiro, Monica Morales, Michael Morrison, Leigh Murdick, Christopher Nahas, Evan Nash, Alison Neilson, Brett Ni, Kyle Nobbe, Kwame Okra, Arthur Okura, Dan O'Toole, Nick Panei, Samir Patel, Snehal Patel, George Pavlis, Erin Pearse, Nicholas Pekarek, Sitong Peng, Lindsey Peniston, Cole St. Peter, Frank Phu, Kim Ming Phun, Dustin Plotkin, Yeshaya Pollack, Amanda Popham,Justin Pribble,Joshua Price,Claudio Rebelo, Todd Remias, Fillard Rhyne, Dana Royal, Maurice Rubinraut, Philip Sandager, Trisha Sanghvi, Vikas Saraf, David Schenck, David Schmitz, Dan Schobe 1,Simon Schurr, Melissa Sharma, Jaren Shigeta, David Sidney, Chris Siew, Yaacov Silber, Carl Simon, Professor Diane Skrzydlo, Lindsey Smith, SPM Soalan, Corey Spitzer, Rob Stoddart, Alison Stroop, Jesse Sulfridge, Shine Sun, Ana Swierczewski, Nicholas Sundgaard, Jamie Swanson, Susan Szpakowski, Khoon Yu Tan, George Tankov, David Tate, Sudeshna Thomas, Nan Tian, Shelly Tillman, Geoff Tims, Terrence To, Mayer Toplan, Leon Travis, Chaim Tropper, Sam Tsang, Siyu Tu, Paul Uchida, Nadia Vaughan, Jim Vegeais, Edward Wang, Koe Hien Wang, Chris Washburn, Leighton Weese, Sal Wentum, Brant Wheeler, Garrett Williams, Karl Willman, Jamie Wong, Victor Wong, Jeff Wood, Nathan Wmx:larL, Mark Woods, Karissa Wysocki, Rodger Yan, Haoxiu Yang, David Yanick, Dennis Yano, Philip Yourno, Eric Yskes, Binlu Yu, Simon Yuen, Ben Zaugg, Aaron Zeigler, Jeffery Zeitler, Cheng Zhang, Sandy Zhang, Vanek Zhu, Julia Zvenigomdsky. I'd also like to thank Professor Warren Luckner for pointing out several errors in the former MFE manual, and Professor Kr1:ysztof Ostaszewski for pointing out an error in that manual.

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Lesson I

Introduction to Derivatives Reading: Derivatives Markets 1.1, 1.2, 1.4, 1.5 This course deals with pricingderivahve securities. A derirnlive security, usually just called a derirntire, is a financial instrument whose value depends on another security. An example is an option: an agreement allowing the buyer of the option to buy or sell an asset at a specific price on a specific day. To trade financial as.ets, there are four steps:

1. Buyer and seller must find each other. Financial markets, such as stock exchanges, facilitate this. 2. Trade is cleared, meaning that both sides must specify their obligations, to pay or to hand m·er a~et. Trades on financial markets are cleared thmughcleari11glw11ses. The clearinghouse matches the buyers and sellers and keeps track of their obligahons and payments. For derivative securities, there are often obligations between the buyer and seller after the sale date. The clearinghouse acts as the intermediary beh\~en the buyer and seller to facilitate these payments. 3. Trade is settled. Usually one side pays, the other side provides the asset. 4. Ownership records are updated. An alternative to trading through financial markets for large buye15 and sellers isorer-lhe-co1111/l'T (OTC) trading. Advantages of OTC are:

1. Easier to trade large quantity directly, avoiding exchange fee and market tumult. 2. OTC may create custom financial a~ets not available on financial markets. 3. Can trade many financial assets in a single transaction. Four measures of market size and activity are

1. Tradi1tgwhmre is the number of units that change hands in a period (a day or a year). For stocks a unit is one share; for options a unit is one option, but typically a stock option has 100 shares of stock underlying it. So trading volume isa somewhat arbitrary measure. Trading volume is a measure of marketachvity. 2. Market ro/ue of a company is the value of a company on an exchange based on the price of its stock. If a company has 100 million shares of stock outstanding and the market price of the stock is $.50, then the company's market value isS billion. (More generally, market value is the price of an asset.) 3. Notimwl rnhie is the value of a derivative relative to some underlying a~t. For example, an ophon may pay the excess of the prevailing interest rate over 6% on a nohonal value of 1,000,000. If the prevailing interest rate is 7%, this option pays (0.07 - 0.06)(1,000,000) = 10,000. (A more precise discu~ionof this type of option, an interest rate cap, will occur later in the course.) Notice that the notional value is just a reference value; there is no implication that someone has a loan for 1,000,000. Notional value is a useful measure of the market size of derivatives. 4. Opeu i11IITes/is the number of contracts for which there isa future obligahon for one party to perform. Jt is a useful measure of the market size of derivatives. Derivatives serve the following purposes: lFMStudyManuaJ-i"edilion Copynght02!118ASM

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1. INTRODUCTION TO DERIVATIVES

1. Risk management. They allow an individual or company to pass on risk to another entity. For example, one can use an option to guarantee the price of a needed item in the future. Eliminahng risk in this manner, by guaranteeing a buying or selling price, is called hedging. Insurance is a derivative. For example, auto collision and comprehensive insurance guarantees the price of a car if it is damaged or stolen. 2. Speculation.This is the opposite of risk management. One may believe that the price of an asset will increase, and by using ophons,can make a bet on the price increase with little im·estment. 3. Reducedtransactioncosts. For example, rather than selling an asset, which involves transachoncost:s,

one may use a derivahve to achieve the same effect. 4. Regulatory arbitrage. Selling an asset may involve realizing a taxable capital gain. Achieving an equivalent result by using an option may postpone taxes. Also, one may maintain voting rights in the underlying stock by using an option instead of selling the stock. There are three perspechves for derivatives: 1. End user. This is the one who buys the derivahve: the individual or the corporation. 2. Market-maker.This is the dealer, the one who sells the derivative.

3. Economicobserver. This is the gm·emment regulator, or the economist who is understanding analyzing the markets.

and

The stock market is liquid because of market-make!';, who stand ready to buy or sell. They earn their living by selling at a price higher than they buy at. The bid price is the amount they will pay for an asset and theofferpriceorask price is the price you can buy an asset for. The difference between these two is the bid-askspread. Nohce that "bid" and "ask" is from the market-maker's perspective. The bid price is what the market-maker bids for the asset, not what the buyer bids. 1A If the bid price on a stock is 29.75 and the ask price is 30.25, how much would you pay for 100shares? Ex.AMPLI!

ANswl!R: 100(30.25)

=~

There are many kinds of order for stocks. A marketorderpays the market price (the ask price) to buy the stock immediately, or sells at the bid price immediately. A limit orderspecifies the maximum buying price or the minimum selling price, and may not be fulfilled immediately if that price is not available. A stop loss sales order specifies that the stock is sold if the price decreases to the specified amount. The actual sales price may be less since the stock is sold at the market price. A long position in a financial instrument is one with a posihve number of units, one in which the instrument was bought. A short position is one with a negative number of units, one in which the instrument was sold. The words "long" and "short" are also used as verbs and nouns. To long a stock means to buy it; to short it means to sell it. The short is the person who sold something short and the long is the pe15on who bought something. Short selling serves the following three purposes: 1. Speculation.If you think the price of an asset will drop, you short it. 2. Financing. When you short sell, you receive cash for the asset you sold. It is a way of borrowing money. (However, if you don't own the asset, you may need the cash for collateral, as we will discuss.) 3. Hedging. Shorting an asset you own is a way to protect yourself against a price drop. IFMStudyManual-i"edilion CopynghtC2!118ASM

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EXERCISESFOR LESSON I

How does short-selling a stock work? Suppo,e you short-,ell stoi::kto Bob. A broker lends you shares of a stoi::k. Typically the owner of the stoi::kdoesn't even know about this lending. You sell the stoi::kto Bob and receive the price. If the stock pay,; dividends, you must pay those dividends to the lender. These dividend;; are taxable to the lender and tax-aserail'. In general, the lea,e rate is the annual cost of holding ana~et as a percentage of the as.et value, Since you bormwt~d the stock, the lender requires collateral to assure that you will purcha,e the stock back. Typically the collateral is greater than the initial value of the stoi::k,to cm·er potential price increa,es. The extra amount is called the haircut. The lender pay,; interest on this collateral, but the interest rate may be less than the market interest rate. The interest rate paid by the lender is called the ll'Po ratefor bonds andtheshorlll'balt>forstocks.

Exercises 1.1. 5200 options, each of them on 100 shares of stoi::k,expire at some time in the future. The current price of the stock is 33 and the current price of the ophon is 25.50. Calculate (a) The market value of the options. (b) The nohonal value of the ophons. (c) The open interest. 1.2 Suppo.e every owner of an option on 100 shares of stock trades the ophon in for hvo options on shares of stock each. How would this affect the market value, notional value, and open interest of the options 7 ~

1.3. Which of the following are hedges and which are speculations

7

(a)

An investor expects the price of a dollar relative to a euro to appreciate, and buys dollars.

(b)

A U.S. company will receive a payment in euros in 6 months and buys an option to ,ell euros for dollars at the end of 6 months.

(c) Acom farmer short .ells com three months before the harvest. 1.4. A 3-year zero-coupon bond with maturity value 1000 has an annual effective interest rate of 6%. The bond is short-sold, with collateral of 110% of its value. After one year, the position is closed: the bond is bought back. At that time, the annual effective interest rate of the bond is 7%. There are no commissions paid when buying or ,elling the bond. The interest rate paid on the loan to provide the collateral is 6%. Net profit at the end of one year is 7.32. Determine the repo rate. 1.5. 100 shares of a nondividend paying stock with bid price 52.25 is shorted. Collateral is 120% of the stock's value, the price at which it may be purchased from the market-maker. The investor pay,; 5% effective annual interest on cash borrowed to ,et up the collateral. The short rebate is 4%. At the end of one year, the position is closed. At that time, the bid price is 50.75. At both the beginning and the end of the year, the bid-ask spread is 0.50. Determine the net profit on the short sale.

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1. INTRODUCTION TO DERIVATIVES

[Introductory Derivatives Sample Question 24] Determine which of the following statements is NOf a typical reason for why derivative securities are used to manage financial risk. (A) (B)

(C) (D) (E)

Derivatives Derivatives Derivatives Derivatives Derivatives

are are are are are

used used used used used

asa means of hedging. to reduce the likelihoOO of bankruptcy. to reduce tran;;achon costs. to sahsfy regulatory, tax, and accounting constraints. as a form of insurance.

Solutions (a) The market value is5200(25.50) = 1132,bOOj. (b) The nohonal value is 5200(100)(33) = 117,lbO,OOO 1(c) The open interest is 5200, since there are 5200 contracts with future obligations. 1.2 The market value is unaffected; two of the new options are worth the same as one old ophon. The notional value is unaffected, since this is based on the underlying stock. The open interest, however, is doubled, since open interest is a count of the number of contracts. 1.3. The investor is speculating, but the other two are hedging. The U.S. company wants to a~ure getting a fixed number of dollars, and the com farmer wants to a~ure receiving a fixed price for the com. 1.4. The original price of the bond is 1000/1.06 3 = 839.62 and the ending price is 1000/1.0?2 = 873.44. Thus 873.44 - 839.62 = 33.82 was the net amount spent repurchasing the bond. The cost of the loan to provide the collateral is 0.06(0.1)(839.62) = 5.04. Total interest on the collateral must therefore be be 33.82+5.04 +7.32 = 46.18. The collateral is 1.1(839.62) = 923.58, so the repo rate is46.18/923.58 = ~1.5. The price the stock is sold for is the bid price, or 100(52.25) = 5225. Collateral is based on the ask price, so it is 100(52.75)(1.2) = 6.330.The investor must borrow 6.330-5225 = 1105. Thus the investor pays 1105(0.05) = 55.25 interest and receives 6330(0.04) = 253.20 interest. At the end of the year, the investor pays (50.75 +0.50)(100) = 5125 for the stock. Adding all the transactions together. 5225-55.25+253.20-5125=~

(B) is not a typical reason to usederivahve securihes.

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Lesson 2

Project Analysis Reading: Carparale Fil1mrce8.5, JFM 21-18 2-3 This lesson begins the corporate finance part of the course. From here up to and including Lesson 13, when not told othenvise, assume that interest rates are annual effective.

2.1 NPV When a company considers embarking on a project, it must verify that this project will meet the company's financial goals. The measure we will u,e for this is NPV, or net present value. To compute the NPV we calculate the free cash flows of the project. The free cash flows are the cash amounts generated by the project itself, both positive and negative, year by year. Cash flows do not include non-cash accounting items, such as depreciation.' The free cash flows also do not include cash flows from financing used to support the project. If a loan is taken to pay the project's initial expenses, neither the loan nor interest on the loan is part of free cash flows. Free cash flows are purely cash generated by the project itself. The NPV is the present value, at the start of the project, of the project's free cash flows. At what interest rate is the NPV calculated 7 Usually the NPV is calculated at the interest rate the company must pay to finance the project. In other words, the NPV is calculated at the interest rate that has to be paid to investors in order to get them to invest in the project. This interest rate is called the cost of capital. We will discuss what the cost of capital should be in the following lessons. Ex.AMPLI! 2A A life insurance company is considering developing a new Universal Life product. It will cost $5 million, payable immediately, to develop this product, and developing the product will take a year. The company estimates free cash flows in subsequent years will be $-1 million at the end of the first year, followed by $1 million per year for 5 years, and will then dec~ase at a compounded rate of 10% per year after that. 1 L l 'r ~ C The company's cost of capital is 12%. Calculate the NPV of the project.

ANswl!R: The NPV generated during the first 6 years, in millions, is -5 - 1.~22 +

~~;~2 = -5.797194 + ~-~2(~\:;; 1

= -2.923488

After 6 years, free cash flows form a geometric series with first term 1/1.12 7 and ratio 0.9/1.12. The NPV generated after year 6, in millions, is 1/1.12 7 1 - o.9/1.12 Total NPV is -2,923,488 + 2,302,869

=1-620,620 (

=2 •302869

~

_I_ I -

~

O.\L -+-0-I

,.p ... 'I-I°""'"''for insurance pmduct5. they indude chang,,5 in r.,s,,rws. A rompany must s,,t aside cash to supp 0. Otherwise they destroy the value of the company.

If we assume that free cash flows are constant, they form a perpetuity. As you learned in Financial Mathematics, the present value at of an immediate perpetuity of 1 per year is 1/i, where i is the interest rate. If the free cash flows are 1 in the first year and grow at compounded rate g, then their present value

NPV =

i:

(l + g)"-l ""'\ (1 +1)" 1/(l+i) 1-(1 +g)/(1 +i) 1/(1 +i) 1 (i-g)/(1+i)

=t=g

(J)

(2.1)

Quiz 2-1 A company is considering a project. This project will require an investment of 10 million immediately and will generate free cash flows of 1 million per year at the end of one year, increasing at a compounded rate of 3% per year perpetually. The cost of capital is 9%. Calculate the NPV of the project.

2.2 Project analysis 2.2.1 Break-even analysis Companies analyze the risk in a project. One way to analyze the risk is to vary the assumptions u,ed to calculate the NPV with the changed assumptions. Break-.even analysisconsists of determining the value of each assumption parameter for which the NPV is 0, assuming that the other assumption parameters are at theirba,elinevalues. Calculation of fRR is an example of break-even analysis. JRR,.the internal rate of return, is an alternative profit measure to NPV. The !RR is the interest rater such that the present value of free cash flows at r is 0. Assuming the usual pattern of negative free cash flows initially followed by positive free cash flows, !RR is the highest interest rate for which the NPV is at least 0. Thus JRR is the highest interest rate for which the company breaks even. A similar analysis can be done for the other parameters. A break-even analysis calculates the breakeven level of number of sales, expenses, sales price, level of cash flows per year, and any other parameter. Ex.AMPLI! 28 A project requires an immediate investment of 19 million. It is expected to generate free cash flows of 2 million per year at the end of the first year, growing 2% per year perpetually. The cost of capital is12%.

Perform a break-even analysis on the rate of growth of free cash flows.

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2.2. PROJECT ANALYSIS

ANSWER:

Let g be the growth rate. We want to solve -15 + ~ = 0 2 -19+--aQ 0.12-g 2 -=19 0.12-g 2

0.12-s=w

g =[0.014741

®

Quiz 2-2 A project to develop a new product requires an immediate investment of 9 million. It will then generate free cash flows of 1 million per year starting with the end of the first year, until the product becomes obsolete and cannot be sold. The cost of capital is 10%. Perform a break-even analysis on the number of years the product must sell.

2.2.2

Sensitivityanalysis

Sensitivity analysis consists of calculating the change in the NPV resulting from a change in a parameter. Typically one sets the parameter to its value in the worst po~ible case and the best po~ible case, and calculates the NPV for both cases. This analysis shows which parameters have the greatest impact on the NPV. ExAMl'LE 2C A project to develop a new product requires an immediate im·estment of 15 million. Free cash flows generated by this project are 20% of sales. Sales are expected to be level, and to continue for a certain number of yea 15, at which point the prOOuct becomes obsolete. The best and worst cases for each assumption are:

Annual sales($ million) Number of years Cost of capital

Worst case

Baseline

Best case

0.16

0.12

7 0.08

Perform a sensitivity analysis on the three factors listed in the table. Which factor is the NPV most sensitive to? We'll do all calculations in millions. For annual saless:

ANSWER:

NPV=-15+sa:,;io.12=-15+s

(

1-1/1.lii) --0.12

which is -$0.581 million for s = 4 and $6.629 million for s = 6, a variation of $7.210 million For number of years 11: NPV=-15+5a111012=-15+5--which is -$2.991 million for For cost of capital r:

11 =

3 and 7.818 million for

NPV=-15+5a_51,=-15+5 lFMStudyManuaJ-i"edilion Copynght02!118ASM

11

(

1-1/1.12") 0.12

= 7, a variation of$10.809 million. 1-1/(1 --,-(

+r) 5)

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2. PROJECT ANALYSIS

=

=

which is $1.371 million for r 0.16 and $4.964 for r 0.08, a variation of$3.593 million. We see that number of years of sales is the assumption to which NPV is most sensitive.

2.2.3 Scenario analysis Often parameters are correlated and should not be analyzed separately. For example, increasing the price of a product may lower sales. Scenario analysis consists of calculating the NPV for various scenarios. A scenario may vary hvo parameters in a consistent manner, leaving the other parameters unchanged if they are uncorrelated.

2.3 Riskmeasures In the previous section we analyzed risk by varying parameters. An alternative method for analyzing risk is to assign a number to the project indicating its riskiness. This section discusses such risk measures. Each of these risk measures is a function from a random variable to a real number. The random variable may be profits, returns on im·estment, or aggregate loss amounts paid by an insurance company. Notice that the direction of risk for aggregate loss amounts is the opposite of profits or returns: the risk is that profits or returns are low and that loss amounts are high.

2.3.1 Four risk measures We will discuss four risk measures: variance, semi-variance, VaR, and TVaR.

variance Variance is a popular risk measure and will be used in mean-variance portfolio theory, which we discuss starting in Lesson 5. If R is the random variable for the return on an investment, the mean return is /J and the variance is (2.2) An equivalent risk measure is the square root of the variance, or the standard deviation,,. the notation SD(R) for the standard deviation.

We'll also use

SD(R) =~=a The standard deviation of the rate of return is also called the wlatility of the rate of return. The variance may be estimated from a sample using the formula

t

1, so we only integrate up to 1. We'll integrate by parts twice. ANSWER: We integrate min(0,x -1)2

The sum of the downside ,emi-variance and the upside ,emi-variance is the variance: El(X - /J)2]

=E[(min(0,X - /J) + max(0,X - /J)}2] =Elmin(0,X - /J)2] + Elmax(0,X - /J)2] + 2Elmin(0,X

- /J)max(0,X - /J)]

The first term of the last expression is the downside ,emi-variance. The second term is the upside ,emivariance. The third term is 0, since 0 is either the minimum or the maximum, so one factor is always 0. However, the semi-variance doesn't have many of the nice properties of variance. For example, there is no easy formula for the semi-variance of a sum of two random variables. The ,emi-variance may be estimated from a sample by (2.4)

~ Quiz 2-3 You are given the following sample:

~

5

10

15

20

25

Calculate the sample semi-variance.

Volue-at-Risk (VaR)

The VaRof a random variable X at level a is the 100a percentile of the random variable. For a continuous random variable, it is x such that Pr(X :S.x) a. For profits or rates of return, where the risk is that X is low, a is picked low, with values like 0.05, 0.025, 0.01, 0.005. For aggregate insurance losses, where the risk is that Xis high, a is picked high, with values like 0.95, 0.975, 0.99, 0.995. The VaR is calculated by inverting the cumulative distribution function:

=

VaR,(X)

=F)/(a)

(2.5)

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10

2. PROJECT ANALYSIS

Ex.AMPLI!

2E Profits have a distribution with the following density function: f(x)

= (l }x) 4

x >0

Calculate VaR of profits at the 0.01 level. ANswER: Integrate f(x) to obtain the cumulative distribution function, then invert that function at 0.01.

F(x) 1 1- (l+x)3

r

3dt

=Jo (1+1)4=1 -

1 (1 + x)3

=0.01

l+x=~=l.003356 X

=I0.0033561

To estimate VaR from a ~mple, the ~mple is ordered from lowest to highest, and then the 100a percentile is selected. This percentile is not well- 1000

othenvi,e Calculatethevalue-at-riskat0.5%. 2.12. Los.es on an insurance are distributed as follows: Greater than

Less than or equal to

Probability

1000 2000 5000 10000

1000 2000 5000 10000 20000

0.45 0.25 0.22 0.05 0.03

Within each range lo~es are uniformly distributed. Calculate the tail value-at-risk for lo~es at 95%. 2.13. Profits X have the following cumulahve distribution funchon: F(x) = 1-e-x/lOOO

X>0

Calculatethetailvalue-at-riskat5%. 2.14. For a simulation with 100 runs, the largest 20 values are 920 948

920 952

922 959

925 962

926 969

932 976

939 989

940 1005

943 1032

945 1050

EstimateTVaR at 95% from this sample. 2.15. Consider the risk measure g(X) = El X2]. Assume it is used only for nonnegative random variables. Which coherence properties does it sahsfy 7 2.H,. Consider the risk measure g(X) = El v'xj. Assume it is u,edonly for nonnegative random variables. Which coherence properties does it sahsfy 7 Finance and Investment sample questions: 27,34,35,42

Solutions 2.1. The pre.ent value of the investment is 12 + (1 -1/1.1 5)/0.1 = 15.791 million. The present value of the free cash flows is 1/(0.1-0.02) = 18.75 million. The NPV is 18.75-15.791 = 12.959 millionj. 2.2. At time 1, the pre.ent value of the cash flows from the widgets is 1,800,000am +200,000(/a)m110

am = l- ~:;· (Ja}m lFMStudyManuaJ-i"edilion Copynght02!118ASM

= 6.144567

6.1445670~10/1.110 =22.89134

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EXERCISESOLUflONS FOR LESSON 2

So the present valueofthecashflowsattime1 is6.144567(1,&J0,000)+22.89134(200,000) = 15,638,489. Discounting to hme0 and subtracting the investment, the NPV is 15,638,489/1.1-10,000,000 = IS4,2H1,8081. 2.3.

Present value of im·estment and fixed expenses is 20 + a,i = 20 +

1-1/1.12 7

~

= 24.563757 million

Present value of net profit from sale of 1 desk per year is 300(1- ~:1~ So to break even, 24,563,757/1369 = ~desks

127 ) = 1369

per year must be sold.

2.4. Let r be the cost of capital. At time 1, the present value of future free cash flows is 1/(r - 0.03) in millions. Thus at time 0 the present value of these cash flows is 1/((1 + r)(r -0.03)). We want r such that -8-

2

J+r

1 + (1 + r)(r -0.03) = O

-8(1 + r)(r -0.03)-2(r-0.03) 8r 2 +9.76r-1.3 =0

+ 1= 0

2.5. The present value at annual effective rate r of cash flows for 11 years at the end of each year starhng at 1 and growing at a rate of g is 1

l+r~

n-1 (l+g)l

1+r

1 1-(W,-)" = 1+r1-(1+g)/(1+r)

1-(-W,)"

~

In the following, all numbers are in millions. For the free cash flows in fi15tyear assumption, the NPVs of theworstandbestcasesare: 10 10 -8+ 1.1(1 ~\ ~ ) = -0.42788

~~\1}

10 10 -8+ 1.3(1 ~\ ~ ~~~~) ) = 0.94&117 with difference 1.37675. For the rate of growth assumption, the NPVs of the worst and best cases are: 1-(1/11) 10) -8 + 1.2 __ • = -0.62652 ( 0.10 -8+1.2(1-(1.05/1.1)10) 0.10-0.05

=0.92777

with difference 1.55429 For the number of years of free cash flows assumphon, the NPVsof the wo15t and best cases are: 3 -8 + 1.2(l -(l.0 /1.1)7) = -1.67634 0.10-0.03

-8 + 1.2(1 ~\1~0~~~~)13) = 1.85~ IFMStudyManual-i"edilion CopynghtC2!118ASM

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16

2. PROJECT ANALYSIS

withdifference3.52694. The most sensitive a~umption 2.6.

I

is number of yean ].

We will ignore the investment cost, which is the same in all scenarios. For annual units sold, the NPVs of the worst and best cases are:

1,000,~;1;50)(0.8) 1,:()(),~1;50)(0.8)

=S,000,000 12,000,000

with difference 4,000,000. For price per unit, the NPVs of the worst and best cases are:

1,200,~;1/5)(0.8)

=B.000,000

1,200,~1;60)(0.8)

10,240,000

with difference 2,240,000. For expenses, the NPVs of the worst and best cases are:

1,200,~~:0)(0.77) 1,200,~'.~~S0)(0.85)

9,240,000

=10,lOO,OOO

withdifference960,000. Annu.il units sold has the highest sensitivity. 2.7.

I

I

.f=

1+3+7+~5+25+39

,,;u= (1-1s)

2 +(J-

15

2 2 1s) + (7-15) 6

EB

2.8. A normal distribution is symmetric. So the downside semi-variance and the upside semi-variance are equal,and the downside semi-variance is therefore half the upside semi-variance, or 50. The downside standard deviation is ../soa::c~2.9. This is a beta distribuhon. If you recognize it and are familiar with beta, you know that the mean is 2/3. Otherwise it is not hard to calculate:

The downside semi-variance is

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EXERCISESOLUflONS FOR LESSON 2

210.

We need the first percentile of profits. Let it be x. Then e-lOOO/x =0.01 1()()()

-----:;-= -ln0.01 x=-1~:1 211.

=lli[ill

Let x be the VaR. Then 2

(1~)

=0.995

1 ~

=0.997497 x

=o.~CX:97 =I1002.s1 I

212 The 95 th percentile of losses is the point with a 5% probability of losses above that point. Since the top interval has probability 3%, we need a subset of the (5000, 1CXXJO] interval with probability 2%. That interval has probability 5%, so we need the top 2/5 of the interval, making 8000 the 95th percenhle. The expected value of losses given that they're abm·e 8000 can be calculated using the double expectahon formula: TVaR(X) = ElX IX> 8000] = Pr(X 5. 10000 I X > 8000) ElX I 8000 < X 5. 10000] + Pr{X > 10000) ElX IX > 100JO] By uniformity, ElX 5. 1CXXJO I X >&JOO]= 9000 and ElX IX > 10000] = 15000. So TVaR(X) = 0.4(9000) + 0.6(15000) = ~ 213.

The 5 th percentile of X is e-x/lOOO =0.95 X

= -1000ln0.95 =51.2933

The straightforward way to calculate the conditional expectation is to integrate x over the density function and then divide by 0.05, the probability of X < 51.2933. The density function is 0.001e-x/lOOO_ ("1'2933

Jo

1512933

0.001xe-x/1000dx = -xe-x/1000 O

(511.933 + Jo e-x/lOOOdx

: -51.2933e-llll5l1.93J-100Qe-Oll5l2 S1 er(T-t)_ For a nondividend paying stock, the left side of this inequality is the expected price of the stock; the right side is the forward price. The forward price is less than the expected price of the stock. The same statement holds for dividend paying stocks, as you can easily verify. One who buys a forward is assuming the risk of stock price volahlity. In return for assuming this risk, the buyer pays less than the expected price of the stock. The buyer cams the risk premium on the stock without making any investment in the stock. One can earn the risk premium on an as.et with no investment. 14A A stock's current price is SOand its 6-month forwani price is 52. Assuming investors are risk-averse, in what range does the expected value of the stock price at the end of 6 months fall? Ex.AMPLI!

ANswl!R: The forward price is less than the expected future price, so the expected future price must be greater than 52. D

14.4 Synthetic forwards It is possible to u,e stocks and bonds to create a combination that has the same payments as a forwani. To create a synthehc long forward, one buys a stock and borrows the price of the stock. Borrowing the price of the stock can be done by ,elling a zero-coupon bond whose price is the price of the stock and which matures at expiry of the forward. To create a synthetic short forwani, one sells the stock and buys the same zero-coupon bond. A market-maker who deals with forwards will off.et the forward with a synthetic forward. If he longs the forward, he shorts the synthetic forwani, and if he shorts the forward, he longs the synthehc forward. The latter .et of transactions: buying the stock, selling a bond, and ,elling a forward is called a cash-and-carry. A cash-and-carry is risk-free. If profit is made on this combination, this arbitrage is called a cash-and-carry arbitrage. If one instead sells the stock, buys a bond, and buy,; a forward, that is called a reiersecash-and-carry.

One can also create a synthetic bond by buying a nondividend paying stock and shorting a forward on the stock. An investor who does that pays the price of the stock inihally, and at the end receives the forward price for the stock. This creates a synthetic zero-coupon bond. If one bought a dividend-paying stock, then one would receive some cash flows before expiry, so it would be like a bond paying coupons. The rate of return on a synthehc bond is called the implied reporote.

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EXERCISESFOR LESSON 14

157

Exercises 14.1. [Former FM Sample Question 18](This q11es/io11 uus deleted by the SOA, so you may skip it, but you should be able to do it.) A jeweler buys gold, which is the primary input needed for her products. One ounce of gold can be used to produce one unit of jewelry. The cost of all other inputs is negligible. She is able to sell each unit of jewelry for 700 plus 20% of the market price of gold in one year.

In one year, the actual price of gold will be in one of three possible states, corresponding following probability table: Market Price of Gold in one year Probability 750perounce 850 per ounce 950perounce

to the

0.2 0.5 0.3

The jeweler is considering using forward contracts to lock in 1-year gold prices, in which case she would charge the customer (one year from now) 700 plus 20% of the forward price. The 1-year forward price for gold is 8.50per ounce. Calculate the increase in the expected 1-year profit, per unit of jewelry sold, that results from buying forward the 1-year price of gold. (A) 0

(B) 8

(Cl 12

(D) 20

(E) 32

14.2 [Introductory Derivatives Sample Question 68] For a non-dividend-paying stoi:.:kindex, the current price is 1100 and the 6--month forward price is 1150. Assume the price of the stoi:.:kindex in 6 months will be 1210.

Which of the following is true regarding forward positions in the stoi:.:kindex 7 (A) (B)

(C) (D) (E)

Long Long Long Short Short

position position position position position

gains gains gains gains gains

50 60 110 60 110

14.3. [Introductory Derivatives Sample Question 70] Investors in a certain stoi:.:kdemand to be compensated for risk. The current stoi:.:kprice is 100. The stock pays dividends at a rate proportional to its price. The dividend yield is 2%. The continuously compounded risk-free interest rate is 5%. Assume there are no transaction costs. Let X represent the expected value of the stoi:.:kprice 2 years from tOOay. Assume it is known that X is a whole number. Determine which of the following statements is true about X. (A) (B)

(C) (D) (E)

The only possible value of Xis 105. The largest possible value of Xis 106. The smallest possible value of Xis 107. The largest possible value of X is 110. The smallest possible value of Xis 111.

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158

14. FORWARDS

14.4. A nondividend paying stock's current price is 100. The 2-year forward price of the stock is 110. Determine the continuously compounded risk-free interest rate. 14.5. Fora stock index (i) The current price is 88. (ii) The4-month forward price is 89. (iii) The continuously compounded risk-free rate is 0.04. Determine the continuously compounded dividend rate of the index. 14.6. [Introductory Derivatives Sample Question 10]StockXYZ has a current price of 100. The forward price for delivery of this stock in 1 year is 110. Unless otherwise indicated, the stock pays no dividends and the annual effechve risk-free interest rate is10%. Determine which of the following statements is FALSE. (A) (B)

(C) (D) (E)

The hme-1 profit diagram and the time-1 payoff diagram for long positions in this forward contract are identical. The time-1 profit for a long position in this forward contract is exactly opposite to the time-1 profit for the corresponding short forward position. There is no comparative advantage to inveshng in the stock versus investing in the forward contract. If the 10% interest rate was conhnuously compounded instead of annual effective, then it would be more beneficial to invest in the stock, rather than the forward contract. If there was a dividend of 3.00 paid 6 months from now, then it would be more beneficial to invest in the stock, rather than the forward contract.

14.7. [Introductory Derivatives Sample Question 73]Thecurrent price of a non-dividend-paying stock is 100. The annual effective risk-free interest rate is 4%, and there are no transaction costs. The stock's two-year forward price is mispriced at 108, so to exploit this mispricing, an investor can short a share of the stock for 100 and simultaneously take a long posihon ina two-year forward contract. The investor can then invest the 100 at the risk-free rate, and finally buy back the share of stock at the forward price after two years. Determine which term best describes this strategy. (A) Hedging

(B) Immunization

(C) Arbitrage

(D) Paylater

(E) Dive15ification

14.8. [Introductory Derivatives Sample Question 20] The current price of a stock is 200, and the continuously compounded annual risk-free interest rate is 4%. A dividend will be paid every quarter for the next 3 years, with the first dividend occurring 3 months from now. The amount of the first dividend is 1.50, but each sub,equent dividend will be 1% higher than the one previously paid. Calculate the fair price of a .J..year forward contract on this stock. (A) 200

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(B) 205

(Cl 210

(D) 215

(E) 220

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EXERCISESFOR LESSON 14

159

[Introductory Derivatives Sample Question 52] The ask price for a share of ABC company is 100.50 and the bid price is 100. Suppose an investor can borrow at an annual effechve rate of 3.05% and lend (i.e., save) at an annual effective rate of 3%. Assume there are no transaction costs and no dividends. Determine which of the following strategies does not create an arbitrage opportunity. 14.9.

(A) (B)

(C) (D) (E)

Short sell one share, and enter into a long one-year forward priceof102.50. Short sell one share, and enter into a long one-year forward priceof102.75. Short sell one share, and enter into a long one-year forward price of 103.00. Purchase one share with borrowed money, and enter into one share with a forward price of 103.60. Purchase one share with borrowed money, and enter into one share with a forward price of 103.75.

contract on one share with a forward contract on one share with a forward contract on one share with a forward a short one-year forward contract on a short one-year forward contract on

14.10. [Introductory Derivatives Sample Question 37] A one-year forward contract on a stock has a price of $75. The stock is expected to pay a dividend of $1.50 at two future hmes, six months from now and one year from now, and the annual effective risk-free interest rate is 6%.

Calculate the current stock price. (A) 70.75

[Introductory Stock XYZ: 14.11.

(B) 73.63

(Cl 75.81

(D) 77.87

(E) 78.04

Derivatives Sample Question 51] You are given the following information about

(i) The current price of the stock is 35 per share. (ii) The expected continuously compounded annual rate of return is8%. (iii) The stock pays semi-annual dividends of 0.32 per share, with the next dividend to be paid two months from now. The continuously compounded annual risk-free interest rate is 4%. Calculate the current one-year forward price for stock XYZ. (A) 34.37 14.12.

(B) 35.77

(Cl 36.43

(D) 37.23

(E) 37.92

For a stock, you are given:

(i) The current price is 50. (ii) The3-month forward price is 52. Determine the 12-month forward price. 14.13.

You are given the following information about BUK stock:

(i) The spot price of a stock is 65. (ii) The stock pays no dividends. The effective annual risk-free interest rate is 0.06. Calculate the annualized forward premium for a 3-month forward on BUK stock.

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14. FORWARDS

14.14.

You are given the following information about ADZ stock:

(i) The spot price of a stock is 95. (ii) The stock pays quarterly dividends of 1.50. (iii) The next dividend is payable 2 months from now. The continuously compounded risk-free interest rate is 0.03. Calculate the annualized forward premium for a 6-month forward on ADZ stock. 14.15.

[Introductory Derivatives Sample Question 6] The following relates to one share of XYZ stock:

The current price is 100. •

The forward price for delivery in one year is 105.

•

P is the expected price in one year. Determine which of the following statements about Pis TRUE.

(A) P < 100

(B) P=100

(C) 100 II> Ill

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(B) I> Ill> II

(C) II> I> Ill

(D) Ill> I> II

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(E) Ill> II> I

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EXERCISES FOR LESSON 16

179

16.12. [Introductory Derivatives Sample Question when the market price of the index is also :().

35] A customer buy,; a 50--strike put on an index

The premium for the put is 5. Assume that the option contract is for an underlying 100 units of the index. Calculate the customer's profit if the index declines to 45 at expiration. (A) -1000

(D) 500

(Cl O

(E) 1000

16.13. [Introductory Derivatives Sample Question 42] An investor purchases a non-dividend-paying stock and writes a /-year, European call option for this stock, with call premium C. The stock price at hme of pun::haseand strike price are both K. Assume that there are no transaction costs. The risk-free annual fon::e of interest is a constant r. Let S represent the stock price at time I.

S > K. Determine an algebraic expression for the im·estor's profit at expiration. (A) (B)

(C) (D) (E)

Ce'r C(1+rt)-S+K Ce' 1 -S+K Ce' 1 + K(1- e' 1 ) C(1+r) 1 +K[1-(1+r)

1

16.14. [Introductory Derivatives two options, A and B: (i) (ii) (iii) (iv)

]

Sample Question 44] You are given the following information about

Opbon A is a one-year European put with exerci,e price 45. Opbon Bis a one-year American call withexen::ise price 55. Both options are based on the same underlying as.et, a stock that pays no dividends. Both options go into effect at the same time and expire at I = 1.

You are also given the following information about the stock price: (i) (ii) (iii) (iv)

The The The The

initial stock price is 50. stock price at expiration is also 50. minimum stock price (from I 0 to I maximum stock price (from I 0 to I

= =

=1) is 46. =1) is 58.

Determine which of the following statements is true. (A) (B)

(C) (D) (E)

Both Both Both Only Only

options options options option option

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A and Bare "at-the-money" at expiration. A and Bare "in-the-money" at expirahon. A and Bare "out-of-the-money" throughout each ophon's term. A is ever "in-the-money" at some hme during its term. Bis ever "in-the-money" at some time during its term.

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180

16. OPTIONS

16.15. [Introductory Derivatives Sample Question 48] For a certain stock, Investor A purchases a 45strike call ophon while Investor B purchases a 135-strike put option. Both ophons are European with the ~me expirahondate. Assume that there are no transaction costs.

If the final stock price at expirahon is S, Investor A's payoff will be 12. Calculate Investor B's payoff at expiration, if the final stock price is S. (A) 0

(B) 12

(Cl 36

(D) 57

(E) 78

16.16. [Introductory Derivatives Sample Question 49] The market price of Stock A is:(). A customer buys a :()-strike put contract on Stock A for 500. The put contract is for 100 shares of A

Calculate the customer's maximum possible loss. (B) 5

(A) 0

(Cl 50

(D) 500

(E) 5000

16.17. [Introductory Derivatives Sample Question 61] An im·estor purchased Option A and Option B for a certain stock today, with strike prices 70 and 80, respechvely. Both options are European one-year putophons.

Determine which statement is true about the moneyness of these options, based on a particular stock pnce. (A) (B)

(C) (D) (E)

If Option If Option If Option If Option If Option

A is in-the-money, then Option Bis in-the-money. A is at-the-money, then Option B is out-of-the-money. A is in-the-money, then Option B is out-of-the-money. A is out-of-the-money, then Option B is in-the-money. A is out-of-the-money, then Option Bis out-of-the-money.

16.18. [Introductory Derivatives Sample Question 66] The current price of a stock is 80. Both call and put ophons on this stock are available for purcha,e at a strike price of 65. Determine which of the following statements about the,e options is true. (A) (B)

(C) (D) (E)

Both the call and put options are at-the-money. Both the call and put options are in-the-money. Both the call and put options are out-of-the-money. The call option is in-the-money, but the put option is out-of-the-money. The call option is out-of-the-money, but the put ophon is in-the-money.

16.19. [Introductory Derivatives Sample Question 62] The price of an as.et will either rise by 25% or fall by 40% in 1 year, with equal probability. A European put option on this asset matures after 1 year. Assume the following: Price of the asset tOOay: 100 Strike price of the put option: 130 Put option premium: 7 Annualeffectiveriskfreerate:

3%

Calculate the expected profit of the put option. (A) 12.79

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(B) 15.89

(Cl 22.69

(D) 27.79

(E) 30.29

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EXERCISESFOR LESSON 16

181

16.20. [Introductory Derivatives Sample Question 75] Determine which of the following risk management techniques can hedge the financial risk of an oil producer arising from the price of the oil that it sells. I. Short forward posihon on the price of oil II. Long put ophon on the price of oil Ill.

Long call ophon on the price of oil (D) 1,11,andlll

(A) I only (B) II only (C) Ill only (E) The mrrect answer is not given by (A), (B), (C), or (D). 16.21. A stoi:.:khas price 51 at hme I in years. You are given: (i) (ii) (iii) (iv) (v)

So =40 The stock pays no dividends. The stock's continuously compounded annual growth rate is 0.12. Therisk-freerateis0.05. a =0.25.

You write a 6--month European call option with strike price 45. Calculate the value-at-risk for the payoff on the option at a 0.05.

=

Solutions

=

16.1. The payoff is 90 - 72 18. The accumulated cost of the premium is 10.50e0 !O{Ol.5) payoff, the profit, is 18 -10.77

=~-

=10.77. The net

16.2 The profit on the short forward is 55- S. The profit on the put is max(0, 50- S)-4e 1104 . This is not greater than 55 - Sunless max(0,50 - S) 0, or S ~ 50, and then

=

-4e 004 > 55-5 S > 55+4e Therangeis[s

004

=59.16

> 59.lb].

16.3. The profit on the forward is S - 60. The profit on the call option is max(0, S - 50) -18e 1104 . If the maxis0,then -18e 1104> $-60 $ 55. That leaves 45 < K < 55. First investor's profit is 55 - K - 3.20(1.03) and second investor's profit is (K + 10) - 55 -3.80(1.03). We equate the;e.

= =

55 - K - 3.20(1.CD) (K + 10) - 55- 3.&1(1.03) 51.704 - K K - 48.914 K=~

16.7. The 40-strike call produces a pm fit at least as high as the 45-strike call if the extra payoff is at least the difference in premiums raised with interest, (6.22- 4.Cll)(l.08) = 2.31. This happens when S ~ 42.31. Otherwise, the45-strike call prOO.ucesthe higher profit. The 35-strike call prOO.ucesa profit greater than the 45-strike call if the extra payoff is greater than (9.12 -4.08)(1.Cll) = 5.44. This happens when S > 40.44. The answer is (C). 16.8. The profit for being long is 1000-$-74.20(1.02) and the profit for being short is S-1000+74.20(1.02). F.quahng these,weget 25 = 2000 - 2(74.20)(1.02) = 1848.63 ThusS=~.(B) 16.9. The expected sales price of an ounce of gold with the put is 0.3(950) + 0.7(900) = 915. Costs are &JO+ 100e006 =906.18. Profit is915-906.18 =[Iill. (D) 16.10. If you short a forward contract, you must pay the stock price minus the contract's forward price, and the stock price is unlimited. If you short a call option, you must pay stock price minus a fixed amount, once again unlimited. If you short a put option, you must pay the fixed price minus the stock price, or at most the fixed price. (B) 16.11. Option I got 2, by exercising when the stock was 18. Opbon II got 3, by exercising when the stock was 28. Opbon lll got 4, since 30 - 26 = 4. (E) 16.12. The payoff at expiration is (50 - 45)(100) = 500. The premium is 5(100) = :()(I. For some reason, this question ignores the interest on the premium, and calculates the profit as 500 - :()(I = [ill(C), but a more exact calculation would be 500 - :()(1(1+ i)r. 16.13. The investor receives C inihally for the call and pays K for the stock. These grow at interest to (C - K)e' 1 . At the end, the im·estor hands m·er the stock and receives K to settle the call. Profit is (C - K)e' 1 + K

=Ce'

1

+ K(1 - e' 1 )

(D)

16.14. American versus European makes no difference. Opbon A is in the money if the stock price is less than 45. That never happens. Opbon Bis in the money if the stock price is greater than 55. That does happen. (E) IFMStudyManual-i"edilion CopynghtC2!118ASM

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EXERCISESOLUflONS FOR LESSON 16

183

The payoff is 12 if S =45 + 12 =57. Then the payoff forB is 135- 57= ~- (E) The only loss from buying a put option is the premium. There is never an obligation to pay anything else. So the maximum loss is~. (D) 16.17. A put option is in the money when the stock price is less than the strike price. If the stock price is less than 70, it is certainly less than 80. (A) 16.15. 16.16.

The call option would pay 15 if exercised immediately, so it is in-the-money. The put option will not pay unless the stock price goes below 65, so it is out-of-the-money. (D) 16.19. If the price rises by 25%, it is 125 and the option pays 5. If it falls by 40%, the price is 60 and the option pays 70. Expected profit is expected payoff, minus put premium accumulated with interest. 16.18.

0.5(5 + 70) - 7(1.CD)= ~

(E)

I guarantees that oil will be sold at a fixed price. II guarantees that oil will be sold for no less than the strike price. lll offe15 no protection on price decreases and only guarantees that oil can be bought at no more than a certain price. (E) 16.21. Since you wrote the option, your profits decrease as the stock price increases. The 5th percentile of the call payoff will occur when the stock's price is at its 95 th percentile. For the stock's growth, the annual lognormal parameters are /J = 0.12- 0.5(0.252) = 0.08875 and a = 0.25. In 6 months or 0.5 yea!';, the 95th percentile is elll&l7S{05)+!6 45(o25)vifs = exp(0.3352) = 1.3982. The 95th percentile of 5112,the stock price after6 months, is 40(1.3982) = 55.93 and the 95th percentile of the call is 55.93 -45 = ~16.20.

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16. OPTlONS

184

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Lesson 17

Option Strategies Reading: Derivatives Markets 3 In this lesson, we discuss ways to combine op hons and other assets. These combinahons hedging, speculation, or other reasons.

may be u,ed for

17.1 Options with Underlying Assets In this sec hon, we'll discuss strategies involving buying/ selling an option and the underlying a~t, which we'll assume is a stock. In each strategy, the option offsets the stock: if we buy the stock, the ophon is short in the stock, and if we sell the stock, the option is long in the stock. The option can be a call or a put. Sotherearefourvariations: 1. Long put, long stock

2. Long call, short stock 3. Shortcall,longstock 4. Shortput,shortstock Let's first discuss the first hvo variations. The;e are insurance strategies. If you own a stock, you may buy a put. The put guarantees that you will be able to sell the stock for at least the strike price. If the stock price is higher than the strike price, you sell it for that higher price; otherwise, you exen::ise the put option. The put, in the presence of the underlying stock, is called a floor; it sets a floor for the price you get selling the stock. Figure 17.1 shows the payoff and profit diagram;; for a floor,assuming you sell the stock on the expiry date. The profit diagram assumes the stock's original price was 50 and that the risk-free interestrateis0. If you short a stock, you may buy a call. The call guarantees that you will be able to buyback the stock for at most the strike price. If the stock price is lower than the strike price, you buy it back for its price;

:y~off ~·······················

;;o~fit 10

........ ................

~

0

30 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2

~o

30

40

50

-10 ••••••••••••••••••••••••••

2 60 Sr - ~0

30

40

· •••• ·

50

60 Sr

Figure 17.1: Payoff and profit for floor of 40

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17. OPTION STRATEGIES

-20 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -~;y~off -30 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · -40 · · · · · · · · · · · · · · ·

-S~O

30

40

10 ..............................

.

:;oL••····························· 0

-10 «J Sr - 2~0

50

30

40

50

«J Sr

Figure 17.2: Payoff and profit for cap of 40

otherwi,e you exercise the call option. The call, in the pre.ence of the underlying short stock, is called a cap; it caps the price for buying the stock. Figure 17.2 shows the payoff and profit diagrams for a cap. The profit diagram assumes the stock's original price was 30 and that the risk-free interest rate is 0. Both of the,e strategies are insurance strategies. Ex.AMPLI! 17A An investor buys a nondividend paying stock for 65. To pmtect the investment, the investor buys a 1-year floor at 55, with premium 2. The annual effective risk-free interest rate is 0.04. Calculate the investor's profit if the stock price at the end of one year is (a) 75 and (b) 45.

=

ANswER: The initial investment is 65 + 2 67. If the stock price at the end of one year is 75, the put expires worthless and the investor receives 75. The profit is 75 - 67(1.04) If the stock price at the end of one year is 45, the investor exercises the option and receives 55 for the stock. The profit is 55 - 67(1.04) l-14.6Bj.

=~-

=

Now let's discuss the last two variations. If one writes a call option, one assumes unbounded liability, since there is no limit to how high the stock price can increase. Writing an option with no protection is called naked writing. One can hedge the option by buying the underlying asset. If one writes a call and owns the underlying asset, that strategy is called a rorered written call. At call expiry, no additional cash will be needed to pay off the call. Figure 17.3 shows the payoff and profit diagrams for a written cm·ered call, assuming the stock is sold at expiry of the option. The profit diagram assumes the stock's original pricewas30andthattherisk-freeinterestrateis0. If one writes a put option, one assumes liability up to the strike price, since the stock price may get as low as 0. To offset this obligation, one may sell the underlying asset short. This strategy is called a rorered written put. The payoff and profit diagrams for a covered put are shown in Figure 17.4. The profit diagram assumes the stock's original price was 50 and that the risk-free interest rate is 0. Ex.AMPLE 178 The price of a nondividend paying stock is 65. A market-maker writes a 1-year put option on the stock with strike price 60 fora premium of 3. To hedge this put option, the market-maker sells the stock short. The annual effective risk-free int(>rest rare is 0.03. Calculate the profit of the market-maker if at the end of the year the stock price is (a) 75 and (b) 50.

=

ANswER: The market-maker initially receives 65 + 3 68 for the stock and put option. If the stock price at the end of the year is 75, the option is worthless and the market-maker pays 75 to buy back the stock. Profit is 68(1.03) - 75 If the stock price at the end of the year is 50, the put option is exercised and the market-maker pays 60 for the stock. Profit is 68(1.03) - 60

=~-

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17.2. SYNTHETIC FORWARDS

187

:;yL=off

i;oL=fit

~···············

10

~·······

........................

0

20 1

~o

-10 ••••••••••••••••••••••••••••••••

30

40

50

2 60 Sr - ~0

30

40

50

60 Sr

Figure 17.3: Payofl and profit for written cc,.,ered call of 40

Payoff

Profit

-40 -30b20b

················

10

-50 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · -60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 - ~0

30

40

50

0

~0 ~ 40

-10

ffi s, - 2

50

ffi s,

Figure 17.4: Payofl and profit for covered put of 40

17.2 Synthetic Forwards If one buys a European call expiring at time T for strike price Kand ,ells a European put expiring at hme T for strike price K, then one is guaranteed to receive the underlying asset at time T for the price of K. If K is the forward price, this strategy isa synthetic forward. It follows that the inihal payment must be 0. Let C(S, K, T) be the price of the call and P(S, K, T) the price of the put. Then C(S, K, T)- P(S, K, T) 0. If K is not the forward price, then this strategy is an r?tf-marketforward. To buy an off-market forward, the buyer must pay the present value of the difference from the forward price. The present value is e-rr(Fo,r - K), which can also be written as

=

e-rr(Fo,r - K)

=e-rr(Se(r-h)r

It follows that C(S, K, T) - P(S, K, T)

- K): Se-6r - Ke-rt

=Se- 6r -

Ke-rt

This equahon is called put-call parity. We will examine this equation in detail in Lesson 18. For now, what you should take away is that many different strategies are equivalent. Buying a call and selling the underlying stock-a cap--is the similar to buying a put, since C(S, K, T) - Se- 6r

=P(S, K, T) -

Ke-rr

so they differ only by a risk-free investment, lending Ke-rr_ IFMStudyManual-i"edilion CopynghtC2!118ASM

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17. OPTION STRATEGIES

Ex.AMPLI! 17C A stock index's current price is 1300. It pays continuous dividends at a rate of 0.02. The continuously compounded risk-free inbc'rest rate is 0.05. Calculate the price for a one-year off-market forward with price 1310.

ANswER: The forwani price is 1300eoll5-ll02

=1339.59. The price of the off-market =e--0°5(29.59) =~

forward is

e-'r(1339.59 -1310)

17.3 Bear Spreads, Bull Spreads, and Collars Option strategies involving two options may involve buying an option and selling another option of the ~me kind (both calls or both puts), buying an option of one kind and ,elling one of the other kind, or buying two options of different kinds (which would be selling two options of different kinds from the seller's perspective). We begin our discussion with spreads: buying an option and selling another option of the ~me kind.

17.3.1 Spreads: buying an option and selling another option of the same kind An example of a spread is buying an ophon with one strike price and ,elling an ophon with a different strike price. Such a spread is designed to pay off if the stock moves in one direchon, but subject to a limit. Bull spreads A bull spnmd pays off if the stock mm·es up in price, but subject to a limit. To create a bull spread with calls, buy a Ki-strike call and sell a K2-strike call, K2 > Ki. Then at expiry time T, 1. If Sr 5: Ki, neither ophon pays. 2. If Ki < Sr 5: K2, the lower-strike option pays Sr - Ki, which is the net payoff. 3. If Sr > K2, the lower--strikeoption pays Sr - Ki and the higher-strike option pays Sr - K2, so the net payoff is the difference, or Ki - Ki. A call option's price decrea.es as the strike price increa.es, since a higher strike price makes it more expensive to buy the stock, making the option less valuable. Thus a bull spread with calls alway,; has an initial cost. Figure 17.5 shows the payoff and profit for a :AJ-60bull spread with calls. The profit is the payoff translated down by the cost of the spread accumulated with interest. Surprisingly, one can create a bull spread with puts that is equivalent to a bull spread with calls. To create a bull spread with puts, buy a Ki-strike put and ,ell a Krstrike put, K2 > Ki. Thenatexpiry time T, 1. If Sr 5: Ki, the lower-strike option pays Ki - Sr and the higher-strike ophon pays K2 - Sr, for a net payoff of Ki - K2 < 0. 2. If Ki < Sr 5: K2, the lov,,,:,r-strikeop hon is worthless and the higher--strike option pays K2 - Sr, so the net payoff is Sr - K2 < 0. 3. If Sr > K2 both options are worthless. A put ophon's price increases as the strike price increases, since a higher strike price allows sale of the stock at a higher price, making the ophon more valuable. Thus a bull spread with puts always has a negahve inihal cost. The payoffs to the purcha,er of a bull spread with puts are non-positive, but they increa,e with increasing stock price. Since the position has negahve cost, you will gain as long as the absolute value of the net payoff is lower than the net amount received at incephon, and the absolute value of the net payoff decrea.es as the stock price increa.es. IFMStudyManual-i"edilion CopynghtC2!118ASM

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17.3. BE.J\R SPREADS, BULL SPREADS, AND COLLARS

Payoff

Profit

20

20

10

10

-10

-10

Figure 17.5: Payoff and profit on bull spread with calls of strike prices 50 and 60

Payoff 20

10

-10f---~

-

2

~o

40

50

60

70

80 Sr

Figure 17.6: Payoff on bull spread with puts of strike prices 50 and 60

Figure 17.6 shows the payoff for a 50-60 bull spread with puts. The profit is the payoff translated up by the inihal cash flow accumulated with interest. The diagram is identical to the one for a bull spread of calls, and is not shown here. Bear spreads A bear sp~ad pays off if the stock price moves down in price, but subject to a limit. To create a bear spread with puts, buy a Ki-strike put and sell a Ki-strike put, K2 > Ki. Then 1. If Sr

~

K2, neither ophon pays.

2. If K2 > Sr

~

Ki, the higher-strike option pays K2 - Sr.

3. If Sr < Ki, the higher-strike option pays K2 - Sr and the lower-strike ophon pays Ki - Sr for a net payoffofK2-Ki. There is an initial cost to a bear spread with puts. Diagrams of the payoff and profit on a bear spread with puts are shown in Figure 17.7. The profit diagram is the payoff diagram translated down by the initial cost accumulated with interest. One can also create a bear spread with calls. Buy a K2-strike call and ,ell a Ki-strike call, K2 > Ki. Thffi

1. If Sr ~ K2, the higher-strike option pays Sr - K2 and the lower-strike option pay,; Sr - Ki for a net payoff of Ki - K2 < 0. IFMStudyManual-i"edilion CopynghtC2!118ASM

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17. OPTION STRATEGIES

190

Net profit 20

Payoff

20 lQf---~

10

-10

- 2~0

-10

40

so

60

70

so5 r

- 2~0-0-~40~~,o~~"'~-,=o-~,.,=

5,

Figure 17.7: Pa)Qfl and profit on bear spread with puts of strike prices 50 and 60 Payoff

20 10

-10 - 2~0

40

50

60

70

80 Sr

Figure 17.8: Payoff on bear spread with calls of strike prices 50 and 60

2. If K2 > Sr ;:,_Ki, the higher-strike op hon is worthless and the lower-strike op hon pays Sr - K 1 for a net payoff of K1 - Sr < 0. 3. If Sr< Ki, both options are worthless. There is an inihal negative cost to a bear spread with calls. However the payoffs for a bear spread with calls are non-positive. They increase with decreasing stock price. Figure 17.8 graphs the payoff ona bear spread with calls. The profit is the payoff translated up by the initial proceeds accumulated with interest. The profit graph is idenhcal to the one for a bear spread with puts and is not shown here. A bull spread from the perspective of the purchaser is a bear spread from the perspective of the writer. Ratio spreads A mtio spreadinvolves buying 11 of one option and selling m of another option of the same type where m 'F-11. It is possible to select m and 11 to make the net initial cost of this strategy zero. However, by doing so, one does not bound the gain or loss from the stock price moving in either direction. Ex.AMPLI! 17D An investor buys a ratio spread of 1-year European calls. He buys 1 call option with strike price 40 and sells2 call options with strike price SO. Option prices are

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17.3. BE.J\R SPREADS, BULL SPREADS, AND COLLARS

191

Profit 20

10

-10 - 2~0

40

50

60

70

80 Sr

Figure 17.9: Profit on ratio spread of E)(ample 17D

Strike price

Call option premium

40 50

10 5

Determine the investor's profit if the ending price of the underlying stock is (a) 45, (b) 55, (c) 65.

=

ANswER: There isno initial cost,since thepurchasedoptioncosls 10and the sold options sell for2(5) 10. The profit is the payoff. When the ending stock price is 45, only the 40--strike option pays off, and the profit is Ii]. When the ending stock price is 55, all options pay off; the investor ~ceives 15 and pays 2(5) 10 for a profit of [i]. When the ending stock price is 65, the investor receives25 and pays2(15) 30 fora profit of[::Ij. We see that the investor pays more the higher the stock price is. Figure 17.9 graphs the profit. D

=

=

Bo)( spreads A box spread is a four-ophon strategy consisting of buying a bull sp~ad with strikes K1 and K2 (K2 > K1) and buying a bear spread with strikes K2 and K1. If both spreads use the same type of option, one spread cancels out the other spread. For example, if both spreads are with calls, the bull spread buys a K1 call and sells a K2 call, while the bear spread buys a K2 call and ,ells a K1 call. The dealer earns a lot of commissions and the investor hasn't accomplished anything. To make the box spread intereshng, the bull and bear spreads should be of different types. For example, if the bull spread is calls and the bear spread is puts, a box spread consists of buying and ,elling the following:

Strike

Bull S read

Bear S read

K1 K2

Buy call Sell call

Sell put Bu ut

Assuming European ophons, whenever you buy a call and ,ell a put at the same strike price,exercise by one of the parties is certain (unless the stock price is the strike price at maturity, in which ca,e both options are worthless), so it is equivalent to a forward. Thus a box spread consists of an agreement to buy the stock for K1 and ,ell it for K2, with definite profit K2 - K1. If priced correctly, there will be no gain or loss regardless of the stock price at maturity. The only purpo.e of a box spread is as a financing arrangement,or perhaps to take advantage of a tax loophole. &LE

17E Consider a box spread consisting of the following 1-year European options: a long call

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17. OPTION STRATEGIES

192

Figure 17.10: Pa)Qff and profit on a collar with strikes 40 and 60

Net profit 20 10

-10

-"'w

30

40

50

60

70

Figure 17.11: Profit on collared stock, with stock price 50 and collar strikes 40 and 60

and short put with strike price 40 and a short call and long put with strike price 60. The continuously compounded risk-free rate is 4%. What is the price of this box spread? ANswER: The long call and short put result in buying the asset for 40, and the short call and long put result in selling the asset for 60, so the investor receives a net of 20 at the end of the year. The present valueof20is20e--0o 4 =~

17.3.2 Collars: buying one option and selling an option of the other kind Ina collar, you sell a call with strike K2 and buy a put with strike K1 < K2. The difference K2 - K1 is called thecoUarwidth. A collar's payoff increases as the price of the underlying stock decreases below K1 and decreases as the price of the underlying stock increa,e; above Ki. Between K1 and K2, the payoff is 0. Diagram;; of the payoff and profit of a collar are shown in Figure 17.10. The profit is the payoff translated down by the cost of the collar accumulated with interest. The cost may be positive or negative. A person who own;; a stock may buy a collar with strikes below and abm·e the current stock price, resulting in a collaredstock. The result is similar to a bull spread. Figure 17.11 is a profit diagram for a collared stock. IFMStudyManual-i"edilion CopynghtC2!118ASM

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17.4. STRADDLES, STRANGLES, AND BUTTERFLY SPREADS

193

Note that if collar width is 0, exercise of the option is guaranteed at expiry hme T, unless Sr = K1 in which case both options are worthless. There is no risk, and if the ophons are priced fairly no profit or loss is possible. One may ,elect the strikes so that the net cost of the collar is 0. This type of collar is called a zero-cost collar. To arrange this, the put strike should be less than the forward price and the call strike should be greater. Assume that the stock is not dividend paying and that the risk-free rate is greater than 0. Then the forward price is greater than the current stock price. One may ,et the put strike equal to the current stock price, and the call strike will be higher. In this case, the zero-cost collar costs nothing and guarantees that at expiry you will receive at least the stock price. Sounds great! An investment that cannot lose money and that may make money! The catch is that a risk-free investment would earn r, so merely guaranteeing that an investment won't lose money still leads to a negative profit. If the put strike is set equal to the forward price, then a zero-cost collar requires the call strike to also equal the forward price, and the resulhng collar will be a forward. Ex.AMPLI! 17F An investor owns a nondividend paying stock with current price SO. The investor buys a zero-cost one-year collar comisting of a put option with strike price SO and a call option with strike price 55. The annual effective risk-free rare is 0.05. Calculate the investor's profit if the price of the stock at the end of the year is (a) 45, (b) 52, (c) 60.

ANswl!R: The collar has zero cost, but the stock costs SO,so - subtract 50(1.05) = 52.50 from the payoff. If the ending stock price 51 = 45, the payoff is SO from the put. If the ending stock price is 52, the payoff is 52. If the ending stock price is 60, the payoff is 55 from the call buyer. The profits are (a) 50-52.50

=~

(b) 52-52.50=~ (c) 55-52.50=~

17.4 Straddles, Strangles, and Butterfly Spreads We'll now discuss strategies that speculate on volatility. Straddles: buying two options of different kinds Straddle A straddle consists of a purchased call and put, both of them at-the-money (K = So) and with the same expiration date. The payoff is ISr - Sol,growing with the absolute change in stock price, so this is a bet on volatility: the more volatile the stock price, the higher the expected gain. This strategy always has a significant cost; you have to pay for both options. Figure 17.12 graphs the payoff and profit ona straddle. An investor can bet on low volatility by writing a straddle. Strangle To lower the inihal cost, you can buy a put with strike price K1 and buy a call with strike price K2 with K1 < So < K2, where So is the inihal stock price. This strategy is called a strangle. A strangle has no payoff if the ending stock price is behv~en K1 and K2, but pays if the stock price change is greater. Figure 17.13 graphs the payoff and profit on a strangle. Butterfly spreads A butterfly spread is a written straddle with additional purchased options to limit the maximum loss. A symmetric butterfly spread includes a long put with strike price K - c and a long call with strike price K + c, where K is the strike price of the straddle and c is any posihve constant. So a symmetric butterfly spread has the following options: IFMStudyManual-i"edilion CopynghtC2!118ASM

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17. OPTION STRATEGIES

194

Figure 17.12: Payoff and profit for straddle with strike price 50 Payoff

20

Profit 20

10

10

-10

-10

Figure 17.13: Payofl and profit for strangle with strike prices 40 and 60

=K - c. =K. Short call, strike price K2 =K. Long call, strike price K3 =K + c.

• Long put, strike price K1

• Short put, strike price K2 • •

This portfolio is a bull spread of puts with strikes K1 and K2 plus a bear spread of calls with strikes K2 and K3. An equivalent portfolio can be created by replacing the puts with calls or the calls with puts, and in that case the strategy would only have one type of asset; for example, the following butterfly spread is equivalent to the one abm·e: • Long call, strike price K1

=K -

• Two short calls, strike price K2 • Long call, strike price K3

c.

=K.

=K + c.

Hence it is considered a "spread", a strategy with only one type of asset. Figure 17.14 graphs the profit on a symmetric butterfly spread. An asymmetric butterfly spread is constructed with the following rules: IFMStudyManual-i"edilion CopynghtC2!118ASM

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17.4. STRADDLES, STRANGLES, AND BUTTERFLY SPREADS

195

N;~pmfit

10

_,:

-20~--,=o--~,~o--=,o--~"'~-~,u--~soSr Figure 17.14: Profit on butterfly spread. European call options with strikes 40, 50, and 60 are used

1. Purchase

11 bull

spreads with strike prices K1 and K2.

2. Purchase m bear spreads with strike prices K3 and Ki.

4.

11 and

m selected so that there is no payoff for Sr

~

K1 or Sr;::,_K3.

= =

For a symmetric butterfly spread, m 11 1. Let's work out the relationship between m and for a butterfly spread with calls, by using the fact that it pays nothing for Sr;::,_K3.

11 in

general

• For the 11 bull spreads with strikes K2 and K1, the payoff is K2 - K1 when the expiry stoi::kprice is K3 > K2 > K1. • For them bear spreads with strikes K3 and K2, the payoff is K3 - K2 when the expiry stoi::kprice is K3 > K2. Adding up the payoffs:

~

= K3-K2

m

K2-K1

The payoff on a butterfly spread is O when Sr < K1. It then increases with slope 11 until it reaches its maximum at Sr = K2, at which point it is 11(K2- K1). It then decreases with slope m until it reaches Oat Sr= K3,and isO for all Sr> K3. There is no free lunch. The butterfly spread makes money at K2, so the three ophons in a butterfly spread must be priced in such a way that the butterfly spread loses money if the final stoi::kprice is below K1 or above K3. In other words, the butterfly spread must have a posihvecost. In Lesson 19 on page 246, we will see that this implies that option prices as a funchon of strike prices must be convex. Ex.AMPLI! 17G A butterfly spread of calls has strike prices 35, 40, 55. It consists of n bull spreads and m bear spreads. Determine m and n.

ANswl!R: We have

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17. OPTION STRATEGIES

196

Payoff

Profit

20

20

10

10

-10

-10

Figure 17.15: Payoff and profit diagrams for asymmetric butterfly spread of E)(ample 17G

The smallest integral values form and II are 11 : 3, m = 1. Figure 17.15 shows payoff and profit diagrams for this spread. The profit diagram assumes that the price of the spread is 7.33. D

Exercises 17.1. An investor buys a three-month cap of 60 on a stock. The price of the call option is 3.69. The current price of the stock is 50. The annual effective risk free rate is0.04 Determine the range of prices for the stock at the end of three months for which the im·estor makes a positive profit. 17.2 An im·estorbuys a one-year floor of 35 on a stock. The price of the put ophon is 1.25. Let S be the price of the stock at the end of the year. The continuously compounded risk-free interest rate is 5%. Determine the range of S for which the floor's profit is higher than the profit without the floor. 17.3. [Introductory Derivatives Sample Question 13] A trader shorts one share of a stock index for .:A.I and buys a 60-strike European call option on that stock that expires in 2 years for 10. Assume the annual effectiverisk-freeinterestrateis3%. The stock index increasesto75after2years. Calculate the profit position. Profit (A) -22.64 (B) -17.56 (Cl -22.64 (D) -17.56 (E) -22.64

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on your combined position, and determine an alternative name for this combined Name Floor Floor Cap Cap "Written" Covered Call

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EXERCISESFOR LESSON 17

17.4. [Introductory Derivatives about options is true. (A) (B)

(C) (D) (E)

17.5.

Sample Question

46] Determine which of the following statements

Naked writing is the prachce of buying ophons without taking an offsethng position in the underlying asset. A covered call involves taking a long position in an asset together with a written call on the same asset. An American style option can only be exen::ised during specified periOOs, but not for the entire life of the ophon. A Bermudan style ophon allows the buyer the right to exerci,e at any time during the life of the ophon. An in-the-money op hon is one which would have a positive profit if exercised immediately. [Introductory

Derivatives

Sample Question 47] An investor has written a cm·ered call.

Determine which of the following represents the im·estor's position. (A) (B)

(C) (D) (E)

Short Short Short Long Long

the call and the call and the call and the call and the call and

short the sta:k long the stock no posihon on the stock short the stock long the stock

17.6. A sta:k index's current price is 1000. The index pay,; continuous dividends pounded rate of 0.02. The continuously compounded risk-free interest rate is 0.07. Calculate the amount payable immediately forward contract with price 1000.

17.7. [Introductory Derivatives Sample Question about futures and forward contracts is false. (A) (B)

(C) (D) (E)

at an annual com-

by an investor who buys a one-year short off-market

69] Determine which of the following statements

Frequent marking-to-market and settlement of a futures contract can lead to pricing differences beh\~en a futures contract and an otherwise identical forward contract. Over-the-counter forward contracts can be customized to suit the buyer or ,eller, whereas futures contracts are standardized. Users of forward contracts are better able to protect themselves from credit risks than users of futures contracts. Forward contracts can be u,ed to synthetically switch a portfolio invested in stocks into bonds. The holder of a long futures contract must place a frachon of the cost with an intermediary and provide assurances on the remaining purchase price.

17.8. [Introductory Derivatives cerning the price of jet fuel.

Sample Question 74] Consider an airline company that faces risk con-

Select the hedging strategy that best protects the company against an increase in the price of jet fuel. (A) (B)

(C) (D) (E)

Buying Buying Buying Selling Selling

calls on jet fuel collars on jet fuel puts on jet fuel puts on jet fuel calls on jet fuel

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198

17. OPTION STRATEGIES

17.9. You are given the following prices for 1-year European call options: Strike price

Call option premium

55 60 65 70

15.00 11.25 8.66 6.05

Consider the following two strategies: I. 55-65 bull spread with calls. II. 60-70 bull spread with calls. The annual effective risk-free interest rate is 0.02. Let S be the price of the underlying stoi::kat the end of one year. For which values of S does Strategy JI yield higher profit than Strategy I? 17.10. An investor shorts a 45-55 bear spread consisting of European puts. Determine the minimum and maximum payoffs to the investor at expiry of the bear spread. 17.11. [Introductory Derivatives Sample Question l] Determine which statement about zero-cost purchased collars is FALSE. (A) (B)

(C) (D) (E)

A zero-width, zero-cost collar can be created by .etting both the put and call strike prices at the forward price. There are an infinite number of zero-cost collars. The put option can be at-the-money. The call option can be at-the-money. The strike price on the put opbon must beat or below the forward price.

17.12. [Introductory Derivatives Sample Question 3] Happy Jalapenos, LLC has an exclusive contract to supply jalapeno peppers to the organizers of the annual jalapeno eating contest. The contract states that the contest organizers will take delivery of 10,000 jalapenos in one year at the market price. It will cost Happy Jalapenos 1,000 to provide 10,000 jalapenos and tOOay's market price is 0.12 for one jalapeno. The continuously compounded annual risk-free interest rate is 6%. Happy Jalapenos has decided to hedge as follows: Buy 10,000 0.12-strike put options for 84.30 and ,ell 10,000 0.14-strike call options for 74.80. Both options are one-year European. Happy Jalapenos believes the market price in one year will be somewhere between 0.10 and 0.15 per jalapeno. Determine which of the following intervals represents the range of possible profit one year from now for Happy Jalapenos. (A) (B)

(Cl (D) (E)

-200 to 100 -110to190 -100 to 200 1~to3~ 200to400

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EXERCISESFOR LESSON 17

199

17.13. [Introductory Derivatives Sample Question 17] The current price for a stock index is 1,000. The following premiums exist for various options to buy or sell the stock index six months from now: Strike Price Call Premium Put Premium 120.41 51.78 950 93.81 74.20 1!100 1,0.:() 71.80 101.21 Strategy Strategy Strategy 1,000-strike

J is to buy the 1,0.:()-strikecall and to sell the 950-strikecall.

JJ is to buy the 1,0.:()-strike put and to sell the 950-strike put. Ill is to buy the 950-strike call, sell the 1,000-strike call, sell the 9.:()-strike put, and buy the put.

Assume that the price of the stock index in 6 months will be behveen 950 and 1,050. Determine which, if any, of the three strategies will have greater payoffs in six months for lower prices of the stock index than for relatively higher prices. (A) None (B) I and II only (C) I and Ill only (E) The correct answer is not given by (A), (B), (C), or (D). 17.14. [Introductory Derivatives Sample Question 39] ~termine creates a ratio spread, assuming all options are European. (A) (B)

(C) (D) (E)

Buy a one-year call,and Buy a one-yearcall,and Buy a one-yearcall,and Buy a one-yearcall,and Buy a one-yearcall,and

(D) llandlllonly

which of the following strategies

sell a three-year call with the same strike price. sell a three-year call with a different strike price. buy three one-year calls with a different strike price. sell three one-year puts with a different strike price. sell three one-year calls with a different strike price.

17.15. [Introductory Derivatives Sample Question 67] Consider the following investment strategy involving put options on a stock with the same expiration date. i) Buy one 25-strike put ii) Sell two 30-strike puts iii)

Buy one 35-strike put

Calculate the payoffs of this strategy assuming stock prices (i.e., at the time the put options expire) of 27 and 37, respectively. (A) -2and

2

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(B) 0and0

(C) 2and0

(D) 2and2

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(E) 14and0

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17. OPTION STRATEGIES

200

17.16. [Introductory Derivatives Sample Question 15] The current price of a non-dividend paying stock is 40 and the continuously compounded annual risk-free rate of return is 8%. You enter into a short position on 3 call options, each with 3 months to maturity, a strike price of 35, and an option premium of 6.13. Simultaneously, you enter into a long position on 5 call options, each with 3 months to maturity, a strike price of 40, and an option premium of 2.78. All 8 options are held until maturity. Calculate the maximum possible pm fit and the maximum possible loss for the entire option portfolio. (A) (B) (C) (D) (E)

Maximum Profit 3.42 4.58 Unlimited 4.58 Unlimited

Maximum Loss 4.58 10.42 10.42 Unlimited Unlimited

17.17. An im·estorconstructs a raho spread on a stock using 1-year call option;;. She buy,; one 75--strike option and sells hvo 90-strike options. You are given: (i) The continuously compounded annual risk-free interest rate is 0.03. (ii) Call option prices are 7.19 for a 75-strikeoption and 3.12 for a 90-strike option. Let S be the price of the stock at the end of one year. Determine the range of S for which profit is posihve. 17.18.

[Introductory

(i) (ii) (iii)

Derivatives

Sample Question 43] You are given:

An investor short--sells a non-dividend paying stock that has a current price of 44 per share. This im·estor also writes a collar on this stock consisting of a 40--strike European put ophon and a 50-strike European call option. Both option;; expire in one year. The prices of the option;; on this stock are: Strike Price 40 50

(iv) (v)

Callo tion 8.42 3.86

Put o tion 2.47 7.42

The continuously compounded risk-free interest rate is 5%. Assume there are no transaction costs.

Calculate the maximum pm fit for the m·erall position at expirahon. (A) 2.61

(B) 3.37

(Cl 4.79

(D) 5.21

(E) 7.39

17.19. [Introductory Derivatives Sample Question 50] An im·estor bought a 70--strike European put option on an index with six months to expiration. The premium for this option was 1. The im·estor also wrote an &I-strike European expiration. The premium for this option was 8.

put ophon on the same index with six months to

The six-month interest rate is 0%. Calculate the index price at expiration that will allow the investor to break even. (A) 63

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(B) 73

(Cl 77

(D) 80

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(E) 87

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EXERCISESFOR LESSON 17

201

17.20. [Introductory Derivatives Sample Question 55] Box spreads are u,ed to guarantee a fixed cash flow in the future. Thus, they are purely a means ofbormwing or lending money, and have no stock price risk.

Consider a box spread ba,ed on two distinct strike prices (K, L) that is u,ed to lend money, so that there is a posihvecost to this tran;;achon up front, but a guaranteed positive payoff at expiration. Determine which of the following .ets of transactions is equivalent to this type of box spread. (A) (B)

(C) (D) (E)

17.21.

A long position ina (K, L) bull spread using calls and a long posihon ina puts. A long position in a (K, L) bull spread using calls and a short posihon using puts. A long position ina (K, L) bull spread using calls and a long position ina puts. A short position in a (K, L) bull spread using calls and a short position using puts. A short position in a (K, L) bull spread using calls and a short posihon using puts.

(K, L)bear spread using

in a (K, L) bear spread (K, L)bull spread using in a (K, L) bear spread in a (K, L) bull spread

You are interested in borrowing $10,000 for one year by using a box spread.

You are given the following option prices: Strike price

Call premium

Put Premium

30 40

8.06 3.13

1.31 5.80

The continuously compounded

risk-free interest rate is 0.06.

One unit of the box spread consists of a long 30-40 bull spread of calls and a long 40-30 bear spread of puts. Calculate the number of units of the box spread needed to achieve the financing goal, and determine whether they are bought or sold. 17.22.

Helen owns a collared stock.

You are given: (i) (ii) (iii) (iv) (v) (vi)

The The The The The The

price of the stock is 60. collar has strike prices 55 and 70. collar expires in one year. stock pays no dividends. continuously compounded risk-free interest rate is 0.02. price of a 55-70 bull spread with calls is 6.29.

Determine the price of the collar.

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17. OPTION STRATEGIES

202

[Introductory Derivatives Sample Question 60] Farmer Brown grows wheat, and will be selling his crop in 6 months. The current price of wheat is 8.50 per bushel. To reduce the risk of fluctuation in price, Brown wants to use derivatives with a 6-month expirahon date to sell wheat behveen 8.60 and 8.80 per bushel. Brown also wants to minimi~ the cost of using derivatives. The annual risk-free interest rate is 2% compounded continuously. 17.23.

Which of the following strategies fulfills Farmer Brown's objechves 7 (A) (B)

(C) (D) (E)

Short Long Long Long Long

a forward contract a call with strike 8.70 and short a put with strike 8.70 a call with strike 8.80 and short a put with strike 8.60 a put with strike 8.60 a put with strike 8.60 and short a call with strike 8.80

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EXERCISESFOR LESSON 17

203

17.24. An investor has a short collar on a stock consisting of a 40-strike put and a SO-strike call. Determine which of these graph;; represents the payoff diagram for this position at the hme of expiry of the option;;. (A)

i-(B~) ----------~

20

30

40 :AJ Stock Price

60

70

(Cl

20

30

40 50 Stock Price

60

70

20

30

40 50 Stock Price

60

70

(D)

20

30

40 :AJ Stock Price

60

70

20

30

40 :AJ Stock Price

60

70

(E)

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17. OPTION STRATEGIES

204

17.25. [Introductory Derivatives Sample Question 59] An investor has a long posihon in a nondividend-paying stock, and addihonally, has a long collar on this stock consishng of a 40-;;trike put and:(1-strikecall.

Determine which of these graphs repre.ent:s the payoff diagram for the m·erall position at the time of expiration

of the op hons.

(B)

(A)

20

30

40 :(I Stock Price

60

70

(Cl

20

30

40 50 Stock Price

60

70

20

30

40 50 Stock Price

60

70

(D)

20

30

40 :(I Stock Price

60

70

20

30

40 :(I Stock Price

60

70

(E)

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EXERCISESFOR LESSON 17

205

17.26. [Introductory Derivatives Sample Question 8] Joe believes that the volatility of a stock is higher than indicated by market prices forophons on that stock. He wants to speculate on that belief by buying or selling at-the-money options. Determine which of the following strategies would achieve Joe's goal. (A) (B) (C) (D) (E)

Buyastrangle Buyastraddle Sell a straddle Buy a butterfly spread Sell a butterfly spread

17.27. An im·estor is speculahng on the volahlity of an index. The current price of the index is 1500. The investor buys a 3-month straddle. You are given: (i) (ii) (iii) (iv)

A 3-monthat-the-money call option costs 70.45. A 3-monthat-the-money put option costs 59.96. The current price of the index is 1~. Theeffechveannual risk-free rate is 0.06.

Let S be the price of the index at the end of 3 months. Determine the range of values of S for which the investor's profit is positive. 17.28. [Introductory Derivatives Sample Question 16] The current price of a non-dividend paying stock is 40 and the continuously compounded annual risk-free rate of return is 8%. The following table shows call and put ophon premiums for three-month European of various exercise prices: Exercise Price Call Premium Put Premium 35 6.13 0.44 40 2.78 1.99 45 0.97

'·"'

A trader interested in speculating on volatility in the stock price is considering two investment strategies. The first is a 40-strike straddle. The second is a strangle consisting of a 35-strike put and a 45--strike call. Determine the range of stock prices in 3 months for which the strangle outperforms the straddle. (A) (BJ (Cl (D) (E)

The strangle never outperforms the straddle. 33.56 < Sr li), the call option must be out-of-the-money. (D) 17.12. The lowest sale price is 0.12(10,000) = 1200 due to the put options, and the highest ~le price is 0.14(10,000) = 1400 due to the call options. Costs are 84.30 - 74.80 = 9.50 inihally for the options and 1000 at the end to provide the jalapenos, or a total of9.SOe 006 + 1000 = 1010.09 when accumulated with interest. Then 1200 -1010.09 = 190 and 1400 -1010.09 = 390. Profit ranges from 190 to 390. (D) 17.13. J and II are bear spreads, which have greater payoffs up to a limit for lower stock prices. JJJ is a box spread, whose payoff is constant. (B) 17.14. Ratio spreads involve buying m options and ,elling 11 'F-m options, all ophons of the same type (call or put) and expiry date. Only (E) meets the.e conditions. 17.15. If the ending price is 27, the 35-strike put pays 8 but the investor pays 3 on each of the 30-strike puts, for net payoff of 2. If the ending price is 37, all the ophons are worthless. (C) 17.16. Since you have 5 long options and only 3 short ones, profit is unbounded; abm·e 40, every increase of x in the price of the stock results in 2x more profit. The loss is the option premium cost accumulated with interest plus the net payoffs of the calls. The highest net payoff is when the stock price is 40, since above 40, the 5 long options pay more than the 3 short ophons. When the stock price is 40, the loss is (5(2.78)-3(6.13))e- 008(l/ 4) +3(5) = 10.42

(C)

17.17. The initial cost is 7.19- 2(3.12) = 0.95; accumulated with interest, it is0.9Se 003 = 0.9789. Thus profit is positive only if the payoff is at least 0.9789, requiring S > 75.9789. If S > 90, profit is (S- 75)-2($-90)-0.9789 = 104.0211-S. This turns nonpositive for S ~ 104.0211. Thus profit is positive for( 75.9789 < S < 104.0211]. 17.18. The written collar consists of a short put and a long call. You gain 2.47 from ,elling the put and pay 3.86 for the call, for a net cost of -1.39. You also receive 44 from ,elling the stock short. This inihal profit accumulates to (44 -1.39)e 005 = 44.79. 1 stoct~~~:re~h~ :fu~~~;:a~et;!:t:/~4~;;~ a~~-r~;t: ;;:c:/~h:~t:k 7ss~i~~~r:;;;~;rt~: higher price, except that if the price is abm·e 50 you can u,e the call to bound your payment at 50. But these results are less profitable than paying 40 for the stock. 17.19. The investor spent 1 and received 8 for a gain of 7. At the end, for index price S, the im·estor will pay &I - S if 70 5: S 5: 80. (Otherwise he will pay O for S > 80 and 10 for S < 70.) Setting S = [fil makes this payment 7. (B)

::a

17.20. A box spread consists of a (K, L) bull spread plus a (K, L) bear spread. It doesn't matter whether the spreads use puts or calls. Thus only (A) and (D) are box spreads. In (A), the box spread consists of buying a bull and bear spread. A bull spread of calls and a bear spread of puts cost money, so (A) is lending. (D) consists of ,elling a bull and bear spread, so the investor in (D) receives money initially; it is equivalent to borrowing. (A) 17.21. The cost of the long bull spread is 8.06 - 3.13 = 4.93 and the cost of the long bear spread is 5.80-1.31 = 4.49. Totalcostis4.93+4.49 =9.42. Toborrow$10,000,youmust~10,000/9.42 = lt()t,1.57] units. 17.22. Let S be the stock price at the end of one year, and compare the payoffs of the collared stock and the bull spread:

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EXERCISE SOLUflONS FOR LESSON 17

209

Collared stock 55:55 55 < 5 < 70 5 ~ 70

Bull spread

55

0

70

5-55 15

s

We see that the payoff on the bull spread is 55 less than the payoff on the collared stock. The collared stock requires an inihal investment of 60 for the stock plus the price of the collar. The bull spread requires an investment of 6.29, its price. Let P be the price of the collar. (60 + P)e-1102

=55 + 6.29e-002

61.2121 +Pe- 1102 :61.4171

P

=(61.4171 -

=

61.2121)e--OOl ~

=

=

17.23. The forward price of wheat is Se-'1 8.S0e-002 (o5 ) 8.59, which is less than the desired price of 8.60, so shorhng a forward contract won't achieve the goal. A long put with strike 8.60 guarantees a price of 8.60, and the price of the put can be offset with a short call with strike8.&l, which limits the sales price to 8.80 at most. So (E) is the cheapest strategy. 17.24. A collar consists of a short call and a long put. A short collar has a long call and a short put. Thus the payoff decreases as the stock price goes below 40, and increa.es as the stock price goes above 50. It is flat in behveen. (D) 17.25. A collar limits gain and loss for the underlying asset. It consists of a purchased put, allowing sale of the asset for the strike price if the price decreases below its strike price (40 here), and a written call, which requires sale of the as.et for the strike price if the price increases abm·e its strike price (SOhere). The payoff upon closing the position is 40 if the stock price is below 40, 50 if te strike price is above 50, and the stock price if it is behveen 40 and 50. Thus (E) is the graph showing the payoff. 17.26. Buying a straddle involves buying at-the-money put and call options and profits from high volatility. (B) Strangles do not u,e at-the-money option;;. Selling a straddle benefits from low volatility. Butterfly spreads use options not at-the-money. 17.27. The cost of the straddle is 70.45 +59.96 130.41. Accumulated with 3 month;; interest, the cost is 130.41(1.061125) 132.32. The index price must mm·e up or down by at least this amount to generate a positive profit. The range of S generating a profit is [ S < 1367.68 or S > 1632.32] 17.28. The straddle consists of buying a 40-strike call and 40-strike put, cost 2.78 + 1.99 4.77. The strangle costs 0.44 + 0.97 1.41. The difference accumulated with interest is (4.77 -1.41)e- 002 3.43. The strangle outperforms the straddle if the stock price changes by less than 3.43, which mean;; it stays behveen 40 - 3.43 : 36.57 and 40 + 3.43 : 43.43. (D) 17.29. The long puts have strike price 30 and are part of the bull spread and the long calls have strike price 50 and are part of the bear spread. The raho of the former to the latter is

=

=

=

=

K3-K2 K2-K1

= 50-42

=3_

42-30

3

=

=

So !(30) [!I] long calls are needed. 17.30. This is a symmetric butterfly spread profit diagram. A butterfly spread combines a long and short bull spread or bear spread. (A) combines a long 90-100 bull and a short 100-110 bull with puts; (B) does the same with calls. (C) combines a long 90-100 bull spread with puts and a short 100-110 bull spread with calls. (E) is similar to (B), since buying a share of stock and a 90 put is similar to buying a 90 call; if the ending price is abm·e 90, you have the stock, and if it is below 90, you have 90. (D) results in unbounded profit if the stock price goes abm·e 110, not the profit diagram shown.

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17. OPTION STRATEGIES

210

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Lesson 18

Put-Call Parity Reading: Derirntires Markets 9.1 In this Jes.son and the next, rather than presenhng a mOOel for stocks or other assets so that we can price ophons, we discuss general properties that are true regardless of model. In this Jes.son, we discuss the relahonship betwt~en the premium of a call and the premium of a put. Suppose you bought a European call option and sold a European put option, both having the same underlying as.set, the ~me strike, and the same time to expiry. fn this entire section, we will deal only with European options, not American ones, so henceforth "European" should be understoOO. As above, let the value of the underlying as.set be Sr at time t. You would then pay C(K, T)-P(K, T)at time 0. lntereshngly, an equivalent result can be achieved without using ophons at alJ! Do you see how 7 The point is that at hme T, one of the two options is sure to be exerci,ed, unless the price of the as.et at time T happens to exactly equal the strike price(Sr = K), in which ca,e both options are worthless. Whichever option is exen::ised, you pay K and receive the underlying as.et:

• If Sr> K, you exerci,e the call ophon you bought. You pay Kand receive the asset. • If K > Sr, the counterparty exercises the put option you sold. You receive the asset and pay K. • If Sr= K, it doesn't matter whether you have Kor the underlying as.et.

Therefore, there are two ways to receive Sr at time T: 1. Buy a call option and .ell a put option at time 0, and pay Kat time T. 2. Enter a forward agreement to buy Sr, and at time T pay fo,r, the price of the forward agreement. By the no-arbitrage principle, the.e two way,; must cost the same. Discounhng to time 0, this means C(K, T) - P(K, T) + Ke-,r = Fo,re-,r

I Put-Call

C(K, T)-

Parity

I

P(K, T) = e-'r(Fo,r-

K)

(18.1)

Here's another way to derive the equation. Suppo.e you would like to have the maximum of Sr and Kat hme T. What can you buy now that will result in having this maximum? There are two choices: • You can buy a hme-T forward on the as.et, and buy a put option with expiry T and strike price K. The forward price is Fo,r, but since you pay at time 0, you pay e-,rFo,r for the forward. • You can buy a risk-free investment maturing for Kat time T,and buy a call option on the as.et with expiry T and strike price K. The cost of the risk-free investment is Ke-,r_ Both methods must have the same cost, so

which is the same as equation (18.1). The Put-Call Parity equation lets you price a put once you know the price of a call. Let's go through specific examples. IFMStudyManual-i"edilion Copynght"2018ASM

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18. PUT-CALL PARJTY

212

18.1 Stock put-call parity

=Soe'r. Equation =So - Ke-,r

For a nondividend paying stock, the forward price is Fo,r C(K, T)- P(K, T)

(18.1) becomes

The right hand side is the present value of the asset minus the present value of the strike. Ex.AMPLI! 18A stock for 45 at the call option Determine months.

A nondividend paying stock has a price of 40. A European call option allows buying the the end of 9 months. The continuously compounded risk-free rate is 5%. The premium of is 2.84. the premium of a European put option allowing selling the stock for 45 at the end of 9

ANswER: D:Jn't forget that 5% is a continuously compounded rate.

=So- Ke-,r =40 _45e-(005)(o75) =40 -45(0.963194) =-3.34375 P(K, T) =2.84 +3.34375 =16.18375 j

C(K, T) - P(K, T) 2.84 - P(K, T) 2.84 - P(K, T)

A convenient way to express put-call parity is with prepaid forwards. put-call parity formula becomes I

C(K, T)-

P(K, T)

=F~r -

Ke-rT

Using prepaid forwards, the

(18.2)

I

Using this, let's discuss a dividend paying stock. If a stock pay,; discrete dividends, the formula becomes C(K, T)- P(K, T) So - PVo,r(Divs)- Ke-,r (18.3)

=

188 A stock's price is 45. The stock will pay a dividend of 1 after 2 months. A European put option with a strike of 42 and an expiry date of 3 months has a premium of 2.71. The continuously compounded risk-free rate is 5%. Determine the premium of a European call option on the stock with the same strike and expiry. Ex.AMPLI!

ANswl!R: Using equation (18.3), C(K, T) - P(K, T)

=So -

PVo,r(Divs) - Ke-,r

PVo,r(Divs), the present value of dividends, is the present value of the dividend of 1 discounted 2 months at5%,or(1)e- 005 16. C(42,0.25)-2.71 45-(l)e- 005 16 -42e--0o5(o25)

= =45- 0.991701- 42(0.987578) =2.5XJO C(42,0.25) =2.71 + 2.5300 =~

®

Qu .. iz 18-1 A ;;.·tock'sprice is 5). The stock will pay a dividend_ of 2 after 4 months. A European call option 'Wlth a strike of 5J and an expiry date of 6 months has a premium of 1.62. The continuously compounded nsk-freerateis4%. Determine the premium of a European put option on the stock with the same strike and expiry. IFMStudyManual-i"edilion CopynghtC2!118ASM

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18.2. SYNTHETIC

213

STOCKS AND TREASURIES

Now let's consider a stock with continuous dividends at rate 0. Using prepaid forwards, put-call parity becomes C(K, T) - P(K, T) Soe- 6 T - Ke-,T (18.4)

=

ExAMl'LE

18C You are given:

(i) Astock'spriceis40. (ii) The continuously compounded risk-free rate is 8%. (iii) The stock's continuous dividend rate is 2%. A European 1-yearcall option with a strike of 50 costs 2.34. Determine the premium for a European 1-year put option with a strike of 50. ANSWER:

Using equahon (18.4), C(K, T) - P(K, T)

2.34 - P(K, T)

=Soe-6T - Ke-,T =40e--Oll2- 5(k-Oll'! =40(0.9&11987)- :AJ(0.923116.3)=-6.94787

P(K, T) = 2.34 +6.94787 =19.287871

~

W

Quiz

~i~~ ::o:::

;:::~

57.

(ii) The continuously compounded risk-free rate is 5%. (iii) The stock's continuous dividend rate is 3%. A European }-month put option with a strike of 55 costs 4.46. Determine the premium of a European 3-month call option with a strike of 55.

18.2 Synthetic stocks and Treasuries Since the put-call parity equahon includes terms for stock (So) and cash (K), we can create a synthehc stock with an appropriate combinahon of options and lending. With continuous dividends, the formula

C(K, T)-

P(K, T)

=S0 e- 6T -

Ke-,T

So= (C(K, T)-P(K,

T) + Ke-'T)e 6T

(18.5)

For example, suppose the risk-free rate is 5%. We want to create an investment equivalent to a stock with conhnuous dividend rate of 2%. We can use any strike price and any expiry; let's say 40 and 1 year. We have So= (C(40, 1)- P(40, 1) +40e- 005 je002

= =

So we buy e002 1.02020 call options and sell 1.02020 put options, and buy a Treasury for 38.8178. At the end of a year, the Treasury will be worth 38.817& 005 40.808. An ophon will be exercised, so we will pay 40(1.02020) 40.808 and get 1.02020 shares of the stock, which is equivalent to buying 1 share of the stock originally and reinvesting the dividends. lFMStudyManuaJ-i"edilion Copynght02!118ASM

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18. PUT-CALL PARJTY

214

If dividends are discrete, then they are assumed to be fixed in advance, and the formula becomes C(K, T) - P(K, T)

= So - PV(dividends) - Ke-,r

So= C(K, T)-

P(K, T) + PV(dividends) + Ke-,r

(18.6)

~

For example, suppose the risk-free rate is 5%, the stock is 40, and the periOO is 1 year. The dividends are 0.5 apiece at the end of 3 months and at the end of 9 months. Then their pre.ent value is 0.5e-005(025)+0.5e--OO.'i(ll75)

=0.97539

To create a synthetic stock, we buy a call, sell a put, and lend 0.97539 + 40e- 1105= 39.0246. At the end of the year, we'll have 40 plus the accumulated value of the dividends. One of the opbons will beexen::ised, so the 40 will be exchanged for one share of the stock. To create a synthetic Treasury', we rearrange the equation as follows: C(K, T) - P(K, T) Ke-,r

= $0 e- 6r - Ke-,r = $0 e- 6r - C(K, T) + P(K, T)

(18.7)

We buy e- 6 r shares of the stock and a put option and sell a call opbon. Using K = 40, r = 0.05, 0 = 0.02, and 1 year to maturity again, the total cost of this is Ke-,r = 40e- 1105= 38.04918. At the end of the year, we sell the stock for 40 (since one option will be exen::ised). This is equivalent to invesbng in a one-year Treasury bill with maturity value 40. If dividends are discrete, then they are assumed to be fixed in advance and can be combined with the strike price as follows: Ke-,r + PV(dividends) = So - C(K, T) + P(K, T) (18.8) The maturity value of this Treasury is K +CumValue(dividend;;). For example, suppose the risk-free rate is 5%, the stock is 40, and the period is 1 year. The dividends are0.5 apiece at the end of 3 month;; and at the end of9 months. Then their present value.as calculated above, is 0.5e-005{0l.5)+Q.5e--OO.'i(ll75) =0.97539 and their accumulated value at the end of the year is 0.97539e005 = 1.02540. Thus if you buy a stock and a put and sell a call, both opbon;; with strike prices 40, the investment will be Ke--0os+ 0.97539 = 39.0246 and the maturity value will be 40 + 1.02540 = 41.0254.

~ ~

~;1:.:

ai:~~!~~~~

t;f t~

1 0 n!~~rre':~~~o=;!~ i;t:skn;;:i~ ~ :1~:: 1:~t:~;=n~~~~ s:~~ of 45. The stock pays continuous dividends at a rate of 1%. The synthetic investment should duplicate 100 shares of the stock. Determine the amount you should invest in Treasuries.

18.3 Synthetic options If an option is mispriced based on put-call parity, you may want to create an aibitrage. Suppose the price of a European call based on put-call parity is C, but the price it is actually selling at is C' < C. (From a different perspective, this may indicate the put is mispriced, but let's assume the price 'CrrotingasyntheticTrro.suryisc.allOO.arom,ttsim,. a put. isc.alleda ,,,,,...,.....,,,,~,,./m,

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18.4. CURRENCY OPTIONS

215

of the put is correct based on some model.) You would then buy the underpriced call option and sell a synthesized call option. Since C(S,K,t)

=Se_,., 1

-

Ke-rt+ P(S,K,t)

you would ,ell the right hand side of this equation. You'd sell e-M shares of the underlying stock, ,ell a European put ophon with strike price Kand expiry I, and buy a risk-free zero-coupon bond with a price of Ke-rt, or in other words lend Ke-rt at the risk-free rate. These transactions would give you C(S, K, t). You'd pay C' for the option you bought, and keep the difference.

18.4 Currency options Let C(xo, K, T) be a call ophon on currency with spot exchange rate 2 xo to purcha,e it at exchange rate Kat time T, and P(xo, K, T) the corresponding put option. Putting equation 18.2 and the last formula in Table 15.1 together, we have the following formula: (18.9) where rt is the "foreign" risk-free rate for the currency which is playing the role of a stock (the one which can be purchased for a call option or the one that can be sold for a put option) and rd is the "domestic" risk-free rate which is playing the role of cash in a stock ophon (the one which the option owner pays in a call ophon and the one which the option owner receives in a put option). 18D You are given: (i) The spot exchange rate for dollars to pounds is 1.4$/£. (ii) The continuously compounded risk-free rare for dollars is 5%. (iii) The continuously compounded risk-free rare for pounds is 8%. A 9-month European put option allows selling £1 at the rateof$1.50/£. A 9--monthdollardenominated call option with the same strike costs $0.0223. Determine the premium of the 9-month dollar denominated put option. ExAMPLI!

ANswl!R: The prepaid forward price for pounds is xoe-r1r

=1_4e-OC6(075)=1.31847

The prepaid forward for the strike asset (dollars) is Ke-r,r

=1.5e-o05(o73) =1.44479

=1.31847-1.44479 =-0.12632 =-0.12632 P(1.4, 1.5,0.75) =0.0223 +0.12632 =I0.148621

C(xo, K, T) - P(xo, K, T) 0.0223 - P(1.4, 1.5,0.75)

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18. PUT-CALL PARJTY

216

('i'\

\V

~i~

0 Tu: :;:e~c=ge rate for yen to dollars is ~V/$. (ii) The continuously compounded risk-free rate for dollars is 5%. (iii) The continuously compounded risk-free rate for yen is 1%. A 6-month yen-denominated European call option on dollars has a strike price of 92Y/$ and costs Y0.75. Calculate the premium of a 6-month yen-denominated European put option on dollars having a strike priceof92V/$. Quiz

A call to purcha,e pounds with dollars is equivalent to a put to ,ell dolla15 for pounds. However, the units are different. Let's see how to translate units. Ex.AMPLI! 18E The spot exchange rare for dollars into euros is $1.05/€. A 6-month dollar denominated call option to buy one euro at strike price $1.1/€1 costs $0.04. Determine the premium of the corresponding euro--denominated put option to sell one dollar foreuros at the corresponding strike price.

ANswER: To sell 1 dollar, the corresponding exchange rate 'WOUidbe $1/€fr, so the euro-denominated strike price is = 0.9091€/$. Since we're in effect buying 0.9091 of the dollar-denominared call option, the premium in dollars is ($0.04)(0.ml) = $0.03636. Dividing by the spot rate, the premium in euros is ~=1€0.034631

fr

Let's generalize the example. Let the domestic currency be the one the option is denominated in, the one in which the price is expressed. Let the foreign currency be the underlying asset of the option. Consider a call ophon, with the following parameters: 1. The spot rate is xo units of domestic currency. 2. The strike price is K units of domestic currency. Then the call premium is C(xo, K, T) units of domestic currency. The call option allows one to buy 1 unit of foreign currency for K units of domeshc currency. To identify the currency used to price an option, we'll used for domestic and f for foreign. Our call premium is Cd(xo, K, T). Now let's create an equivalent put option. This will allow one to sell K units of domeshc currency for 1 unit of foreign currency. But a single unit of a put option allows selling 1 unit, so one unit of the equivalent put ophon must allow selling 1 unit of domestic currency for 1/ K units of foreign currency. Moreover, the spot price of domestic currency is 1/xo in foreign currency. So we need KPd(1/xo, 1/ K, t) indomeshc currency to equate to Cd(Xo, K, T) indomeshc currency.

Since the put option's price should be expressed in the foreign currency, the left side must be mulhplied by xo, resulting in KxoPj (_!_,2-,T) '" K

= Cd(Xo,K,

T)

where PI is the price of the put option in the foreign currency. Note that if settlement is through cash rather than through actual exchange of currencies, then the put options KxoPJ(1/xo, 1/ K, T) may not have the same payoff as the call option Cd(xo, K, T). The put options KxoPJ(1/xo, 1/ K, T) pay off in the foreign currency while the call option Cd(xo, K, T) pays off in the domestic currency. The payoffs are equal based on exchange rate Xo,but the exchange rate xr at time T may be different from Xo.

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EXERCISESFOR LESSON 18

®

217

Quiz 18-5 T-he spot rate __ for y_endenominated _inpounds sterling is 0.005£/-Y. A 3-month_po=d_ -denom_;nated put option has strike 0.0048£/Y and costs £0.0002. Determine the premium m yen for an equivalent 3-month yen-denominated call option with a strike ofY208j-

Exercises Put-call parity for stock options 18.1. [CAS8-S03:18a] A four-month European call option with a strike price of 60 is selling for 5. The price of the underlying stoi:.:kis 61, and the annual continuously compounded risk-free rate is 12%. The stock pays no dividends. Calculate the value of a four-month European put option with a strike price of 60. 18.2

For a nondividend paying stock, you are given:

(i) ltscurrentpriceis30. (ii) A European call option on the stock with one year to expiration and strike price 25 costs 8.05. (iii) The continuously compounded risk-free interest rate is 0.05. Determine the premium of a 1-year European put option on the stock with strike 25. 18.3. A nondividend paying stoi:.:khas price 30. You are given: (i) The continuously compounded risk-free interest rate is 5%. (ii) A 6-month European call option on the stoi:.:kcosts 3.10. (iii) A 6-month European put option on the stoi:.:kwith the same strike price as the call option costs 5.00. Determine the strike price. 18.4. A stock pays continuous dividends proportional to its price at rate ti. You are given: (i) The stock price is 40. (ii) The continuously compounded risk-free interest rate is 4%. (iii) A 3-month European call option on the stoi:.:kwith strike 40 costs 4.10. (iv) A 3-month European put option on the stoi:.:kwith strike 40 costs3.91. Determine ti. 18.5. For a stock paying continuous dividends proportional to its price at rate ti= 0.02, you are given: (i) The continuously compounded risk-free interest rate is 3%. (ii) A 6-month European call option with strike40 costs 4.10. (iii) A 6-month European put option with strike40 costs 3.20. Determine the current price of the stoi:.:k.

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18. PUT-CALL PARITY

218

18.6. A stock's price is 45. Dividends of 2 are payable quarterly, with the next dividend payable at the end of one month. You are given: (i) (ii)

The continuously compounded risk-free interest rate is 6%. A 3-month European put option with strike SOcosts 7.32.

Determine the premium of a 3-month European call option on the stock with strike 50. 18. 7.

A dividend paying stock has price 50. You are given:

(i) (ii) (iii)

The continuously compounded risk-free interest rate is 6%. A 6-month European call option on the stock with strike 50 costs 2.30. A 6-month European put option on the stock with strike 50 costs 1.30.

Determine the present value of dividends paid m·er the next 6 months on the stock. 18.8. Consider European options on a stock expiring at time I. Let P(K) be a put ophon with strike price K,and C(K) be a call option with strike price K. You are given P(:()) - C(55) = -2 P(S5) - C(60) = 3 P(«J) - C(SO)= 14

(i) (ii) (iii)

Determine C(60) - P(50). 18.9.

[CASB·S00:26] You are given the following:

(i) (ii) (iii) (iv) (v)

Stock price=$.50 The risk-free interest rate is a constant annual 8%, compounded conhnuously The price of a 6-month European call ophon with an exercise price of$48 is $5. The price of a 6-month European put ophon with an exen::ise price of$48 is $3. The stock pays no dividends

There isan arbitrage opportunity puts and calls.

involving buying or ,elling one share of stock and buying or selling

Calculate the profit after 6 months from this strategy. 18.10. (i) (ii) (iii) (iv) (v)

[Introductory

Derivatives

Sample Question 2] You are given the following:

The current price to buy one share of XYZ stoi:.:kis 500. The stock does not pay dividends. The annual risk-free interest rate.compounded continuously, is 6%. A European call option on one share of XYZ stoi:.:kwith a strike price of K that expires in one yearcosts66.59. A European put option on one share of XYZ stoi:.:kwith a strike price of K that expires in one year costs 18.64.

Using put-call parity, calculate the strike price, K. (A) 449

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(B) 452

(Cl 4&1

(D) 559

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(E) 582

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EXERCISESFOR LESSON 18

219

18.11. [Introductory Derivatives Sample Question 5] The •

One share of the

•

The

rs index currently

rs index does

rs index has the following

characteristics:

sells for 1,000.

not pay dividends.

Sam wants to lock in the ability to buy this index in one year for a price of 1,025. He can do this by buying or selling European put and call options with a strike price of 1,025. The annual effective risk-free interest rate is 5%. Determine which of the following gives the hedging strategy that will achieve Sam's objective and also gives the cost today of establishing this position. (A) (B)

(C) (D) (E)

Buy the Buy the Buy the Buy the Buy the

put and put and put and call and call and

sell sell sell sell sell

the the the the the

call, call, call, put, put,

receive 21.81 spend 23.81 no cost receive 21.81 spend 23.81

18.12. [Introductory Derivatives Sample Question 14] The current price of a non-dividend paying stock is 40 and the continuously compounded annual risk-free rate of return is 8%. You are given that the price of a 35--strikecall option is 3.35 higher than the price of a 40-strike call ophon, where both options expire in3 months. Calculate the amount by which the price of an otherwise equivalent 4()-strike put option exceeds the price of an othenvi,e equivalent 35-strike put option. (A) 1.55 18.13. [Introductory price is 100.

(B) 1.65

(Cl 1.75

(D) 3.25

(E) 3.35

Derivatives Sample Question 41] XYZ stock pay,; no dividends and its current

Assume the put, the call and the fonvard on XYZ stock are available and are priced so there are no arbitrage opportunities. Also, assume there are no transaction costs. The current risk-free annual effective interest rate is 1%. Determine which of the following strategies currently has the highest net premium. (A) (B)

(C) (D) (E)

Long a six-month Long a six-month Long a six-month Short a six-month Long a six-month

100-strike put and short a six-month 100-strike call fonvard on the stock 101-strike put and short a six-month 101-strike call forward on the stock 105-strike put and short a six-month 105-strike call

18.14. [Introductory Derivatives Sample Question 40] An investor is analyzing the costs of two-year, European options for aluminum and zinc at a particular strike price. For each ton of aluminum, the h~atedfor C0111Jl'lrit>11CI') Ust> tht> fallou~11gi11farmatio11 far qut>slimrs18.29 a11d18.30:

You are given: (i) The spot exchange rate is 1.5$/£. (ii) The continuously compounded risk-free rate in dollars is 6%. (iii) The continuously compounded risk-free rate in pounds sterling is 3%. (iv) A 6--month dollar-denominated European put option on pounds with a strike of 1.5$/£ costs $0.03. 18.30. Determine the premium in pounds of a 6-month pound-denominated dollars with a strike of (1/1.5)£/$.

European put option on

18.31. [1999 Sample 2:44] You purchased a share of XYZ Corporation at 25, and it has now increased to 45. You are given: (i) the annual risk free rate is 5 (ii) the price for a three month call option with an exercise price of 35 is 13; (iii) XYZ does not pay dividends; and (iv) you want to lock ina sale price of at least 35 for the next three months. What is the cost of the option that achieves this result? (A) 1.58

(B) 2.58

(Cl 3.58

(D) 4.58

(E) 5.58

18.32. The spot exchange rate of dollars foreuros is 1.2$/€. A dollar-denominated put optiononeuros has strike price$1.3. Determine the strike price of the corresponding euro-denominated call option to pay a certain number of euros for one dollar. 18.33. You are given: (i) The spot exchange rate of dollars foreuros is 1.2$/€. (ii) A one-year dollar-denominated European call option on euros with strike price $1.3 costs 0.05. (iii) The continuously compounded risk-free interest rate fordolla15 is 5%. (iv) A one-year dollar-denominated European put optiononeuros with strike price $1.3 costs 0.20. Determine the continuously compounded risk-free interest rate for euros. 18.34. You are given: (i) The spot exchange rate of yen foreuros is 110Y/€. (ii) The continuously compounded risk-free rate for yen is 2%. (iii) The continuously compounded risk-free rate for euros is 4%. (iv) A one year yen- Sr - K. If Sr ;::,_ K the sum of the payoffs not including interest is the same as the payoff with early exercise, but if Sr < K the payoff is greater this way.

This illustrates two of the disadvantages of early exercise. You lose protechon against the price of the stock going below the strike price. The call option gives you an implicit put option on the stock. And you loseinterestonthestrikeprice. The algebraic proof that early exercise of an American call ophon on a nondividend paying stock is not rational is by using put-call parity. Although put-call parity only applies to European ophons, an American ophon is worth at least as much as a European option. By put-call parity, at time I, an ophon that expires at hme T satisfies: CEu,(Sr, K, T- t) = PEu,(Sr, K, T- t) + Fi,r(S) - Ke-,(r-t) but for a nondividend paying stock, Fi,r(S) = 51, so CEur(Sr,K, T- t) = PEur(Sr,K, T- t)+ Sr - Ke-,(r-t)

= PEur(Sr,K, T- t)+ (Sr - K) + K (1-e-,{r-t))

(19.1)

;::,_ Sr-K

while exercising the option is worth Sr - K, so the European ophon for the remainder of the periOO is worth more than the exercise value. The American option for the remainder of the periOO is worth at least as much as the European option, so early exercise is not rational. 'Selling a stod.short ,n.,anss,,lling a stod. that you do not own. You borrow thestc,clands,,11 it. Later on, you mt>Stbuy it baclsothatyouc.anreturnit.Whilethestod.i5lmrr t costs less than C(S, K, I). Then C(S, K, I) is too expensive, so sell it and buy C(S, Ker(r-t), T). You immediately get a cash flow of C(S, K, t) - C(S, Ke'(r-r), T). Now let's evaluate your future profit under all possible scenarios: 1. Al time t, Sr 5: K. Then C(S, K, t) is worthless. You keep the profit you already made. In addition, you may makeaddihonal profit if Sr> Ke'(r-t)_ 2. Al lime t, Sr > K. Borrow the stock and deliver it and collect the strike price K. Alternatively, pay off Sr - Kand sell the stock short, achieving the same result, since you get Sr for the stock and pay Sr - K, leaving you with K. Invest Kat the risk-free rate, so you'll have Ke'(r-t) at hme T. Then: (a) If Sr 5: Ker(r -t), your accumulated cash will beat least enough to buy back the stock, and you'll get additional profit to the extent the stock price is lower than Ke'(r-t)_ IFMStudyManual-i"edilion CopynghtC2!118ASM

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19. COMPARJNG OPTIONS

240

TimeO: lluyC(S,Kr,Tl SellC(S,K,,1) Net receipt of C(S, K,, t)- C(S, Kr, I)

NO

TimeT: lluysrnckforSr Net payment Sr
Ki and C(S, K2, T) < C(S, Ki, T) - (K2 - Ki)- To create an aibitrage maximizing the initial gain, ,ell one Ki-strike call and buy one Ki--strikecall. The immediate gain is C(S, Ki, T) - C(S, Ki, T), which is greater than Ki - Ki. If none of the ophons are exercised you have a gain. Suppo.e the ophon you sold is exercised at time I :ST. Then there are two possibilihes: 1. Ki < St ::!, Kt You pay Sr - Ki, which is less than K2 - Ki. Since you got more than K2 - Ki inihally, you have a net gain. 2. St > Kt Exercise the ophon you bought. You pay Sr - Ki and receive 51 - Ki on the ophon you bought, for a net payment of K2 - Ki. Since you got more than K2 - Ki initially, you have a net gain. An arbitrage with minimal initial gain is illustrated in the following example: Ex.AMPLI!

19D Two 1-year American call option5 on the same stock are priced as follows: Strike price

Premium

40 45

10 4

The continuously compounded risk-free inrerest rare is 0.08. You take advantage of arbitrage by buying one 45--strikecall and selling c 40-strike calls, where lowest possible value that results in no possibility of loss when interest is ignored. Afrer one year, the stock price is 46. Determine your profit including interest.

c is the

ANswl!R: American option5 can be exercised at any time, so the inrerest rare plays no role in the solution. Your initial gain is 10c - 4. We already saw above how to perform an arbitrage for c 1. Therefore, the lowest value of c that will work cannot be more than 1. Now consider the payoff at the end of one year. We'll con5ider the three cases:

=

1. S1 = 45 The call you bought is worthless and you have to pay Sc on the calls you sold.

=

2. S1 > 45 If 51 45 + k, k > 0, then you collect k on the call you bought and pay c(S + k) on the calls you sold. The net payment is Sc+ k(c -1), but this is less than Sc since c < 1, so this case costs you lessthanthefirstcase.

=

3. S1 < 45 If 51 :S 40 then all options are v,,orthless and there is no payoff. If 51 40 + k, 0 < k < 5, then you pay kc, which is less than Sc, so this case costs you less than the first case.

So the worst case is 51 =45 when you pay Sc at expiry. We want to select c so that there is no possibility of loss, so let's set the total of initial gain and final cost equal to 0. (10c-4)-Sc

=O Sc =4

c=!

=

You sell 4/5 of a 40-strike call. Your initial gain is 10(4/5) - 4 4. At expiry, your gain is (46 - 45) (!)(46 - 40) = -3.8. With inrerest, your net gain is 4eOC6- 3.8 = 0.5.33148j.

I

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246

19. COMPARJNG OPTIONS

To generalize this answer, the worst case is when the KI options are exerci,ed when Sr = Ki. Your initial gain is cC(S, K1, T)- C(S, K2, T) and the payment at time I is c(K2 - K1). Solving fore to make the sum(}:

cC(S,K1, T)c

C(S,K2, T)-c(K2

C(S,K~:tt~(;~

- Ki)= 0

-Ki)

A stronger statement regarding slope can be made for European options, as discussed in the following qmz.

®

Quiz 1~. The slope inequalities discussed in this subsubsection, equations (19.4.1), ignored interest. V\'e must ignore interest if we want them to apply to Amencan as "Wellas European options, smce for American options it is impossible to determine when the options will be exercised. All we can say is that if the option you sold is exercised, you can exercise the option you bought at the same time and effect an arbitrage in that manner. For European options (or American call options on non dividend paying stocks), for which we know the exercise time and can consider interest, we can make stronger statements. 1. Suppose C(S, K2, T) and C(S, K1, T) are two European options on the same stock, K2 > K1, and the continuously compounded risk-free interest rate is r. Determine the maximum possible value for C(S, K1, T) - C(S, K2, T). 2. Redo example 190, except that instead of selecting c as the lowest possible value that results in no possibility of loss when interest is ignored, select c as the lowest possible value that results in no possibility of loss, taking interest into account.

Convexity

Looking at the graphs in Figures 19.4 and 19.5, we see that they are bowl-shaped, or concave up. The put premium increases more quickly with higher Kand the call premium decreases more slowly with higher K. Option premiums are comex. The rate of decrease in call premiums as a function of K decreases. The mte efincrrose in put premiums as afunction of K increases Of the three properties, this one is the hardest to understand. So let's start with a specific example. Although the following example is phrased as a European option, everything applies equally well to American options, since if one option is exercised early, the other two options could be exercised at the same time. Suppose there are 40--strike, 50--strike,and 80--strikecall options on the same stock with the same expiry date T. Let portfolio A have one :(]-strike call option. Portfolio B will have a 40-strike call options and b &I-strike call options. We want to select a and b so that portfolio B pays at least as much as portfolio A in all cases. What is the smallest a, and the smallest b to go along with that a, such that portfolio B will alway,; pay at least as much as portfolio A? To answer the question, let's analyze the situation at expiry. There is no loss in generality, since the following logic can be used if the 50--strikeoption is exercised early, by exercising the other two options early (if the stock price is high enough). So let Sr be the stock value at expiry. We must consider three cases: Sr ~ 40, 40 < Sr ~ 80, and Sr > 80. 1. Sr ::!,40 is trivial since none of the options pays, so B's payout equals A's payout in that case no matterwhat(a,b)is. 2. 40 < Sr::!, 80. For40 < Sr ~ 80, what is the worst possiblecase 7 The80-strike call pays nothing, so we must compare a 40-strike calls with one SO-strikecall. One 40-strike call alway,; pays more than IFMStudyManual-i"edilion CopynghtC2!118ASM

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19.4. DIFFERENT STRIKE PRICES

247

Payoff

40

Figure 19.6: Demonstration that Portfolio B, consisting of 0.75 of a 40-strike call option and 0.25 of an SO-strike call option, always pays more than Portfolio A, which consists of one SO-strikecall option

one SO-strike call in this range, so a ~ 1. Think of Figure 16.1: the SO-strike call's payoff abm·e ~ is a straight line with slope 1, whereas the payoff of a 40-strike calls above 40 is a straight line with slope a ~ 1, so the exce~ of a 40-strike calls over one SO-strike call decreases as Sr increases. The worst po~ible case in this range is therefore Sr = 80. To make the payoff equal for Sr = 80, we need 40a = 30, or a = 0.75. With a = 0.75, portfolio B pays at least as much as portfolio A for Sr ~ 80. 3. Sr > 80. For Sr > 80, if the payoffs on the hvo portfolios increase at the same rate as Sr increa.es, then portfolio B will always pay at least as much as portfolio A, since they already pay the same at 80, by the last paragraph. Consider the change in payoff as Sr increases by E. For portfolio A, the payoff increases by E. For portfolio B, the payoff increa.es by E(a + b). So we need a+ b = 1. This meansb=0.25. Figure 19.6 shows the resulting Portfolio B when (a,b) = (0.75,0.25). We see looking at the figure that b was set so that the line going from (40,Q)hits the Portfolio A line at Sr = 80, at which point the &I-strike ophon pay,; off. With (a, b) = (0.75,0.25), portfolio B pay,; at least as much as portfolio A in all cases, but it pays more when 40 < Sr ~ &I. Portfolio A pays nothing below~- and pays le~ than portfolio B abm·e ~ but below &I. So portfolio B must cost at least as much as portfolio A That means that0.75C(S,40, T) +0.25C(S,80, T) ~ C(S.~. T). The generalization of this logic is that if K3 > K2 > K1 and we express K2 = aK1 + bK3 with a+ b = 1, then C(S, K2, T) ~ aC(S, K1, T) + bC(S, K3, T). In other words, the graph of call prices as a function of K is below a straight line, or convex; if you draw a straight line from one point of the graph to another, the graph will always be below that line. And this is true for puts as well. Algebraically, for K3 > K2 > K1: C(S, K3, T) - C(S, K2, T) > C(S, K2, T) - C(S, Ki, T) K3-K2

P(S,K3,T)-P(S,K2,T)

-

K2-K1

> P(S,K2,T)-P(S,K1,T)

K3-K2

K2-K1

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19. COMPARJNG OPTlONS

248

An equivalent way of expressing convexity is that the price of the K2-strike option is le~ than the linearly interpolated prices of the other two options: C(S, K , T) :S. (K2 - K1)C(S, K3,;~ ~ ~~3 - K2)C(S, K1, T) 2 P(S, K , T) :S. (K2 - K1)P(S, K3, ;~ ~ ~~3 - K2)P(S, Ki, T) 2 In terms of derivatives,

®

a2C~;:, T)

~O

(}2p~;;,T)

~O

Quiz 1.·.9-7Foro. ptions wit.h the same style (Europ.eanor American) and expiry date, you are given: (1) 40-stnke put options on a stock have pnce 1. (11) 70-stnke put options on a stock have pnce 10. Determine, based on convexity of option prices, the highest possible price for a 60-strike put option, and determine the number of 40- and 7(}-strike put options needed to guarantee a payoff at expiry at least as large as the payoff of one60-strike put option. If there is a mispricing, the option in the middle will be overpriced relahve to the other two options. Therefore, ,ell the ophon in the middle and buy the ones at the two extremes. An arbitrage that maximizes inihal gain is the one we have just discu~ed. To create such an arbitrage with calls, sell the K2--strikecall. Buy K1--strike and Krstrike calls such that a linear combinahon of K1 and K3 equals Ki. In other words, buy~ K3--strikecalls and~ Ki-strike calls. Since there is a mispricing, the price of the Krstrike call is greater than the linearly interpolated price of the other hvo calls, so you have an immediate gain. If the K2--strikecall option is never exerci,ed, you come out ahead. Let's go through the cases in which the K2--strikecall is exerci,ed.

1. K2 0 and we get :: = ::(Sr - K1)-(Sr - K2) = :: = ::(K3 - K1 -x)-(K3

- K2 -x)

=(K3 - K2)- :: =::(x)-(K3 = K3-K: -_(::-K2)(x)

- K2) +x

3

= ::=::(x) which is posihve since both factors are posihve. Don't let the algebra confuse you; Figure 19.6 is what you should think about, and you .ee there that Sr = K3 is the worst ca,e, the point at which the hvo lines meet. The algebra simply describes the line from (40,0) to (80,30) in that graph. 2. St > K1 You pay Sr - K2 and receive :: = :~(Sr - Ki)+::=

::(Sr - K3) =Sr+::=

::(-K1)+

:: = ::(-K3)

= Sr - K2

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19.4. DIFFERENT STRIKE PRICES

249

In the above arbitrage, you sold one K2-strikecall for the price II and you bought p K1-strikecalls and q =~-and paid v. Your gain was 11 - v. To create an arbitrage with minimal initial gain, sell one K2-strike call and buy (11/v)p K1-strike calls and (11/v)q K3-strike calls. You will pay a total of II for the calls you bought, resulting in no initial gain. However, the portfolio you bought will pay at least 11/v > 1 times as much as you will pay on the K2-strikecall. Unlike the direction and slope properhes, which only involve two ophons, convexity involves three options. For direchon and slope properties, when selling the m·erpriced ophon, the range of numbers of the other option to buy for an arbitrage is a closed interval. For convexity, the range of the number of each of the two other options to buy to form an arbitrage isa two-dimensional figure, a triangle. The next example works out the region of arbitrages.

q K3-strike calls, where p =~and

19E For three 6-month American call options on a stock: (i) One with strike price 45 sells for6.30. (ii) One with strike price 44 sells for 7.00. (iii) One with strike price 40 sells for9.50. The option with strike price 44 is m·erpriced based on the convexity property of ophon premiums. You therefore sell it. Determine the maximum and the minimum amount of the other two options you should buy to guarantee a profit. ExAMl'LE

ANSWER: Let x be the number of 40-strike calls and y the number of 45--strike calls. The first condihon we need is no initial im·estment; there is no guarantee of future gain, since it is possible that none of the options will pay off, so we need to assure no net im·estment if we want an arbitrage. To assure no net investment,weneed 9.50x+6.30y 5. 7.00

To assure no loss, we will exercise our options when the 44-strike one is exercised. If the stock price S is less than 44, the 44-strike option will not be exercised. The stock price can be arbitrarily high. For each unit increase in stock price abm·e 45, we must pay an additional 1 on the option we sold. So we must make sure that the ophons we buy also pay an additional 1 for each increase in the stock price abm·e 45. Thatmeansx+y ;::,.1. The final condition takes care of when the stock price is between 44 and 45. In this range, only the 40-strike option pays off. We must make sure that the amount it pay,; is more than the amount we need to pay, or at least that the amount it pays plus the profit we made inihally is more than the amount we need to pay. The worst possible case is when the stock price is 45. The profit we made initially is

7-9.5x-6.3y The amount we need to pay if the stock price is 45 is 1-Sx So we must have

7-9.5x-6.3y ;::,_ 1-Sx 4.Sx +6.3y 5.6 So we have threecondihons: 1. 9.5x +6.3y 5. 7 2. 4.5x +6.3y 5. 6 3. x+ y ;::,_ 1.

Calculate the intersections of the three lines. As you can see from Figure 19.7, x may be between 1/6 and 7/32,while y maybe between 25/32 and 5/6. D

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19. COMPARJNG OPTIONS

250

Figure 19.7: Values of (x, y) allowing arbitrage in E)(ample 19E

19.4.2 Options in the money An option is in the money if it would have a positive payoff if it could be exen::ised. A call ophon is in the money if the strike price is less than the underlying a~t price. A put option is in the money if the strike price is more than the underlying a~t price. An option is out of the money if the price of the underlying asset is different from the strike price in such a way that the ophon doesn't pay off. A call option is out of the money if the strike price is greater than the underlying asset price. A put option is out of the money if the strike price is less than the underlying as.et price. An option is at the moneyif the strike price equals the underlying asset's price. Consider two American call options with the same expiry but with different strike prices, K1 and K2 > K1. Assume both are in the money; in other words, the stock price is greater than K2. Suppose the option with strike price Ki is optimal to exercise. We are going to demonstrate that in that ca,e, the ophon with lower strike price K 1 is also optimal to exercise. The proof goes like this:

1. By the slope property for two options, the price of the option with strike price K1 cannot be greater than the price of the option with strike price Ki plus the difference in the strike prices:

2. Since the ophon with strike price Ki is ophmal to exercise, its value is S - Ki. Plug that into the abm·e equahon: 3. S - K1 is the exerci,e value of the K1 strikeoption,and the inequality we just derived indicates that the option is worth no more than its exerci,e value, so it should be exercised. A similar argument for puts demonstrates that if a put with strike price K1 is ophmal to exercise, then so is an otherwise similar put with strike price K2 > K1.

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EXERCISESFOR LESSON 19

251

Exercises Bounds for Option Prices 19.1. The current price of a stock is 35. Let C(S, K, T) and P(S, K, T) be European calls and puts respechvely on the stock with strike price Kand expiry T. Which of the following statements are true 7 I.

P(S,35,T)

~35e-,r

-35e--,1r

II. P(S,35, T)- C(S,30, T)

~

30e-rr -35e- 6r

JI!. P(S,35, T)- C(S,30, T)

~

35e-rr -35e-6r

19.2

You are given:

(i) Thepriceofastockis35. (ii) The stock pays continuous dividends proportional to its price at an annual rate of0.04. (iii) The continuously compounded risk free interest rate is0.06. (iv) A one-year American call ophon on the stock has a strike price of 32. Determine the lowest possible price for this call option. 19.3. You are given: (i) Thepriceofastockis35. (ii) The stock pays continuous dividends at the annual rate of 0.04. (iii) The continuously compounded risk free interest rate is0.06. (iv) A one-year American put ophon on the stock has a strike price of 34. Determine the lowest possible price for this put option. 19.4. A 182-day American call option ona stock has a strike price of 100. The continuously compounded risk-freeinterestrateis4%. A dividend is payable on the stock on day 91. Determine the lowest dividend for which early exerci,e of the option may be rahonal. 19.5. [2-F00:25] Which of the following statements about options are true? I.

Put/call parity implies that puts and calls should trade at the same price, when the stock price equals theexen::iseprice.

II. Early exerci,e of an American call option only makes ,en,e when there is positive cash flow prior to maturity on the underlying as.et. Ill.

As price volatility increases, call option prices ri,e, and put option prices fall.

(A) ]only

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(B) llonly

(C) lllonly

(D) I and II only

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(E) I and Ill only

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252

19. COMPARJNG OPTlONS

Different strike prices 19.6. You are given the following prices for American call options:

I

Stn:f'"

I

Optm1~ pnce

I

Determine the highest possible price for an American call option with strike price 48. 19. 7. You are given the following prices for European put options: Strike

rice

tion

40 50 Determine the highest possible price for a European put option with strike price 44. 19.8. You are given the following prices for American call options: Strike

rice

tion

40 50 55 To exploit the mispricing, you sell one SO-strike option and buy x 40-strike ophons and y 55-;;trike options, with x and y selected to create an aibitrage. Determine the range of possible values for x. 19.9. You are given the following prices for American put options:

To exploit the mispricing, you will buy ten .:()-strike put options. Let x be the number of 65-strike put options to ,ell and y the number of 75--strikeput options to buy. Determine the range of possible values for x. 19.10. For a stock, you are given: (i) Thestock'spriceis40. (ii) The stock's continuous dividend rate is0.02. (iii) A one-year 35-strike European call ophon has premium 10. (iv) A one-year45-strike European call ophon has premium 2. (v) The continuously compounded risk-free interest rate is 0.05. Determine the lowest and highest arbitrage-free premiums for a 1-year 4()-strike European put ophon on the stock.

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EXERCISESFOR LESSON 19

253

19.11. For a stock, you are given: (i) Thestock'spriceis40. (ii) The stock pays no dividends. (iii) A 6-month30--strike European put ophon has premium 3. (iv) A 6-month40--strike European put ophon has premium 6. (v) A 6-month45--strike European call option has premium 3.5. (vi) The continuously compounded risk-free interest rate is 0.05. To exploit the mispricing, you sell three 40--strike put ophons, buy one 30--strike put ophon, and buy two synthetic 45--strike put ophons created using appropriate amounts of 45-strike call ophons, shares of stock, and risk-free bonds. Calculate your net gain after6 months, including interest, if the stock price is43 then. Additional released exam questions: Advanced Derivahves Sample:2,26, CASJ.-507:12,CAS3--F07:13, MFE/3F-509:12

Solutions 19.1. I. By put-call parity P(S,35, T) - C(S,35, T) 35e-rT _35e-6T_ ✓

= 35e-,r

- 35e- 6r and C(S,35, T) ~ O so P(S,35, T) ~

=

II. By put-call parity P(S,30, T) - C(S,30, T) 30e-rr - 35e- 6r, and P(S,35, T) ~ P(S,30, T). ✓ JI!. Since P(S,35, T)- C(S,35, T) 35e-,r -35e- 6r and C(S,30, T) ~ C(S,35, T), the inequality should be reversed: P(S,35, T)- C(S,30, T) 5: 35e-,r -35e- 6r. X

=

=

19.2. Since it is an American option, it could be exerci.ed immediately for a gain of 35 - 32 3, so it must be worth at least 3, but we can do better. It is worth at least as much as a European call option, which must be worth at least as much as the put-call parity value using a put value of 0, or

19.3. An American put option is worth more than a European put option. By parity to a European call option with value 0, the put ophon is worth at least

That lower bound is not as good as the lower bound of 0, so [illis the only lower bound for the put premium. 19.4. At the very least, early exercise means losing interest on the strike price. Paying at day 91 costs 100, 004 while the present value at day 91 of paying at day 182 is 100e- 0 251'. l, so the lost interest is

so the dividend must be at least 0.995017.

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19. COMPARJNG OPTIONS

254

19.5. I. This is not true if the dividend rate does not equal the risk-free rate; if S 1 Se-M - Ke-rt S(e--,1 - e-'t) 0 only if the parenthesized expression is 0. X

=

=

= K, then

C- P

=

II. Thisistrue. ✓ Ill.

Increases in volahlity increase the prices of both call and put ophons. increase, since the probabilihes that they pay off increase. X

Intuitively, prices should

(B)

19.6. Byconvexity, C(S,48) 19.7.

s. 0.2C(S,40)

+0.BC(S,50)

=0.2(12) + 0.8(3) =]}II

Byconvexity, P(S,44) 5-

G~

=:~)P(S,40)

+ (:~=:~)P(S,50)

=0.6(4) +0.4(9)

=~

19.8. Refer to Example 19E for a similar queshon. We have x + y ;::,_ 1, and to assure no investment, 5x + 3y 5- 4. If the stock price is 55, we need the net amount received initially, 4 - Sx - 3y, to be greater than the payment at the end, 5-15x, so 4- Sx -3y ;::,_ 5-15x, or 10x -3y ~ 1. So the three constraints are 5x +3y 5:4 10x -3y ~ 1 x+y~1 The inter,ection of the first and third constraints is(½,½), and the inter,echon of the ,emnd and third mn;;traintsis (fj,"fJ),sothean;;werislti

::!,x

fj.

::!.

19.9. To have no loss initially, we need the proceeds from the 65-strike puts (12 for each put sold) to exceed the cost of the 50-strike puts (5 for each of the ten pun::hased) plus the mst of the 75-strike puts (15 for each put purcha,ed), or 12x ~ 10(5)+15y 12x-15y ~ 50 To have no loss if the 65-strike puts are exerci,ed when the stock price is less than 50, the worst possible case is if the stock price is 0. (We assume the stock price cannot be negative.) Then the gain on ten 50-strike puts and y 75--strike puts plus the initial gain must exceed the gain on x 65-strike puts.or :(1(10) +75y + 12x -15y -:(] ~ 65x 53x -«ly 5: 450

If the 65--strike puts are exerci.ed, we can exercise the 75--strikeputs simultaneously. To have no loss if the 65--strike puts are exercised when the stock price is beh\'eert 50 and 65 (50 being the worst ca,e), we need the payment on the x 65-strike puts minus the proceeds from they 75-strike puts (since the 50-strike puts may end up being worthless) to be less than the initial gain, or

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15x-25y

5: 12x-50-15y

-3x+10y

~:(]

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EXERCISESOLUflONS FOR LESSON 19

The inter.ectionof

255

the fi15tand ,emnd constraints is 12x-15y =50 53x-60y=4~ 48x-60y

= 200 50 12(:AJ)-~ y=-,-,-=3 X:

110

The inter.ection of the fi15t and third mnstraints is 12x-15y -3x+10y

= 50 50

=

-12x+40y

=200

y

=10

x=~

3

The intersection of the second and third constraints (not needed to answer the exercise's question) is 53x -60y =450 -3x+10y=50 -18x+60y=~ 35x =750 x=~

7

50+3(150/7) y=--10--=7

&I

Figure 19.8 shows the area of possible (x, y). Thus [ 50/3 ::!, x ::!, 50 ]. 19.10. By slope, the difference beh\'eena 35-strike call option and a 40-strike call ophon for a European option cannot be greater than 5 discounted at interest. See Quiz 1%. Therefore the largest difference of price beh\'een the calls is 5e-' 1 = 5e- 1105 = 4.75615. This implies that a 40-strike call option must be worth at least 10-4.75615=5.24385. By convexity, a 40-strike call option must be worth at most the linearly interpolated value of 35--strike and 45-strike, or (½)(10)+ (½)(2)= 6. By put-call parity, the put premium is P(40,40, 1) = C(40,40,1) +40e--0os -40e--0oi = C(40,40,1)-1.15877 Plugging in 5.24385 5. C(40,40, 1) 5. 6, we get [4.08508 ::!,P(40,40, 1) ::!,4.84123 f 19.11. To synthesize a 45--strike put option, by put-call parity, we need P(40,45,0.5) = C(40,45,0.5) +45e--0oi5 - S so to synthesize 2 puts, we buy 2 calls, make a loan for 2(45)e--0o25 = 87.7779, and ,ell 2 shares of stock. The total cash flow of the inihal transactions is Buying 30-strike put -3.0000 Selling 3 40-strike puts 18.0000 Buying245-strikecalls -7.0000 Making loan -87.77"79 1').0000 Selling 2 shares Total IFMStudyManual-i"edilion CopynghtC2!118ASM

0.2221

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19. COMPARJNG OPTIONS

256

40

Figure 19.8: Values of (x, y) demonstrating arbitrage in e)(ercise 19.9

After 6 months, all ophons are worthless. The loan pays back 90 and we pay 2(43) = 86 to buy back the stock. The initial gain is accumulated with interest. Thus the net gain after 6 months is 90-86+Q.2221e002S :~

Quiz Solutions 19•1. The American put ophon must be worth at least as much as a European put option. parity with a call premium of 0:

By put-call

So the American put option must have a price of at least~-

19•2. The value of the put ophon is0.82. The value of the interest on the strike price is

So the total loss by exercising now is 0.82 + 1.4049 = 2.2249. The present value of dividends is 1.5 (1 + e-(004){ 3/12)) : 2.9851. Since the present value of dividends is greater than the Ins.es from early exerci,e, it may be rational to exercise the option early.

19•3. You clearly make money in the boxes where the I option is worthless and the T option pays off. These are the lower left call box and the upper right put box. In addition, the boxes where the I opbon pays off and the T opbon is worthless are, paradoxically, boxes where you make money. For a call, if the t option pays off, you sell the stock short and receive K. At time T, if Sr< Ke'(r- 1), you pay less than Ke'(r -t) to buy back the stock and keep the excess interest, Ke'(r -t) - Sr. For a put, if the t option pays off, you pay Kand receive the stock. At time T, if Sr> Ke'(r-t), you ,ell the stock and have more than enough to pay back the loan of K with interest. So the following are the boxes with checkmarks:

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QWZ SOLUTIONS FOR LESSON 19

257

Calls Sr< K Sr> K I

✓

I

✓

Puts Sr< K Sr> K

I ✓

S, 20. Since the option pays off regardless, it is worth the discounted value of the expected payoff, which is the excess of the forward price of the stock (2Se001) over the strike price. (This works out to e- 1101 (25e110120) 5.20, but you don't need to calculate this.) Since the ophon is worth the ~me for both volatilities, the difference is [Q}. 20.26.

=

II:

f;(r-h)h+oJ,/fi:

d:

ell005--0l

f;(00.5---llll3){025)-+-02-.u}s:

f;ll\05:

1.110711

=e--OOrept the 4 eodiog nodes of the ab~e e>ample.

Quiz 21·3 For a 1-year Amencan put option on a futures contract on gold, you are given: (1) The pnce of the futures contract 1s650. (11) Thestrikepnceis640. (iii) a =0.25. (iv) The continuously compounded risk-free rate is 4%. The option is modeled with a 2-period binomial tree based on forward prices. Determine the put's premium.

21.5 Other assets Stock indices A stock index pays dividends as the underlying stocks pay dividends. It is convenient to mOOelthis with a continuous dividend rate, even though in reality dividends may not be uniform throughout the calendar year. Since we've already done many examples with stocks and continuous dividend rates, there is no need for an additional example. IFMStudyManual-i"edilion CopynghtC2!118ASM

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EXERCISESFOR LESSON 21

295

Table 21.1: What to use for the dividend yield (li) in the binomial trees fo1T11ulas"

Under] in as.set Stock or stock index

Dividend ield Dividend yield

Currency Futures contract

Foreign risk-free rate Risk-freerate\r)

• Always t,s,, the (domestic) risl-f,.,,, rate fm, hUs,,J in formula forp'. See Table 212 fm fmmulu fm futuresrontracts

Table 21.2: Fo1T11ula summary for options on futures contracts

6=~

p

.

F(u-d)

1-d

= -;;=-;j 1

C

~ 11

=e-,hw•c"

if the tree is ba,ed on forward prices

+ (1- p")Cd)

B=C

Summary Table 21.1 has a summary of what to u,e for li in the binomial tree formulas with each underlying as.set. The only formulas which do not follow the pattern of the formulas in the previous lesson's formula summary are the ones for options on future contracts. The,e are given in Table 21.2.

Exercises Options on stock 21.1. [2-F00:32] A nondividend paying stoi:.:khas a current value of 100. In each of the next six-month periOOs, the stoi:.:kprice could rise by 25% or fall by 25%. The risk-free interest rate is 6% per year. What is the price of a one-year European call on this stoi:.:kwith an exerci,e price of 907 (A) 18.5

21.2 (i) (ii) (iii) (iv) (v)

(B) 19.0

For a mulh-period

(Cl 19.6

(D) 21.3

(E) 22.5

binomial tree of stock prices you are given:

The stock's initial price is 43. The continuously compounded risk-free interest rate is 5%. The stock pays dividends proportional to its price at a conhnuous The stock's volatility is 30%. Each period of the tree is 3 months.

rate of 2%.

The tree is constructed ba,ed on forward prices. Calculate the projected stock price at nOOe 1111J11, S""d"·

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21. BINOMIAL TREES-GENERAL

296

21.3. (i) (ii) (iii) (iv) (v) (vi) (vii)

For a 1-year European call option on a stock whose prices are mOOeled with a binomial tree: The tree has 2 period;;. The tree is constructed ba,ed on forward prices. The stock price is 42. Thestrikepriceis35. The continuously compounded risk-free interest rate is 0.05. The stock pays no dividends. a =0.1.

Determine the option's premium. 21.4. (i) (ii) (iii) (iv) (v) (vi) (vii)

For a 1-year European put ophonon

a stock whose prices are mOOeled with a binomial tree:

The tree has 2 period;;. The tree is constructed ba,ed on forward prices. The stock price is 42. Thestrikepriceis42. The continuously compounded risk-free interest rate is 0.04. The stock pays no dividends. ,, =0.1.

Determine the option's premium. 21.5. (i) (ii) (iii) (iv) (v)

Future prices of a stock are modeled by a J.-period binomial tree, with each period being 4 month;;. The tree is constructed ba,ed on forward prices. The stock price is:(). The continuously compounded risk-free interest rate is 0.03. The stock pays dividend;; proportional to its price at a conhnuous a =0.3.

rate of0.06.

A European call option on the stock expiring in one year has strike price 60. Determine the price of the call option. 21.6. Future prices of a nondividend paying stock are mOOeled with a 2-period binomial tree, with each periOO being one year. You are given: (i) (ii) (iii) (iv)

The tree is constructed ba,ed on forward prices. The stock's initial price is 50. The continuously compounded risk-free interest rate is 6%. ,, =0.3.

An American put ophon on the stock with strike price60 expires in 2 years. Determine the price of the put option.

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EXERCISESFOR LESSON 21

21.7.

Future prices of a stock are mOOeled with a 2-periOO binomial tree, each period being one year.

You are given: (i) (ii) (iii) (iv) (v)

The tree is constructed based on forward prices. The initial stock price is SO. The continuously compounded risk-free interest rate is 3%. The stock pays continuous dividends proportional to its price at a rate of 6%. ,, =0.3.

An American call ophon on the stock expiring in 2 years has strike price 60. Determine the price of the call option. 21.8. Future prices of a nondividend-paying periOO being one year.

stock are mOOeled with a 2-periOO binomial tree, each

You are given: (i) The ratio of the stock price at the upper nOOe of a branch to the price at the beginning of the branchisl.3. The ratio of the stock price at the lm\~r nOOe of a branch to the price at the beginning of the branchis0.75. (iii) The initial stock price is 100. (iv) The continuously compounded risk-free interest rate is 10o/o. (ii)

An American put ophon on the stock expiring in 2 years has strike price 120. Determine the price of the option. 21.9.

Future prices of a stock are modeled with a 2-period binomial tree.each periOO being three months.

You are given: (i) (ii) (iii) (iv) (v)

The tree is constructed based on forward prices. The initial stock price is 20. The continuously compounded risk-free interest rate is 5%. The stock pays continuous dividends proportional to its price at a rate of 3%. "=0.5.

Calculate the risk-neutral probability of an increase in stock price in the first period. 21.10.

Future prices of a stock are mo.:leled with a 12-periOO binomial tree, each periOO being one month.

You are given: (i) (ii) (iii) (iv) (v)

The tree is constructed based on forward prices. The stock's initial price is SO. The continuously compounded risk-free interest rate is 4%. The stock pays no dividends. "=0.1

An American call ophon on the stock expiring in one year has strike price 70. Calculate the price of the call option.

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21. BINOMIAL TREES-GENERAL

298

21.11.

Future prices of a stock are mOOeled with a 12-periOO binomial tree,each periOO being one month.

You are given: (i) (ii) (iii) (iv) (v)

The tree is constructed based on forward prices. The stock's initial price is 72. The continuously compounded risk-free interest rate is 8%. The stock pays continuous dividends proportional to its price at a rate of 2%. ,, =0.1

A European put option on the stock expiring in one year has strike price 60. Calculate the risk-neutral probability of a payoff from this option. 21.12. Future prices of a stock are mOOeled with a 2-period binomial tree. The risk-neutral probabilihes of up movements at all nodes of the tree are equal. A European ophon on the stock has the following prices: ---------------

62.8

33.8 ------

Co -------

---------6.4--------

16.6 0

Determine Co, the price of the ophon at the inihal node. 21.13. Future prices of a nondividend periOO being 6 months.

paying stock are modeled with a 2-periOO binomial tree, each

You are given: (i) (ii) (iii) (iv)

The tree is constructed based on forward prices. The initial stock price is 100. The continuously compounded risk-free interest rate is 6%.

,1=0.4.

An American put ophon on the stock expiring in one year has strike price 105. Determine the number of shares in the replicahng portfolio of the put option at the initial nOOe.

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EXERCISESFOR LESSON 21

21.14.

299

Future prices of a stock are modeled with the following 3-periOO binomial tree:

Consider a European call ophon on the stock expiring in one year. The numbers on the ending nodes are the prices of the option at tho,e nOOes. You are also given: (i) (ii)

The risk neutral probability of an up mm·e is 0.45. The continuously compounded risk-free interest rate is 5%.

Calculatethepriceofthecall. 21.15. To compute prices of a 6-month American call option, future prices of the underlying stock are modeled with a 2-period binomial tree with 3 month periods. You are given: (i) In each periOO, the price of the stock are either multiplied by 1.1 or multiplied by 0.9. (ii) The initial stock price is 40. (iii) The stock pays continuous dividends proportional to its price. The dividend yield is 0.03. (iv) The continuously compounded risk-free interest rate is 0.05. Determine the least upper bound of strike prices for which early exerci,e is optimal at least at one nOOe. 21.H,. [CASB·S00:28] An investor, wishing to insure herself against a decrea.e in the value of her stock without incurring the total cost of buying a European put option, makes use of a collar strategy, whereby she sells a European call option and purchases a put option. A~ume the following: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Stock prices change quarterly. The options mature in six months. The current stock price is 50. The call option strike price is 60. The put option strike price is 45. Each quarter, the stock price will either increa.e or decrea.e by 20%. The risk-free interest rate is 5% per annum, compounded continuously. The stock pays no dividends.

Determine the initial cost of the collar.

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300

21. BINOMIAL TREES-GENERAL

21.17.

[CAS8-S01:29a] You are given the following informahon:

(i) The current price of a share of XYZ Company stock is 100. (ii) Over each of the next two six-month periods, the price is expected to go up by 10% or down by 10%. (iii) The risk-free rate of interest is 6% per annum compounded ,emi-annually. (iv) The stock pays no dividends. Determine the value of a one-year American put option with a strike price of 115. Use thefal/011,~1rg i11farmatio11farq11es/io11s 21.18 a11d21.19: The current price of a stock is 10 and it is expected to increase or decrease in price by 10% m·ereach of the next two 6-month periods. The annual risk-free rate with continuous compounding is 6%. The stock pays no dividends. [CAS8-S02:27a] Determine the value of an American put option with a strike price of 10.~ maturing in one year.

21.18.

[CAS8-S02:27b] Determine the number of shares of stock and the amount of bonds earning the risk-free rate to be held at Ta: 0 in the replicating portfolio for the put ophon.

21.19.

[CASB-S03:39] The current price of a nondividend paying sta:k is 50. The stock value either increa.es by 6% or decreases by 5% every six months.

21.20.

The annual continuously compounded risk-free rate is 3%. Determine the value of a one-year European call option with a strike price of 52. 21.21.

[CASB-S04:30] You have the following informahon:

(i) A stock will go up in value by 12% or down in value by 8% ina given year. (ii) The continuously compounded risk-free rate of interest is 5% per annum and the price of the stock is now 60. (iii) There exist options with an exercise price of 70 and maturity of two years. (iv) There are no dividends. Assuming two years until expiration, determine the value of an American call on the stock.

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EXERCISESFOR LESSON 21

21.22. (i) (ii) (iii) (iv) (v)

301

[CASB-S06:25] You are given the following information about European options on stock ABC: Strikepriceis95. Current stock price is 100. Time to expiration is 2 yea!';. The stock pays no dividends. Price of a put is 0.75. This price is calculated using a 2-step binomial model where each step is one year in length.

The stock price tree is shown below: -----121 ----100 ------

110 -----' 35

the call option alway,; pays off. So you're guaranteed to get the stock and pay the strike price. The present value of this transaction is the present value of the stock (42, since there are no dividends) minus the present value of the strike price (3Se-005 : 33.29~3) or42- 33.29303: I B.70b97j. If you chose to go through the binomial tree, you would get Figure 21.5. 21.4. The mm·ement:s based on forward prices are: 11 :

erh+"Vh: e-004(05)+0 1,/ifs: 1.094952

d:

e'h_,,,r,;: eoo4(os}-o1,/ifs: 0.950554

The risk neutral probability is p":

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EXERCISESOLUflONS FOR LESSON 21

305

50.3546 45.9880 0 42

1.0430 -------------------

-------------------

43.7141

-------------------

37.9492 4.0918

39.9233 2.0555

Figure 21.6: Binomial tree for solution to e)(ercise 21.4

Note that 4211d> 42, so the option only pays off at the dd nOOe. Then

Sdd =42(0.950554 2) =37.9492 Po= e--004(1- p")2(42- Sdd) = e--004(1 -0.482330) 2(42- 37.9492) = ~ The full binomial tree, although not needed for the solution, is shown in Figure 21.6. Note that an American option would be worth more, since exercise is optimal at node d. 21.5.

Wecalculate11andd. f':(r-h)h: f':(003--006)/3:f':-001: 0.990050 II: f';(r-h)h+,J'/fi:e(003-006)/3+03-yl/3:

1.177278

d = e--0o3/3--03Yl/3=0.832597 The risk-neutral probability is e(r-h)h -d p" = ~

~:~;~~~

~:!~~::~

= 0.456807 S""" = 50(1.177278 3) = 81.58441, while S""d = ~(1.17727s2)(0.832597) = 57.69832 and the other nodes are even lower, so the call is exercised only at 111111. We pull back from that node 3 times, so the answer is

i

Co= (81.58441 - «J)e--0o3p-3 = 21.58441(0.970446)(0.456807 3) = 1.99bb81 j The resulting binomial tree is shown in Figure 21.7, but there is no need tocalculateall the intermediate values. 21.6. 11 = e006-+03o = 1.43333 and d = e006---01!= 0.7866.3, so e006 - 0.78663 2556 p" 1.43333-0.78663 0.4 The stock prices at the right nodes of the tree are ~e 072 = 102.72166, 50eo 12 = 56.37484, and 50e-0 48 = 30.93917. The values of the put at these nodes are 0, 3.62516, and 29.06083 respectively. Going back one year, the upper nOOe is

P" =e- 006(0.42556(0)+0.57444(3.62516)) lFMStudyManuaJ-i"edilion Copynght02!118ASM

=1.96117

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306

21. BINOMIAL TREES-GENERAL

81.58441 69.29919 ---

:•,~:;

21.58441

~

--------------

9.761797

---------

49 00993 -----------

576b832

50 1.996681

---------------------41.62987 --------------

40.80554 0

---------

34 66092

--------~ ---28.85859

Figure 21.7: Binomial tree for solution to e)(ercise 21.5

102.72166 0 71.66647 -------------1.961172

50 11.96749

-------------39.33139

------------------

56.37484 3.62516

20.66861

---------

---------

30.93917 29.06083

Figure 21.8: Binomial tree for solution to e)(ercise 21.6

while the lower node, before testing for exercising the option, is P:uropean=e- 1106 (0.42556(3.62516)+0.57444(29.06(ll3)) At the lower mx:le, exercising the ophon is worth60-39.33139

=17.17448

: 20.66861, so it is ophmal. Thus the value

of the put at the inihal node is

Po= e- 1106 (0.42556(1.96117) +0.57444(20.66861))

=1n.%7491

The resulting binomial tree is shown in Figure 21.8. 21.7. 11

=e(r-h}+-o1 =1/003--006)-+-03 =ell17 =l.J0996

d

:e e--0°125(0.52506.3)(48.4-K) 00125(0.52-:()63))< 44-48.4e- 0012·\0.525063)

K(l-e-

0.4814(i)K < 18.~266

K e--0°125 (48.4p· +39.6(1-

p")-

K)

K(l -e--00\25) < 44-e--00\25(48.4(0.525063) +39.6(1-0.52~)) 0.012422K Kr,orS > lf. The next example discu~es the lower bound of a stock's price for an op hon that is ophmal to exercise early. ExAMl'LE

(i) (ii) (iii) (iv)

22A For an American call option on a stock

The stock price is initially 60. Thestrikepriceis(i). r :0.05. 0 :0.04.

There are 3 months to expiry. If the value of a3-month put option with strike price 60 is 1 and early exerci,e at this point is ophmal, what is the lowest po~ible current value for the stock? ANSWER: One way to do this is to compare what you gain by early exerci.e, the present value of the future dividends, with what you lo,e, the the present value of interest on the strike price plus the value of the put. The present value of future dividends is S(l -e- 61 ), since the value of the stock is Sand the value of a prepaid forward on the stock, which is the present value of the stock without future dividends, is Se-M, so the difference is the pre.ent value of future dividends.

S (1-e-

61 ):

s(1-e-

0 ·251'. 0 -04

l):0.00995017S

The present value of interest on the strike price is

Therefore to make early exerci,e ophmal, 0.00995017S

~

0.745332 + 1 : 1.745332

s ~ 0~~~~~7 lFMStudyManuaJ-i"edilion Copynght02018ASM

:1175.4071

319

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22. BINOMIAL

320

TREES: UNDERSTANDING

EARLY EXERCISE OF OPTlONS

Table 22.1: Fo1T11ula Summary for Lesson 22

I For an infinitely-lived call option on a stock with

a= 0, exercise is optimal if Sli > Kr.

Alternatively, you can u,e put-call parity. The value of an American call is at least the value of a European call, and by put-call parity that is

C(S, K, T) = P(S, K, T) + Se-ot - Ke-' C(S,60,0.25)

=1 + Se- 001

-

60e--0ous

Early exercise is worth S -60, so to make it optimal S -60 S - 60 0.00995017$

~ ~

s~

1

=0.990049835 ~

58.254668

C(S,60,0.25), or

0.99004983S - 58.254668 1.745332

0~=~~7

=1175.407]

For early exercise of a put option to be optimal, interest on the strike price must be greater than the sum of stock dividends and the value of the implicit call. The textbook has sample graphs of early-exercise boundaries, the lowest stock prices at which early exercise of a call option can be optimal or the highest stock prices at which early exercise of a put option can be optimal. The higher the volatility, the less likely early exercise is optimal because of the increased value of the implicit options. The closer to expiry, the more likely early exercise is optimal because lost interest is lower and the value of the put is lower.

®

Quiz 2...2-1 For an. Ame.rican. call option on a stock: (1) Thestockpnceis5J. (11) Thestr1kepnce1s45. (iii) There are 3 months to expiry. (iv) The stock is about to pay a dividend of D. (v) r =0.04. Determine the least upper bound of values for D such that immediate exercise is definitely not optimal.

Exercises 22.1. For an American call option on a stock, you are given: (i) Thestrikepriceis5J. (ii) The continuously compounded risk-free interest rate is 0.04. (iii) The stock pays continuous dividends proportional to its price at a rate of0.03. (iv) With6 months to expiry, exercise of the option is optimal when the stock price is 80. Calculate the highest po~ible price for a 6-month European put option on a stock worth &I with strike price SO.

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EXERCISESFOR LESSON 22

22.2 (i) (ii) (iii) (iv)

321

For a 1-year American put option on a stock with 3 months left to expiry, you are given: Thestrikepriceis40. The stock pays continuous dividend proportional to its price at a rate of0.02. Three months before expiry, the stock price is 30. Three months before expiry, the value of a 3-monthcall option on the stoi:.:kwith a strike price of 40is0.15.

Calculate the lowest possible continuously compounded risk-free rate so that exercising the ophon three months before expiry may be rational. 22.3. For an American put option on a stock with 6 months to expiry: (i) (ii) (iii) (iv)

Thestrikepriceis40. The price of a European call option with6 months to expiry and strike price 40 is 0.80. The stock pays a dividend of x at the end of 4 months. The continuously compounded risk-free interest rate is 0.06.

Determine the least upper bound of values of x for which it may be ophmal to exerci,e the ophon immediately. 22.4. For an American call option on a stock with 1 year to expiry: (i) (ii) (iii) (iv)

Thestrikepriceis:(J. The stock pays continuous dividends proportional to its price at a rate of0.06. The continuously compounded risk-free interest rate is 0.04. You are given the following table of prices for American put ophons with 1 year to expiry and strike price 50:

I

s:;:tkp~;~:e

I

~

I

i.~

I

{s

I

i.~

I

i6 I

Based on this information, determine the greatest lower bound of stock prices for which it may be optimal to exerci,e the option immediately. Additional released eum questions: SOA MFE-507:4

Solutions 22.1. Since early exerci,e is optimal, the present value of dividends must exceed the present value of interest on the strike price plus the put value P(80, 50, 0.5). The pre.ent value of dividends is 80 (1 - e---OS(lllll)) : 1.191045 The present value of interest on the strike price is so(1-e---0.S( 1104l) =0.990066 So the put cannot be worth more than 1.191045 - 0.990066 = [ 0.200979 (

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22. BINOMIAL TREES: UNDERSTANDING

322

EARLY EXERCISE OF OPTlONS

22.2. Jf early exerci,e is ophmal, then the pre.ent value of interest on the strike price must exceed the present value of dividends plus the value of the implicit call option. The present value of dividends is ~(1-eThe value of the call option isgiwnas

025{0ll2))=0.149626

0.15. The present value of interest on the strike price is

So 40 (1 -

=0.299626 = 0.2:626 =0.0074~ 0.25r =ln(1 - 0.00749064) =0.00751884

e--0

25') ~ 0.149626 + 0.15

1 - e-025r

r=4(0.00751884)=10.03008]

22.3. The implicit call op hon is worth at least as much as a European option, and po~ibly more because it is American. Exercising gains interest on the strike price, but loses the implicit call and the dividend. If exercising is ophmal, the interest on the strike price must exceed the sum of the value of the implicit call and the value of the dividend. Interest on the strike price for 6 months is worth 40(1 - e-0.06(0.5)) 1.1822, 3) xe--OOl_ Therefore and the present value of the dividend is xe--OOO:l/

=

=

1.1822 -0.80X

xe--0°2 >

o

< (1.1822- 0.8)e002 < ~

=

22.4. The gain in exercising is the gain individendson the stock,or 5(1-e- 006) 0.0582355. The loss is the loss in interest on the strike price, 50(1- e--004) 1.96053, plus the loss of the implicit put. Calculate the gain and loss starting at 64 and working down:

=

Stock price 64 63 62

Dividend gain 64(0.058235) 63(0.058235) 62(0.058235)

=3.72707 =3.66883 =3.61060

Interest and put loss 1.96053 + 1.6 1.96053 + 1.7 1.96053 + 1.8

=3.56053 =3.66053 =3.76053

When the stock price is 62, it is not optimal to exercise, since the loss is greater than the gain. We don't know what the put's price is when the stock price is behveen 62 and 63, but it is worth at least 1.7. So the gain is definitely larger than the loss when 0.0582355 > 1.96053 + 1.7, or 5 3.66053/0.058235 62.858. The lowest price for which it may be optimal to exerci,e the option is~-

=

=

Quiz Solutions

=

22-1. The pre.ent value of interest on the strike price is45 (1 - e--Ol.'i(004)) 0.447757. The present value of dividends is D. Therefore, to makeearlyexerci,e non-optimal, dividends must be less than interest, or D 150/80): Pr(~>~)=

Pr(1n(~)

>In(¥))=

1-N(ln(l

5

/g_l -0· 6

42

) = 1-N(0.34768) = lo.364041

To illustrate the lognormal model for stock prices, suppo.e the parameters of the stock price mOOel are /J = 0.15 and a= 0.3, so that ln(S1/So) is normal with parameters m = 0.15and v = 0.3. Figure 23.1 graphs the probability density function of (a) the normal random variable with parameters m and v, and (b) the stock priceafteroneyear,S1. You will notice that while the normal random variable is symmetric amund0.15, the stock price graph is not symmetric around 40e015 = 46.47. Exponentiahng a random variable preserves its percentiles, so the median stock price is 46.47. However, this is not the most likely stock price; 40e111 s-o.3' = 42.47 is the 3iJ) = 48.61, which is greater most likely stock price. Morem·er, the expected stock price is 40 (e015 +ll.'i(o than the median. There is a greater than 50% chance that a lognormal random variable will be less than its mean Ex.AMPLI! 23C St is the price of a nondividend paying stock at time I. St follows a lognormal model. You are given: (i) So =40. (ii) The stock's continuously compounded expected growth rate is a= 0.15. (iii) The stock's volatility o = 0.3. Determine

1. The average price of the stock after one year. 2. The average price of the stock after four years. 3. The median price of the stock after one year. 4. The median price of the stock after four years. 5. E[ln(Si/So)]. 6. E[ln(S4/So)]. ~r ~h~~~ :v,~~~~i.s. By definition of continu0 2. The average price of the stock after4 years can be calculated by multiplying a by 4: ElS4] = Soe4a = 06 40e =172.88481. ::s~;;:~po:d~eel::;;i:;~!e~~:e~;tl~~e:

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23. MODELING STOCK PRICES WITH THE WGNORMAL

328

DISTRIBUTION

q'i(x;0.15,0.3) 0.7 0.6 0.5 04

0.3 0.2

0.1 -0.8

-0.6

-0.4

-0.2

0.2

0.6

04

(a) Probability d"nsity functim. for the cm.tinuously rompowtded

0.8

annual rate of return, ln(S1 /So)

Probability

:~~

o::"I'"•••••~••••••••••••••••••• 0

m

m

w

~

oo

m

~

oo

100

110

51

{b)f'rOOabllityd ~)

(J)

:1-N(ln(lS0/:~-

0 34 - ) :1-N(0.48101)

:[0.315251

Quiz2···3-1 Astock.'spriceSr followsalognormalmodel. You are.given: (1) So: 100. (11) The continuously compounded expected rate of return is 0.12. (iii) The continuously compounded dividend rate is 0.03. (iv) The volatility is 0.4. Determine the value I for which the median price of the stock at time I is 110.

23.2.3 Prediction intervals Using the lognormal model, we can an;;werquestion;; on the distribution of the price of the stock. We can calculate the probability that the price of a stock is in a certain range. We can also calculate predichon intervalsJ for the price of the stock. The term "p-prediction interval" connotes an interval centered at the mean of a random variable for which p is the probability that the random variable is in the interval. 23E A stock's prices follow a lognormal distribution. You are given: (i) a :0.14 (ii) 0 :0.02 (iii) o :0.3

Ex.AMPLI!

'In the sec K)

=N(-ji) =N(j2)

(23.2) (23.3)

PELS1I Sr< K]:

soe~ 23.7.

By formula (23.2), Pr(Sr 40, so the probability of it being exerci,e,:l is 4 3 Pr(So.s > 40) = 1- N(ln( 0/ S)~07 0.24 0.5

56 ) = 1- N(0.34136) = 1-0.63359 =[0.366411

23.10. For this lognormal distribuhon, /J = 0.1 and a= 0.2, so m =/JI=

1 and v = aVI = 0.2Y10. The

23.11. Weuseformula(23.9). Ji

ln(40/40) +0.05 ~ 0.05 + 0.5(0.32 ) 0 = -0.15

_ 0 15

J2 = J1 - aVI = 0.15-0.3

ElS1 I 51 > 40] = Soe"_6 N(~i) N(d2)

= 40 N(0.15) N(-0.15) =40(0.55962)=~ 0.44038 23.12. Weuseformula(23.6). Ji = Jn(40/SO)+ ~~~~°;;o.1s2 /2)(2) = -o.omo 3 J2 = J1 -0.15¥2 = -0.21516

ElS2 I 52 72.7755] = 0e _ 0

-1.345

N(J2) = 0.04998

!:S

~~~ ~;t~:~~~f ~:r:~~~~=tat

93

l) = 83.0440

the stock price is greater than 72.7755, is &1.0440- 50 : 33.0440. The

23.H,. Since you've written the ophon, the worst case is when the ophon pay,; the most, or when the stock price falls the most. The normal parameters are m = 0.12-0.5(0.2 2 ) = 0.10 and v = 0.2. The 10th percentile of the stock's price at the end of one year is SOexp(0.10-2.326(0.2)) =34.70289 We use equation (23.6) to calculate the expected value of the stock given that it is le~ than34.70289.

J _ ln(50/34.70289) + 0.1 + 0.5(0.22 ) 1 0.2 J2 =2.426-0.2=2.226

N(-J1) =0.00763

ElS1 I S1 < 34.70289] = SOe::~~~~:

2.426

N(-J2) =O.OBJ1

63 ) = 32.4076

The expected payoff on the put given that the stock price is le~ than 34.70289 is 50 - 32.4076 = 17.5924 and the lVaR of the payoff isl-17.5924 I.

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QWZ SOLUTIONS FOR LESSON 23

343

Quiz Solutions 23•1. ThemedianofS

1/Soisem,and

m = (a - 0-0.5,1 2)1 = (0.12-0.Q3-0.5(0.4

2 ))t

= 0.011

To make the median 110, we must have l(J()e-OOH:110 0.011 = ln1.1 t = 100ln1.1 =[Iill

23•2. The lognormal parameters are m = 0.1- 0.5(0.252 ) = 0.06875 and v = 0.25. The prediction interval is obtained by adding and subtrachng 1.96v to and from m, or (m -1.96v, m + 1.96v) = (0.06875 -1.96(0.25),0.06875 + 1.96(0.25)) = (-0.42125,0.55875) The prediction interval is (SOe-1142125, 5Qe 11•551175) = I (32.81,87.42)

1-

23•3. We'll use equation (23.3). • ( ln(60/70) + (0.15- 0.05 - 0.5(0.22 ))(0.25)) N(di) = N --~~~~--~0.2v'if25 = N(-1.34151) =I0.08988)

23-4. We use formula (23.10). Notice that K =Soso ln(So/K) =O. J1 = 0.1 + ~:~(0.22) = 0.6

N(d1) =0.72575

J2 =0.6-0.2=0.4

N(d2) = 0.65542

The expected payoff is 01 (0.72575)-100(0.65542) = ~ E[max(O,S1 - K)] = lOOe-

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Lesson 24

The Black-Scholes Formula Reading: Derivatives Markets 12.1-12.2, Appendix 12.A

This lesson is very important. The formula we discuss is used repeatedly throughout the cour,e. The Black-Scholes formula is a closed-form formula for evaluating the value of a European option. We have seen in our study of binomial trees that the value of such an option depends purely on the sum of the prOOuct of the probability of reaching a node at the end of the tree hmes the value of the ophon at that ending mx:le. The fact that European options cannot be exercised early simplifies matters. The Black-Scholes formula can be .een as the limit of the results of evaluation of binomial trees as the number of periods goes to infinity. The assumptions needed for the formula are in Table 24.1. The general form of the formula, the price at time O of a European call option expiring at hme T onan a~t $-the only one you have to know-is General Black-Scholes Formula

(24.1) where

ln(FP(S) / FP(K)) +},1 2T d1 = a../f d2=d1-aYT Notice that by substituting for d1 in the formula for di and simplifying, d1 can also be expressed in a formula that looks like the formula for d1, except that the plus sign becomes minus:

di= ln(FP(S) / FP(K)) - },12T

(24.2)

a-If Table 24. 1: Assumptions of the Black-Scholes formula

• Conhnuously time.

compounded

returns on the stock are normally distributed

• Conhnuously constant.

compounded

returns on the strike a~t

and independent

m·er

(e.g., the risk-free rate) are known and

• Volatility is known and constant. • Dividends are known and constant. • There are no transaction costs or taxes.

• It is possible to short--sell any amount of stock and to borrow any amount of money at the risk-free rate. IFMStudyManual-i"edilion CopynghtC2!118ASM

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346

24. THE BLACK-SCHOLES FORMULA

This alternahve formula is useful if you are given d2 and must calculate something el,e. In equation (24.1), FP indicates a prepaid forward price. Sis the underlying asset---the asset which you may elect to receive at the end of the period. K is the strike asset-the asset you may elect to pay at the end of the periOO. Since forwards all grow at the risk-free rate, you can slate an equivalent formula dropping the P's from the ratios used to calculate d1 and d2, as long as you drop them from both the numerator and denominator of the ratio. For example, you may define d1 as 2

di = ln(F(S)/ K) + ½a T

a-If since the forward price of K is K. In fact, you may use forwards for any periOO (not necessarily the durahon of the option), as long as the numerator and denominator both use forwards for the same periOO. The next few sec hons discuss special ca.es of the general Black-Scholes formula.

24. 1 Black-Scholes Formula for common stock options For call options on common stock with continuous dividends, Sis the common stock and FP(S) = Se- 61 , where O is the continuous dividend rate. The strike as.et is cash, who,e prepaid forward price is FP(K) =Ke-rt, where r is the continuously compounded risk-free rate. We have lnFP(S)=lnS-fiT lnFP(K) = lnK-

rT

So we can write the Black-Scholes formula as (24.3) where d1

ln(S/K) + (r- 0 + ½a2 )T a..!f

d2=d1-aVT

For put ophons, the mies of Sand Kare reversed. The Black-Scholes formula can be written as (24.4) where d1 and d2 have the same definition as for a call op hon. This formula is a special case of formula 24.1 if you exchange the roles of Sand K. Ina call ophon, you may buy Sin return for ,elling K; ina put option, you may "buy" K by ",elling" S. So inequation24.1, replace S with Kand K with S everywhere, including in thedefinihonsof d1 and d2, andafterdoinga little work (such as recognizing that ln(K/S) = - ln(S/K)), you will have formula (24.4). Notice the relationship between formulas (24.3), (24.4)and formulas (23.10), (23.7). For example, to go from formula (23.10) to formula (24.3), 1. Replace the true rate of return on the stock a with the risk-free rate of return r. Then d1 is the same in both formulas, and Se(a-h)TN(d 1) becomes Se(r-h)TN(d 1) in the call formula, and d2 becomes d2 . 2. ~~:;i:enst::=rf~r(~:tt

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6 the risk-free rate. Then Se(,-h)r,\'(d1) becomes Se- r,\'(d1).

1"1~

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24.1. BLACK-SCHOLES FORMULA FOR COMMON STOCK OPTlONS

347

The steps to go from formula (23.10) to (24.3) are necessary becau,e we need to discount the ophon payoff and it is difficult to determine the correct discount rate. We therefore replace the expected payoff under the true measure with the expected payoff under the risk-neutral measure and discount at the risk-free rate. The main insight of Black-Scholes was that an option can be priced by using the risk-neutral measure and discounting at the risk-free rate. 24A For J.-month 52--;;trikeEuropean optiom on a stock, you are given: (i) The stock's price follows the Black-Scholes framework. (ii) The stock's price is SO. (iii) The stock's volatility is 0.4 (iv) The stock's continuous dividend rate is 4%. (v) The continuously compounded risk-free interest rate is 8%. Calculate the premiums for call and put optiom.

Ex.AMPLI!

ANswl!R: We have S = 50, K = 52, r = 0.08, o = 0.4, t = 0.25, 0 = 0.04. Plugging into equation (24.3), the call premium is di

ln(S0/52) + (0.08- 0.04 + ½(0.42))(0.25)

0.4-vQ.25 -0.039221+0.12(0.25) 0.2 = -0.04611 N(di) = N(-0.04611) = 0.48161 d2 = -0.04611-0.4.../oTs = -0.24611 N(d2) = N(-0.24611) = 0.402&1 C(SO,52,0.4, 0.Cll,0.25, 0.04) = 50e-0 25 (oo4)(0.48161) - 52e- 025 (008 l(0.40280) = 23.841 - 20.531 = IIE] Similarly, the premium for a put is N(-d2) = 1- N(d2) = 0.59720 N(-d1) = 1- N(d1) = 0.51839 004 l(0.51839) P(S0,52,0.4,0.08,0.25,0.04) = 52e--025(0 C6l(0.59720)- sae- 0 251'. =30.439-25.662=Kill

®

Quiz24-1 ForaEuropeancalloptiononastock: (i) The stock's price follows the Black-Scholes framework. (11) Thestock'spnce1s60. (iii) The stock's volatility is 0.25. (iv) The stock pays no dividends. (v) The continuously compounded risk-free interest rate is 0.1. (vi) The option expires in 1 year. (vii) The strike price is 65. Calculate the premium for the call option.

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24. THE BLACK-SCHOLES FORMULA

348

Discrete Dividends

The Black-Scholes formula's parameter a is the volahlity of the stock, or ✓var(ln(S1 /So)). As we will learn later, the Black-Scholes model assumes that Sr is continuous. If Sr is not continuous because of discrete dividends, it does not ~tisfy the hypotheses of the Black-Scholes model. Instead, the Black-Scholes mOOel must be applied to the prepaid forward price. Therefore, a is defined as the volatility of the prepaidfimrord. Whenever you see a mentioned in a question relating to the Black-Scholes framework, you must treat ,, as the volatility of the prepaid forward, even if the question does not say so. If a stock pays continuous dividends, then the volatility of the stock equals the volatility of the prepaid forward. If a stock has discrete dividends, the prepaid forward price is S - PV(Div,;). The general formula can then beu.ed. Ex.AMPLI! 248 A stock has quarterly dividends, paid at the end of 3 months and 6 months from now. You are given:

(i) Thestockpriceis42. (ii) Quarrerly dividends are 0.75. (iii) The volatility of a prepaid forward on the stock is 0.3. (iv) A 6--month European put option is written on the stock with strike price 40. (v) The put option, if it is exercised, is exercised on the stock ex-dividend. (vi) The continuously compounded risk-free rate is 0.04. Calculate the put option's premium with the Black-Scholes formula. ANswl!R: The prepaid forward price of the stock is FP(S)

=42-

0.75e--025(o04)-0_75e-oS(oo4)

We can use formula (24.4) by setting S

di=

=40.52231

=40.52231 and 6 =0: ln(40.52231/40) + (0.04 + ½(0.32))(0.5) 0.3-v'o.s

= o.01!;1~~;0425 =o.26149 =N(-0.26149) =0.39686 d2 =0.26149 -0.3--.fo.s =0.04936 N(-d2) =N(-0.04936) =0.48032 N(-d1)

p:

40e- 002 (0.48032) - 40.52231(0.39686)

=18.832 -16.082 =~

®

Quiz 2.·.4-2For a.1-mo_n. th.European call option on a stock, you are given: (1) Thestockpnce1s27. (11) Thestr1kepnce1s30. (iii) The continuously compounded risk-free interest rate is 8%. (iv) The stock pays continuous dividends proportional to il5 price at a rate of 2%. (v) The volatility of the stock is 0.2. Calculate the d1 used in the Black-Scholes formula for the price of this option.

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24.2. BLACK-SCHOLES FORMULA FOR CURRENCY OPTIONS

349

24.2 Black-Scholes formula for currency options For a currency option, the domestic risk-free rate serves as rand the foreign risk-free rate serves as 0. The formula for a call ophon is (24.5) where

d1

ln(x/ K) + (r - 'I+ ½a2 )T a../f

d2=d1-aYT

and xis the current exchange rate, the number of units of domeshc currency per unit of foreign currency. The formula for a put option is (24.6) where d1 and d2 are as defined for a call ophon. This formula is known as the Garman-Kohlhagen model. This is a special case offomtula (24.1). The prepaid forward price of a currency is FP(x)=xe-,,r,

so lnfP(x) = lnx - r1T,and fP(K) = Ke-,r,making

thed

ln(FP(x) / fP(K)) + ½,12T

avf

di=

1

of formula (24.1)

ln(x/K) + rT-

r1T + ½,12T

avf

which is the same as the d1 of formula (24.5). ExAMPLI!

(i) (ii) (iii) (iv)

24C You are given: The spot exchange rate for yen in dollars is 0.009$/V. a =0.05 The continuously compounded risk-free rate for yen is 2%. The continuously compounded risk-free rate for dollars is 4%.

Calculate the Black-Scholes price for a 1-year European dollar-denominated strikepriceof0.010.

di = ln(0.009/0.010) +i_-:-

2 0.02 + ½(0.05 )

call option on yen with a

_ _ 1 68221

N(d1) = N(-1.68221) = 0.04626 d2 = -1.68221 - 0.05 = -1.73221 N(d2) = N(-1.73221) =0.04162

C(0.009, 0.010, 0.05, 0.04, 1,0.02) = 0.009e -1102 (0.04626) - o.010e--004 (0.04162) = o.00J4081 - 0.0003999 = I 0.0000082 I

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350

24. THE BLACK-SCHOLES FORA1ULA

24.3 Black-Scholes formula for options on futures Refer to Section 21.4 for a discu~ion of how options on futures work. The situation here is simpler, since our formulas only apply to European options, where there is no early election. At the time the ophon expires, the payoff on a call option will be the excesseftheforwrml price (11otthe prepaidfarmml price) over the strike price, if this is greater than 0. Compare this to a stock, where the payoff on a call option is the excess of the stock price over the strike price. In the ca,e of a stock, we discount the forward price of the stock, or Se(r-h)r, and the strike price, K, both at the rater, to obtain an expression with Se-hr and Ke-,r_ In the case of a future, we discount the forward price and the strike price at the rater. If the option period and the futures periOOare the same, then the option is equivalent to an ophon on the stock. You don't really need a separate Black-Scholes formula for ophons on futures with the same periOOas the stock. Remember that in the general formula, forwards may be used in all the ratios instead of prepaid forwards, and the forwards don't have to have the same duration as the options. So let's drop all the P's in formula (24.1) to obtain the formula for valuing a European option ona futures contract. Here are the formulas for futures:

C = Fe-,r,\ 1 (d1)- Ke-,r,\ 1 (d2)

(24.7)

P = Ke-,r,\ 1 (-d2)-

(24.8)

Fe-,r,\ 1 (-d1)

where di= ln(F/K)+½a d2=d

2 T

a-If 1 -aVT

In the,e formulas, a is the volahlity of the future, and Tis the duration to expiry of the option, ,wt durahon to expiry of the future. These formulas are called the Blackfomm/a. Let T2 be the futures period and T1 the option period. If the options periOO and futures period are the same, then Fe-,r, = Se-hr, so the Black formula reduces to the Black-Scholes formula. But if T2 > Ti, then F= Se(r-h)r,_ The put premium can be derived from the call premium with put-call parity. When K = F, the call and put premiums are the same. ExAMl'LE 240 A futures contract on silver has a price of 10 for delivery at the end of 1 year. Volatility is 0.25. The continuously compounded risk-free rate is 4%. Calculate the premium for a 1-year Eumpeancall option on the futures contract with strike price 10. ANSWER:

Both the strike price and the futures price are 10. d1 = ½(~::t)

= 0.125

d2 = 0.125-0.25

= -0.125

N(d1) = N(0.125) = 0.54974 N(d2) = N(-0.125) =0.45026 C = l0e--0°4(0.54974)- 10e- 004 (0.45026) = ~ A put ophon would have the same premium.

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EXERCISESFOR LESSON 24

351

Table 24.2: Fo1T11ula Summary for Lesson 24 A~t

Call Premium

Put Premium ln(S/K) + (r-rtfolio

=

s;;.=:o

= (48)~0.1) =

-¥

=

5}J

25.27. 0=

33

8 ~· ) =5.28

a~ r =0.5 a-r=0.5(0.3)

=0.15

y- r = 0.15(5.28) = ~

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25. THE BLACK-SCHOLES FORMULA: GREEKS

4-04

25.28.

By put-call parity, P

= 8.5 - 51 + 45e--0.os= 0.305324

The call 1l is backed out from J.S= S~all =51

~;-a

8

ilraa = (J.S~;S.S) =0.583333 L\.put

= 0.583333-

e-

6 = 0.583333 -1 = -0.416667

Theelashcity of the put option is (51(-0.416667)) /0.305324 = l-b9.598]. 25.29. Since the stock is not dividend paying, C..put= ilc.11- e- 6r = C..:a11 -1.

= -0.26

b,put=0.74-1 lO= S(0.74) 3.42

s = 20(3.42) =92.4324 0.74

-4 = 92.432~-0.26) P= 92.432~:-0.26) =~ 25.30. By put-call parity, Cp>rtfolio= C - P

= 45- 40e--004= 6.56842. The difference in t.'s, 6JX>rtfolio, is

flcaa - 6put = ilcall

-(C..:all

-

e- 6 T) = e- 6 T = 1.

So the elasticity is

Ib.8509b

SL\portfolio = ~: Cport1o1;0 6.56842

J.

Quiz Solutions 25-1. b,put =t.a.a

-0.79 = ll.:,,,a

-e-h(r-1) - e-OIX,{05)

ll.:,,,11 = -0.79+0.9704

= ilca11 - 0.9704 =~

25-2. A tough question. The only Greek we know that has a shape like this, first decreasing and then increasing, is (;J. This is also the only Greek which a~umes both positive and negative values. Since it assumes positive values, it's probably a put theta with a short life, a high r, and a low ti. It could also be a call theta with a short life, a high ti, and a low r. I generated the graph using a put theta with K = 100, r =0.06,ti =0,,1 =0.25,t =2/3.

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QWZ SOLUTIONS FOR LESSON 25

405

25-3. For each option, we'll u,e the formula 6put = C>ca11 - e- 6r. For the first option: .6 :0.53153-e--0025(1l2.5) =-0.46224 For the ,econd option: 6 = 0.30766 - e--0025(05)= -0.67992 For the third ophon:

6 = 0.24825- e- 0025= -0.72706

Then delta for the portfolio is .oport1callportfoln =0.70536-0.47295

=I0.232411

Delta for a put is delta for a call minus e- 6 r. (lnourca,e, e- 6 r = 1, but it doesn't matter.) Subtrachng e- 6 r from the delta of each option does not affect the difference of the deltas. So delta for the put portfolio, (B), is the same as delta for the call portfolio. 25•5.

25-6. 1. The weighted average is

2. The risk premium, y- r = Op:,rth,lio(a- r), soit is 1.0266(0.15-0.05) = lo.10266 ].

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406

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25. THE BLACK-SCHOLES FORMULA: GREEKS

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Lesson26

Delta Hedging Reading: Derivatives Markets 13 The textbook's chapter 13 is a gentle intrOOuction to delta hedging. J recommend reading it if you find the following summary too short. A market-maker is a broker or dealer who sells or buy,; options. A market-maker is not interested in speculating on the market but rather would like to make money through bid-ask spreads. Thus if he sells a call option, he would not like to lose money if the stock goes up in value. To avoid this, he must hedge the investment by buying something that would go up in value if the call goes up in value. The most obvious candidate is the underlying stock. How much of the underlying stoi:;k should he buy 7 Since ti. measures the increase in the ophon's value per unit increase in stock, he should buy D, shares of stock. 6 shares of stock will cost more than 1 call option, so there is an interest cost in the portfolio. We will break our discussion up into the following parts:

1. The m·emight profit on a delta-hedged portfolio 2. The delta-gamma-theta approximation 3. Rehedging 4. Hedging multiple Greeks

26. 1 Overnight profit on a delta-hedged

portfolio

Assume a stoi::kpays no dividends. A delta hedged portfolio consists of selling (or buying) an option, buying .il shares of stoi::k, and borrowing the money needed for the other two transactions. (It can be self-financed.) The overnight profit therefore has three components: 1. The change in the value of the option. 2. .il times the change in the price of the stock. 3. Interest on the borrowed money. The first two in this list are called "mark-to-market , since the option and stoi::kare not usually sold; their values are just recalculated. This is simple enough and there is no need to memorize a formula, but if you wish to .ee a formula for the m·emight profit, we can write one. Let r be the risk-free rate (as usual), So and 51' the stoi::kprices at the beginning and after a day, C(So) and C(S1) the ophon prices at the beginning and after a day. Then the market-maker profit is Profit= -(C(S1)-

C(So)) +.il(S1 - So)-(e-' 1365-1) (.ilSo - C(So))

(26.1)

This formula ignores dividends on the stock. 1Temporarilythesub$aiptswlllbeindaysratherthaninyears IFMStudyManual-i"edilion CopynghtC2!118ASM

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408

26. DELTA HEDGING

Ex.AMPLI! 26A A market-maker sells 100 JO-day European call optiom ona nondividend paying stock and delta-hedges them. You are given that initially (day 0): (i) S 200 (ii) r =0.05 (iii) o =0.4 (iv) K 200 Calculate the market-maker's overnight profit:

=

=

1. If thestock'spricesla:y5 the same on the next day (day 1). 2. If the stock's price increases to 205 on the next day (day 1). 3. If the stock's price decreases to 195 on the next day (day 1). ANswER: Initially:

di=

(0.05+0.5(0.4 2))(30/365) 0.4~

0.09317

=N(0.09317) =0.53712 d2 =0.09317- 0.4~ =--0.02150 N(d2) =N(--0.02150) =0.49142 C(200,200,30/365) =200(0.53712) -200e-:JJ{Oll5/J65)(0.49142) =9.543 Ji= N(d1)

The initial investment by the market-maker is the cost of 6. shares of stock minus the call premium, or 100((0.53712)(200)-9.543)

=9788

The interest cost for one day is (e005 / 3(.5 -1) (9788) = 1.3. 1. If the stock's price remains the same, then the value of the call after one day is (0.05+0.5(0.4 2))(29/365) di=--~~~0.4{i91365

0.09161

N(di) = N(0.09161) = 0.53650

d2 =0.09161-0.4.../i9f36s = --0.02114 N(d2) = N(--0.02114) =0.49157 C(200,200,29/365) = 200(0.53650) -200e- 29(oll5)/ 3(.5(0.49157)= 9.376 The mark-to-market profit on the call is 100(9.543 - 9.376) = 16.7. Subtracting the interest cost, the net profit is Profit= 16.7-1.3 = DI!]. Profit occurred because the option's value decays with time. 2. If the stock's price increases to 205, then the call's value is di = ln(205/200) + (0.05 + 0.5(0.42))(29/365)

0.31061

0.4~ N(di) = N(0.31061) = 0.62195 d2 = 0.31061 - 0.4~ = 0.19787 N(d2) = N(0.19787) = 0.57842 0 C(205,200,29/365) = 205(0.62195)-200e- 291'. 05Jl365 (0.57842) = 12.274 IFMStudyManual-i"edilion CopynghtC2!118ASM

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26.1. OVERNlGHT PROFIT ON A DELTA-HEDGED PORTFOLIO

409

Overnight Profit 20 10

-10 -20

-30 --40 -50 -60

-70 194

196

198

200

202

204

206

208

210

Stock Price after I day Figure 26.1: O,,ernight profit as a function of stock price for a delta-hedged market-maker who has written a rail

The mark-to-market profit on the call is 100(9.543 -12.274) = -273.1. Therefore, the gain on the stock is 100(0.53712)(205 - 200) = 268.6. Total profit, including interest, is -273.1 + 268.6 - 1.3 = ~3. If the stock's price decreases to 195, then the call's value is

di=

ln(195/200) + (0.05 + 0.5(0.42))(29/365) 0.4~

-0.13294

N(d1) = N(-0.13294) =0.44712 d2 = -0.13294 - 0.4~

= -0.24569

N(d2) = N(-0.24569) = 0.40296 C(195,200,29/365) = 195(0.44712) -200e-2'l(Oll5)/J(.5(0.40296)= 6.916 The mark-to-market profit on the call is 100(9.543-6.916) = 262.7. Gain on the stock is 100(0.53712)(195200) = -268.6. Total profit is 262.7 -268.6 -1.3 = ~ If the stock pays continuous dividends proportional to its price, the market-maker earns those dividends (or pays them if the stock is shorted); however, it may not be possible to calculate them without knowing all the intermediate stock prices.

In the preceding example, v,,'{' saw that the market-maker profits on small changes in the stock price but loses money on large changes. Figure 26.1 shows the overnight profit for various stock prices after one day. We see that the market-maker makes money if the stock price changes by less than about 4.19 in either direction and loses money otherwise. The significance of 4.19 is that it is about one standard IFMStudyManual-i"edilion CopynghtC2!118ASM

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26. DELTA HEDGING

410

deviation around the mean; in other words SaYh =200(~)

=4.1874

V365

s + Sa Yh=200 + 4.1874 =204.1874 S - Sa Yh=200- 4.1874 =195.8126 So if the stock mm·ed to 204.19 or 195.81, the market-maker would break even. ExAMPLI!

268 You are given:

(i) Astock'spriceis56.78. (ii) r =0.08. (iii) 0 =0.02.

(iv) o =0.3. A market-maker delta hedges a portfolio including an option on the stock. Determine the two stock prices on the next day for which the market-maker 'WOUidbreak even. ANswER: Add and subtract So..fh to and from 56.78. SoVh s + SoVh

S - SoVh

®

=56.78(~) =o.8916 =56.78 +0.8916 =I s1.6116 I =56.78 -0.8916 =1ss.88841

Quiz 2.·.6-1For a de.lta.-hedged portfolio on. a stock.paying no d1.·vidends,you are given: (1) The stock pnce is 15.00. (11) The continuously compounded nsk-free interest rate 1s 3% (iii) An overnight increase of the stock price to 15.25 would lead to the delta-hedged portfolio neither gaining nor losing money. Determine the stock's volatility. Delta hedging did not prevent profits or los.es because delta itself keeps changing. To minimize the risk, the market-maker would rehedge frequently. To rehedge, the market-maker buys or sells stock so that he owns L\.shares of stock for each option, where L\.is revi.ed at the rehedging date. Assuming that the position is always fully financed, the "cash flow" at the time of rehedging-where cash flow includes mark-to-market gains and los.es----equals the m·emight profit or loss. Ex.AMPLI! 26C In Example 'lfiA, calculate the purchase or sale of stock necessary on the first day to re-delta hedge the position if the stock price is

1. 200onday1. 2. 205onday1. 3. 195onday1. ANswl!R: Let L\.1be L\.on day 1 and let L\obe L\.on day 0. We already calculated L\oin the ans-r to Example 26A as 0.53712. Since L\.1 N(d1) where di is the value at day 1, we've calculated L\.1for all three scenarios as well.

=

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26.2. THE DELTA-GAJ\1i\1A-THETA APPROX/MATTON

411

1. .61 = N(0.09161) = 0.53650, whereas .60 = 0.53712, so the market-maker~ 100(0.53712 0.53650) =lliill] shares. To illustrate the cash flow remarkabm·e: the market-maker would ,ell 0.062 shares for (0.062)(200) 12.4 but would also lend 12.4, so that there would be no cash flow from rehedging. However, the mark-to-market change and the interest on the inihal investment would be cash flow. 2. .61 = N(0.31061) = 0.62195, so the market-makerFurs7100(0.621950.53712) =~shares.

=

3 . .61 = N(-0.13294) = 0.44712, so the market-make~

26.2 The delta-gamma-theta

100(0.53712 - 0.44712) = [IQQJ shares.

D

approximation

A better approximahon of the change in option value than we got using only .6 would be obtained if we brought in the ,econd derivahve r. By Taylor's series, f(x) f(xo) + f'(xo)(x -xo) + ½f"(xo)(x -xo) 2 +error term

=

Jf we let Ebe the change in stock price 51 - So, and remembering that for f(x) = C(S), the call premium, f'(xo) = .6andf"(xo) = r,thisbecomes C(S1) = C(So) +.ilE + ½rt 2 +error term We also have to bring (;J, the derivative with respect to time, into the equation if we advance one day. Before we do so, let's stick with the initial day, day 0. 26D For a 30-day European call option on a stock, you are given: (i) S = 200 (ii) r =0.05 (iii) a=0.4 (iv) K = 200 (This is the same call option as in Example 26A.) Estimate the change in the call option's value if the stock price immediately changes to (a) 205 and (b)195. U.ef=0.01732. ExAMl'LE

ANSWER:

(a) We calculate .6:

d 1

= (0.05+0.5(0.4 2))(30/365)

0.09317

0.4~ .6

=N(d1) = N(0.09317) = 0.53712

If the price moves to 205, the eshmated change is 0.53712(5) + 0.5(0.01732)(52) = ~ The actual price of the call option after the change in the price of stock is

d = ln(205/200) + (0.05 + 0.5(0.42))(30/365)

_ .ts= 0.800 shares and buys -6.40 = 0.644 shares. The net sale is 0.800 - 0.644 = 0.156 shares, and the market-maker receives 35(0.156) = 5.46. The net investment is -3.98 - 5.46 = ~26.9. We now need 0.435-0.622 : -0.187 shares, and have -0.156 shares, so we sell 0.187-0.156 = 0.031 004 1365-1) = 0.0010. shares,receiving (0.031)(40) = 1.24. Theinterestonourprevious investment is9.44 (eTotal investment required is -1.24 - 0.0010 = 1-1.2410 j.

=

26.10. In a bull spread, the purchaser buys a lower-strike option and sells a higher-strike op hon. Therefore, the counterparty, the market-maker, sells a lower-strike option and purchases a higher-strike option. Here, he purchases a 6.5-strike call and sells a 55-strike call. To hedge, he must buy 1lss shares of stock and sell &s shares of stock. The premium and delta for the 55-strike call is di= ln(60/55) + (0.145)(0.25) 0.3v'i.us N(di)=N(0.82174)=0.79439 d2 = 0.82174 -0.3v'o25

_ 0 82174

= 0.67174

N(di)=N(0.67174)=0.74913 .6.= N(di) =0.79439 C(60,55,0.25) = 60(0.79439) - 55e-o25(o1l(0.74913) = 7.479 The premium and delta for the 6.5-strike call is di=

ln(60/65) + (0.145)(0.25) 0.3v'i.us

_ _ 0 29195

N(d1) = N(-0.29195) = 0.38516

d2 = -0.29195-

0.3-v'i.us = -0.44195

N(di) = N(-0.44195) = 0.32926

.6.= N(d1) =0.38516 C(60,65,0.25) = 60(0.38516) - 65e-o25(o1l(0.32926) = 2.236 The net stock purchase is .6.ss-C>(,.5: 0.79439-0.38516 = 0.40923 shares. The net investment required, including the premium received on the sale of the spread, is 60(0.40923)- 7.479 +2.236 = ~ lFMStudyManuaJ-i"edilion Copynght02!118ASM

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EXERCISESOLUflONS FOR LESSON 26

427

26.11. The initial cash flows from the investor's perspechve are purcha,e of option (-4.38), sale of 0.6.5 shares of stock (65) and lending the sum of tho,e two cash flows (lending 60.62). The value of the portfolio after 4 months is 4.6.5- 0.65(110) + 60.62e-110513 = -5.2112. The investor receives 1-5.2112 j upon closing the position. 26.12. This is really a put-call parity question, but in the context of delta hedging. Let 11be the hme the option was bought, 12 the hme it was sold, and T margin expiry. In the following lines, we calculate the factor to accumulate an investment from hme 11to hme 12. C(S,K,t)-

P(S,K, t) = S- Ke--,r 4.25-8.50

= -4.25 =40-

9.30-5.80

= 3 ..:(1= .:()- Ke--r(T-t;)

Ke--,(r-r,)

Ke--r(T-t1)=44.25 Ke--r(T-t;)=46.50 e-r(t,-t,) = :::~~ = 1.05085 At time 11, the im·estor purcha,ed a call and sold short 0.3 shares of stock. The investment at hme l1 was 4.25 - 0.3(40) = -7.75. This im·estment is worth 9.30 - 0.3(50) = -5.70 at hme 12. The profit is the proceeds of -5.70 minus the original investment -7.75 accumulated at interest, -5.70-

(-7.75)(1.0.:(185) = ~

26.13. Let T be original hme to expiry and I current time to expiry. By put-call parity, 2.80-4.90:Ke--rT 5.90-1.30=

_45

Ke--'1 -40

Ke--rT =42.90 Ke--'1 =44.60 e-r(T-I): 44.60 42.90 The amount invested was 100(2.80) + 100(0.57)(45) = 2845. The proceeds at the end are 100(5.90)+100(0.57)(40) =2870 and the profit is 2870- 2845(44.60/42.90) = [-87.74 ]. 26.14. The movements are addihon and subtrachon by SaVh = 40(0.3)/¥52 = 1.66. Thus if the stock mm·es to either 40 + 1.66 =~or 40 - 1.66 = ~ there is approximately no gain or loss. 26.15. For the approximate value,\\~ u,e a delta-gamma approximation. C(Sr+h) = 2.74 + 2(0.62) + ½(22)(0.015) = [IQ!] 26.16. We u,e a delta-gamma approximation. P(Sr+h)= 1.84- 2(0.42) + ½(22)(0.014) = ~

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26. DELTA HEDGING

428

=

=

26.17. t. for the call is C.put+e-0 -0.40+e--OOl 0.5802and r is the same as for the put, 0.02. The change in stock price is -1, so the approximate change in the call price is 0.5802(-1) + ½(0.02)l(-1) 2] 1-0.57021. 26.18. By the delta-gamma appmximahon,

=11(40.5O = 0.0915-

0.0915

Since 6

=

40) + 0.5r(40.5 - 40)2 2 0-~-~-0425)(0.5 ) 0.1724

=N(d1), d1=N- 1(0.1724)=

-0.94472 ln(40/~) +,~-07 +O.S,12 _ _94472 0

- o.1s31 + o.s,, 2

=-0.94472a

o.s,,2 +0.94472,,-0.1531 a= -0.94472

+

=O

\11.1987 =~

26.19. Using the delta-gamma-theta approximation, C(Sr+l/52)

=C(Sr) +EL\+ ½e2r- eh =2.50 + 0.25(0.3) + ½(0.252)(0.05) -*:

~

It is also acceptable to multiply e by ~ instead of i26.20. Using the delta-gamma-theta approximation,

26.21. .6 for the delta-hedged position is Oby definition of delta hedging, but rand e are unchanged by delta hedging. The profit before interest is approximated using the delta-gamma-theta approximation, with e divided by 365 to account for one day:

Interest cost is approximately rh(.6.5- C)

=(0.05)(:Ji;;)(0.3109(55)- 2.42) = (~)(14.6795) =0.002011 =

Net profit is -0.000725 - 0.002011 1-0.002736126.22. Since the ophons have the same strike price and expiry, by equahon (25.1)

and to delta-hedge, we need

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EXERCISE SOLUflONS

FOR LESSON 26

429

e--0.oJS c+1=-

"°'"

d1 = (0.08-0.0;_;;,o.42))(0.5)

o.22981

5 ,\'(0.22981) = e-00!.\0.5~88) llca11= e--OOl

c=o.5; 75. Howt~ver, the stock goes below 75 in 2 consecutive down mm·ements. After that point the value is ~(0.9 2) = 72.9. Three mm·ement:s are then needed to get the stock abm·e 85 at the end, since 72.9(1.12)(0.9) = 79.3881 < 85. So the only path on which there is a payoff is dd111111. IT] 27.15. It requires at least 3 periOO.sto go from 60 to below 45. So the stock must reach 60 in 2 period;;, which is only possible with 2 up mm·ement:s. Then3 down mm·ement:s are needed to get it below 45. [!] 27.16. If not for the barrier, there would be a payoff if there are at least 3 up mm·ement:s (regardless of the order), since 100(1.053)(0.91?) = 106.69 > 104. This happens on 16 of the path;;, since ups and down;; are symmetric, so the number of paths with at least 3 up mm·ements equals the number of paths with at least 3 down mm·ement:s, or half the path;;. For cases with 3 or more up movements, which mean;; 2 or fewer down movements, the barrier only eliminates the case with 2 immediate down movements. An up movement followed by 2 down mm·ements (udd) leads to 100(1.05)(0.96'2)= 96.768 > 96, so even after just one up mm·ement, 2 down mm·ements will not hit the barrier. Therefore, of the path;; with 3 up movements, the only path which doesn't pay off is (dd111111), so the remaining~ paths have payoffs. 27.17. By put-call parity,

C - p = Se-6h

- Ke-rh

C = 1.60+ 50e-ll005-50e-ll015 =2.09~ Then the two barrier calls must add up to 2.0950, so the premium for the 3-month down-and-out call optionis2.0950-0.85=~27.18. The price of an up-an-in call is the price of a call minus the price of an up-and-out call, or 8.51-4.33 =4.18. If you buy an up-an-in put and sell an up-and-in call, the payoff is the strike price minus the stock price whenever the barrier is hit. So the value of such a portfolio is the present value of strike minus stock hmes the probability of gethng it,or Up&ln Put- Up&lnCall = 0.6(Ke-,r - (S - PV(divs))) Up&ln Put - 4.18 = 0.6(75e-llS(Ol) - (&I - 2.5e--02{0l))) Up&ln Put = 4.18 + 0.6( -6.2073) = ~ 27.19. The periOO h = 0.25. In the Cox-Ross-Rubinstein tree 11= e"..r,;. = e03,/ii}.5 = e1115= 1.1618 and d = e_,,..r,; = e- 113 w1s"= e- 1115= 0.8607. The risk-neutral probability of an up mm·ement is . e(r--.l)h_ d e{005---0)(ll25) -0.8607 p =~ = 1.1618-0.8607

e00\25 -0.8607 0 5043 1.1618-0.8607 = -

The barrier of 40 can be reached in 3 ways: 1. 2 immediate downs, since50(0.860l2) = 37.0409. The risk-neutral probability is (1-0.5043) 2 = 0.2457. 2. 1 immediate down, followed by an up and 2 down;;, since 50(0.86073)(1.1618) = 37.04. The riskneutral probability is 0.5043(1 -0.5043) 3 = 0.0614. 3. 1 up followed by 3 downs, risk-neutral probability (0.5043)(1- 0.:()43) 3 = 0.0614. No other path break;; the barrier. The total risk-neutral probability of a rebate is 0.2457 + 2(0.0614) = ~IFMStudyManual-i"edilion CopynghtC2!118ASM

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454

27. ASIAN, BARRJER, AND COMPOUND

OPTIONS

27.20. The fi15tophon pays 54 -50 = 4. The ,econd didn't hit the barrier, so is isn't "in". The third one hit the barrier, so it is "out". The fourth one is worthless becau,e the exercise price is higher than the final price. The fifth one is worthless becau,e the exerci,e price is less than the final price. The sixth one pays 5. The total payoff is 4 + 5 = [2}. 27.21. The up-and-in barrier call pays 60 - 50 = 10. The up-and-out barrier call pays 0. The average of the three months is (80 + 55 + 60)/3 = 65, so the Asian call pay,; 15. The difference behveen the highest and lowest payout is 15 - 0 = ITI]. 27.22. The first three options are equal in value since the stock must reach 40 to have a payoff in all three cases. The fourth is worth less, and the fifth is worth even less than the fourth since it also requires reaching a value of 45 but only pays the excess m·er 45 rather than the excess m·er 40.

27.23. Ill and IV are worthless since no payoff is possible, since the exchange rate would have to go below the strike price 0.5 to make a payoff possible but then the ophons are out becau,e they hit the barrier. V is worth less than I since it is necessary for the exchange rate to not go below 0.45 in V whereas there is no such restriction in I. VI and Vil are the same as J since the exchange rate must in any ca,e go below 0.5 to get a payoff, so the barrier doesn't make the payoff less likely. II is worth more than I by put-call parity, since C = P + (xo - K)e-' 1 and Xo- K = 0.6 - 0.5 = 0.1.

27.24. The agreement pays max(105, 5(1/3)) = 105+ max(0, 5(1/3)-105). term is a call on the stock with strike price 105.

The present value of the max

FP(S(l/3)) = 100-2e--Ol.'i(ll05)=98.0248 di = ln(98.0248/105) + (0.05 + 0.5(0.32))(1/3)

0.3 ✓ 1/J

0.21404

di= -0.21404 - 0.3,/1[3 = -0.38725 N(di) = N(-0.21404) =0.41526 N(di) = N(-0.38725) =0.34929 C(S, 105, 1/3) = 98.0248(0.41526) -105e- 110513 (0.34929) = 4.637 The value of the agreement is 105e-ll05/3 +4.637 = ~27.25. The payoff is max($100, £50) = $100 + max(0, £50 - $100) = $100 + 50 max(0, £1 -$2). The present value of the max term is a dollar-denominated call on pounds with strike price $2. The price of one such callis: di= o.o3-o.og_;s°"S(o.o8

2 )

-0.21

d2 = -0.21 - 0.08 = -0.29 N(di) = N(-0.21) =0.41683 IFMStudyManual-i"edilion CopynghtC2!118ASM

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EXERCISESOLUflONS FOR LESSON 27

455

N(d2) = N(-0.29) =0.38591 C(f,$2, 1) = 2e--0°·5(0.41683)- 2e- 1103(0.38591) = 0.0440 So the value of the agreement in dollars is 100e--0o3 + :AJ(0.0440)= ~27.26. The value of the payment on the first choice is 50 + max(0, 5(1) - :A.I),so the current value is 04 +C(S,50,1). :A.1e--0 The value of the payment on the second choice before multiplying by c is 50 - max(0, 50 - 5(1)) so the current value is 50e- 1104 - P(S,50, 1). Let's calculate the ophon values. d1 = 0.04 - 0.0~.: 0.5(0.42)

0.25

N(di)=N(0.25)=0.59871 di= 0.25- 0.4 = -0.15 N(d2) = N(-0.15) =0.44038 C(S,50,1) =50e-

002

(0.59871)-:A.le--004 (0.44038)=8.187

We can calculate the put using Black-Scholes or by put-call parity. P(S, 50, 1) = C(S,50, 1) + Ke-, - Se- 6 = 8.187 + 50e- 1104 - 50e- 1102 = 7.216 We want 50e- 1104 + 8.187 = c (50e--004 - 7.216), or 56.2265 = 40.8235c, soc= 56.2265/40.8235 = [!}ill. 27.27. By compound option parity PutonPut = CallonPut + xe-'

11 -

P

By making x large, a PutonPut may be worth more than a Put, so I is false. A call on an asset cannot be worth more than an asset, since at best the call gets you the asset, so CallonPut is worth less than a put, and CallonPut - P is negative. Thus from the parity equation, a PutonPut is worth less than xe-' 11, making JJ true. We can replace Pin the parity equation with C + Se-M' - Ke-' 1' using ordinary put-call parity. All of these terms are independent of x, so by making x arbitrarily large, the PutonPut may be worth more than a Call, making Ill false. (E)

27.28. The underlying call value is

d1

ln(42/40) + (0.06- 0.02 + 0.5(0.22))(0.5) 0.2../o.s

0.55713

N(di)=N(0.55713)=0.71128 d2 =0.55713-0.2"/05 =0.41571 N(di)=N(0.41571)=0.66119 C(42,40,0.5) =42e- 0 .0J(0.71128)-40e--0° 3(0.66119) =3.911 By put-call parity, the put on call premium is PutOnCall = CallOnCall-

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C +4e--0.06(1l25) = 0.85-3.911

1"1~

+3.9404 =~

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456

27. ASIAN, BARRJER, AND COMPOUND

OPTlONS

54.45

49.5 ------------

45

------------

0.20037 ---------------

1.91351

44.55 0.45

--------------

4.05224 405

-----------36.45 8.55

Figure 27.2: Underlying put option-------------for e)(ercise 27.30.

27.29. The underlying put value is 2 di = ln(48/50) + 0-~~ 0.02 +0.5(0.3 )

0.14726

= N(-0.14726) = 0.44146 di =0.14726-0.3 =-0.15274 N(-d2) = N(0.15274) = 0.56070 N(-d1)

P(48,.:(], 1): .:(]e--OOl'(0.56070) - 43e-002(0.44146): 5.632

By put-call parity, the put on call premium is Put0nPut = CallOnPut - P + 5.25e-Oll\/4 = 1.05 -5.632 + 5.1718 = ~

27.30.

The risk neutral probability is p":

eOOl-0.9 =0.550251 1.1-0.9

The tree in Figure 27.2 shows the value of the underlying put option. The

II

node is calculated as

P" = e- 001(0.550251(0) +0.449749(0.45)) = 0.20037 The d mx:le is calculated as Pd

:e--OOl(0.5.:(]251(0.45)+0.449749(8.55)) :4.05224

Although the value of the put option at the initial node is shown, it is not necessary to calculate it. The put-on-put ophon is worth 1 - 0.20037 = 0.79963 at II and Oat d. Therefore, its initial value is e- 001(0.550251(0.m6J))

=lo.43sb1 I

Notice how it is worth more if the stock price goes up. Calls-on-puts and puts-on-calls are worth less if the stock price goes up.

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QWZ SOLUTIONS FOR LESSON 27

457

76.05 -------------

26.05

58.5 13.27682

40.95 0

45 6.76675

---------------------------

31.5 0

---------------------------

22.05 0

Figure 27.3: Underlying call option --------------for e)(ercise 27.31.

27.31.

The risk-neutral probability is p" =

e{Oll\-1l03)(05)_0.7 1.3-0.7 0.52519

The tree is drawn in Figure 27.3. The put-on-call ophon is worth Oat mx:le II and0.55 at node d. Therefore its premium is (1-0.52519)(0.SS)e-""1103=ID.253431

Quiz Solutions 27•1. The European put (A) is worth more than any Asian option, since the final price has more volatility than the average price. The geometric average price is le~ than the arithmetic average price. Since the option pay,; 40 minus the average if greater than 0, this makes the geometric average options more valuable than the arithmehc average ones. A daily average is less volahle than a weekly one, so daily average options are worth le~ than weekly average ones. This makes the arithmehc average daily price put option (C) the least valuable one.

.l'J.fi

27•2. The average stock price is¥ = 33 after two ups, = 27 after an up and a down,~ = 22 after a down and an up, and ~ = 18 after two downs. The payoff after two ups is 36 - 33 = 3; the payoff after a down and an up is 24 - 22 = 2; and otherwi,e the payoff is 0. In the tree, 11 = 1.2 and J = 0.8 at all mx:les. The risk-neutral probability of an up is e-07-~o~o~s°"8= o.sso5m Therefore the average payoff is 0.5505032(3) + (1 - 0.550503)(0.550503)(2) = 1.4041 Discounting for 2 years, the value of the option is e-l(00-1)(1.4041)= ~27-3. The variable U can be e)(pre~ed as

S(O) ✓ (S(1)/S(0)) 2 (S(2)/S(1)).

Then

In U = lnS(O) + ln(S(l)/S(O)) +0.5 ln(S(2)/S(1)) which has standard deviation 0.3 Y12 + 0.52 = ~lFMStudyManuaJ-i"edilion Copynght02!118ASM

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458

27. ASIAN, BARRIER, AND COMPOUND OPTlONS

27-4. J is worth more than JJsince the lm\~r strike mean;; a payoff at least as high as JI. VI is worthle~. and V is worth as much as II. lll and IV are both worth less than JJsince they add a condition of barrier 45. They are both worth more than VI since they are not worthless and may pay out under certain circumstances, whereas VI is worthle~. So we end up with Vl 90. (In fact, r would have to be quite negative for this not to be true') The value of the option is then P= (1-p·)e-'Pd = (0.574443)e-'(90-100d) 32 )

= (0.574443)e-r (90-100e'--0 : 0.574443 (90e-r -1(J(Je-03l) and we are given that this equals 8.46. Solving for r,

8 90e-r = _ ;:: +lOOe--032 =87.3422 0 5 43 e-r = 87-:;22 =0.97047 r=-ln0.97047=~ 29.7. ln(X(10)/X(0)) (0.32 )(10)=~(C)

I X(0)

(B)

is normally distributed with parameters (a - 0.5a 2)(10) and (,, 2)(10) =

29.8. Let Q(t) = ln(S(t)/S(t -1)). Then Q(t) are independent for I = 1, 2, 3, and are normal with parameters /J = 0.08- 0.5(0.32 ) = 0.035 and ,, 2 = 0.3 2 = 0.09. We can decompose Gas follows: lnG = lnS(0) + ½(Q(l) + (Q(1) +Q(2)) + (Q(l) + Q(2) + Q(3))) = lnS(0) + Q(l) + fQ(2) + }0(3) The variable Q(1)+jQ(2)+jQ(3)

is normal with /J = 2(0.CD5)= 0.07 anda

So G/5(0) islognormal with the same paramete15 /Janda.

S(O)ell07+1l5(1l14) = 4 0eou =~ 29.9.

2

= 0.09 (1 +

2 @2 + (½} ) = 0.14.

The expected valueofG is (B)

We back out the risk-free rate from the Sharpe ratio. 0.35= o.~3-r r=0.15-0.3(0.35) For the Black-Scholes formula, note that K = S =

(0.045+0.5(0.3 2 ))(2) di= 0.31/2 IFMStudyManual-i"edilion CopynghtC2!118ASM

=0.045

:A) since

the option is at-the-money. 0.42427

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EXERCISESOLUflONS FOR LESSON 29 84.5

50 13.6801

~

-------

44.5

-------

45.5 5.5

65 25

._____________~

35 2.7307._____________ 24.5 0

Figure 29.1: Binomial tree for American call option of question 29.10. Boldface values indicate option is optimal to exercise.

d2 = 0.42427 - 0.3Y2 = 0 N(d1) = N(0.42427) = 0.66431

N(d2)=N(0)=0.5 C(50, 50, 2) = 50(0.66431) - :AJe--0o(S(i)(0.5) =~ 29.10.

(E)

The risk-neutral probability of an up mm·ement is ell04-ll03_0 7 p" = ___ • =0.51675 1.3-0.7

The binomial tree is shown in Figure 29.1. At the upper nOOeat the beginning of the .e(:ond year, the value of the ophon if not elected is e--0°4(p"(44.5) + (1 - p")(5.5)) = e- 1104 ((0.51675)(44.5) + (0.48325)(5.5)) = 24.647 which is less than the exercise value of 25. At the lower nOOeat the beginning of the .econd year, the value of the option is Cd

=e- 1104 ((0.51675)(5.5)) =2.7307

At the initial node, the value of the ophon is C = e- 1104 ((0.51675)(25) + (0.48325)(2.7307)) = 13.68 We conclude that the option premium is~(C) 29.11. ''•hd is not needed. We use formula (25.15). y-

r= Q(a -r)

-0.20-0.05

= Q(0.15-0.05) Q:

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~~;~5 =811

(C)

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29. SUPPLEMENTARY QUESTIONS-DERIVATIVES

498

29.12.

d1=

ln(Ke 05 1) = Pr(lnQ(2) +2lnQ(3))

>0

lnQ(i) is normal with /J = 0.15 - 0.5(0.4 = 0.07 and ,, = 0.4 Let X = lnQ(2) + 2 lnQ(3). Then X is normal with /J = 3(0.07) = 0.21 and a 2 = 0.4 2 (12 + 22 ) = 0.8. Therefore, 2

2

2

)

.

Pr(X > 0) = 1- N(-0·

21

) = N(0.23479) =[0.59281]

v'o.8

29.28.

By put-call parity, Ke-'

For K

1

-

Se- 61

=P(S,K,t)-

(B)

C(S,K,t)

=30, 60: 30e-r 1 -se6(k-rl

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61

-se-6/:

=9-

20

22-6:

=-11 16

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EXERCISESOLUflONS FOR LESSON 29

Subtracting the first from the ,econd, 30e-'

503

1

= 27, so 10e-rr = 9 and

10-x

= -11 +9= -2 50e-rt _ se-61: 30e-rt _ se-61 + 20e-rr

x+ y =12+15

29.29.

=-2

X

y-8=7

=-11+2(9)=7 =[li]

= 12

y=15

(C)

By put-call parity,

The Treasury matures for 10,000, so K = 10,000. We will need 2~ option;; with strike price 40 to get a strike price of 10,000. The stock porhon of the portfolio on the left-hand side is then 250 times Se--02.'i(ooi)= Se- 11005, so we need 250e- 11005 =~shares. (B) 29.30. The replicating portfolio pay,; 15 at the top mxle and Oat the bottom node, since the stock price declines 5 after payment of the dividend to 115 or 75 at the two nOOes. A share of stock will end up with a 40 difference in value (115 vs. 75) at the end of the period. So the number of shares of stock in the replicating portfolio is~ = @}ill.(A) 29.31. Let S(t) be the price of the stock at time I with I measured in years. The put pays off if - 5(3). Let Q(t) = S(t)/5(1-1). Cube the inequality. For the put to payoff, we need S(O)'Qll) (Qll)Ql2)) (Qll)Ql2)Ql3)) > S(O)'Qll)'Ql2)'Ql3)' Q(2r 1Q(3r 2 > 1 Logging, this mean;; the variable T = lnQ(2) + 2lnQ(3), a normal random variable, is less than 0. The parameters of Qare /J = 0.12- O.S(0.32) = 0.075 and a 2 = 0.3 2 = 0.09. The parameters of Tare /J = 3(0.075) = 0.225 and ,, 2 = 0.09 + 4(0.09) = 0.45. The probability that Tis less thanO is

=

-0225) Pr(T < O) = N -· - = N(-0.33541) = ~ ( v'o.45 29.32.

(C)

The volahlity of the difference in rates of return behveen the stock;;, from equation (28.3), is

✓o.25 2 + O.J2 - 2(0.8)(0.25)(0.3) = v'o.i.ms = 0.18028 We can treat this as a call on Q with strike as.et S. Since both stock;; are nondividend paying, r = 0 = 0. The Black-Scholes formula reduces to di= ln(45/4~~~~~(0.0325)

0.74347

d2 = 0.74347 - 0.18028 = 0.56319 N(di)=N(0.74347)=0.77140 N(di)=N(0.56.319)=0.71335 C(Q,S,1) =45(0.77140)-40(0.71335)

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=[Iill

1"1~

(D)

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504

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29. SUPPLEMENTARY QUESTIONS-DERIVATIVES

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Lesson 30

Real Options Reading: Corparale Fi11mrce 22.1-22.4 This lesson is based on the Cor,x;rateFi11111rce textbook. assume that interest rates are annual effective.

In this lesson, unless you are told otherwise,

Often a business has the option of either beginning a project immediately or waiting for more information before deciding whether to embark on the project. Such an ophon is called a realoption. It may be valued using the methods similar to tho,e u,ed for financial options: trees or Black-Scholes.

30. 1 Decision trees For a financial option, we may use binomial trees to value it. Each nOOe leads to two branches. The ,elechon of future stoi::kprices at the ends of the,e branches is ba,ed on volatility. For a real option, there may be more than two scenarios, so we would not restrict ourselves to having only two branches coming out of a mx:le. And the value of the option at the end of the mx:le is based on whatever the real situabon For a financial option, the only decisions to make are whether to buy the opbon, and for an American option whether to exercise the option. For a real opbon, there may be decisions to make within the project other than merely whether to enter or exit the project. Therefore, for real opbons we use decision trees rather than binomial trees. Decision trees have decision nOOes and information nodes. At a decision node one can make a decision and choose which branch to follow. At an information node there is no control-the branch that is selected is a random event, just like ina binomial tree. We'll use the following com·entions that the textbook uses: • Squares are used for decision nodes and circles for informabon nOOes. • Diagonal lines out of information nodes are dashed. • Monetary costs may be placed at the ending nOOes or on the bottom of any line. (However, in our example,\\~ do not place monetary amounts in the tree.) Any nOOe may have more than 2 branches. Here is an example: You can purchase a property for 10 million. You can then build an apartment building with 100 units on it. It will take one year to build, and will cost 20 million paid immediately. You can build the building immediately or you can wait one year. If you wait one year, there will be no expenses during the first year. If you choose to build at the end of one year, it will take one year to build and will cost 22 million paid immediately. Based on rents in the area, the units would generate 600,000 per year of free cash flows in the second year from now, growing 2% per year, perpetually. The area may become more popular in a year, with probability 50%. If the area becomes more popular, then rents would be higher. The units would generate 1,000,000 per year of free cash flows in the second year from now, growing 2% per year. If you choose not to build, you can sell the property at the end of one year for 10 million if the area doesn't become more popular, and for 20 million if the area becomes popular. IFMStudyManual-i"edilion CopynghtC2!118ASM

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30. REAL OPTIONS

More popular

50%,' Build now

50%',

Unchanged

Buy property More popular


0

(E) 5333 with cumulative distribuhon

Claim counts follow a binomial distribution with ma: 2, q : 0.3. You estimate annual aggregate losses using simulation. Use the following uniform numbers on l0, 1) to simulate claim counts for 3 years: 0.7

0.1

0.5

Use the following uniform numbers on l0, 1), in order and as needed, to simulate loss sizes: 0.213

0.627

0.424

0.055

0.121

0.670

The automobile collision cm·erage has an ordinary deduchble of 500 per loss. Determine average annual payments made under this coverage. (A) 418

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(B) 136.3

(Cl 1669

(D) 2045

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(E) 2122

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PRACTICE EXAMS

21.

An equity-linked

annuity pay,; the same return as a stock index.

A guarantee rider is offered. The rider guarantees that after 1 year, the purchaser of the annuity will receive at least a 4% continuously compounded return. You are given: (i) (ii) (iii) (iv) (v) (vi)

The stock index follows the Black-Scholes framework. The volatility of the index is 0.3. Dividends are incorporated in the index. The continuously compounded risk-free interest rate is 4%. The average continuously compounded return of the stock index is 15%. Everyone who purchases the annuity keeps it for at least one year.

Determine the value of the guarantee rider as a percentage of the initial stock index price. (A) 11.0%

22

(B) 12.0%

(Cl 13.0%

(D) 14.0%

(E) 15.0%

The price of a stock at time I, with I in years, is 5(t). You are given:

(i) (ii) (iii) (iv) (v)

The stock's price is lognormally distributed. 5(0) 100. The continuously compounded expected rate of return on the stock is 0.15. The stock pays dividends of0.0SS(t)dl between hmes I and I+ di. The stock's volatility is 0.2.

=

Calculate the expected value of 5(1) given that 5(1) < 90. (A) 81.11

(B) 81.82

(Cl 83.25

(D) 85.19

(E) 87.56

23. Summary stahshcs for the returns on a stock over an 18--yearperiOO, R;,are

tR1=3.2

"

t/f=0.7164

A 95% confidence interval for the return is constructed using the methOO of Car,x;mte Fi11a11ce. Calculate the lower bound of the confidence interval. (A) 0.134

(B) 0.135

(Cl 0.136

(D) 0.174

(E) 0.175

24. Your European friend will give you a gift of the minimum of €100 and $1:() at the end of 5 yea!';. You are given: (i) The continuously compounded risk-free rate for dollars is 0.06. (ii) The continuously compounded risk-free rate for euros is 0.04. (iii) The current exchange rate is €1 $1.55. (iv) A European option to buy $1.50 for€1 at the end of 5 years costs €0.01.

=

Calculate the current value of the gift in dollars. (A) 110

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(B) 113

(Cl 115

(D) 119

(E) 121

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PRACTICE EXAM 8

25. For a project, the equity beta is 1.2 and the debt beta is 0.4. The project is financed with 8 million equity and 2 million debt. The annual effechve risk-free interest rate is 0.03 and the risk premium is 0.1. Calculate the unlewred cost of capital. (A) 0.086

(B) 0.090

(D) 0.130

(Cl 0.110

(E) 0.134

26. You are given the following prices for European call options expiring in three months on a stock: Strike rice 50 55 70

Call 16 14

To exploit the mispricing, you ,ell a call option with strike price55 and buy or ,ell appropriate amounts of the other two call options so that: (i) At expiry, the net value of the position is guaranteed to be non-negahve. (ii) The net excess of the premiums you get m·er the premiums you pay is maximized. The stock price after three months is 60. Determine the net profit ignoring interest. (A) 2.50

(B) 2.75

(Cl 3.00

(D) 3.25

27. Which of the following market anomalies have been ob,erwd

(E) 3.50

7

I. January effect: returns are higher in January.

II. Mondayeffect: retumsarehigheronMonday. Ill. Time-of-Day effect: returns are higher close to market open. (A) None (B) I only (C) II only (E) The correct answer is not given by (A), (B), (C), or (D).

(D) lllonly

28. For a stock index, you are given: (i) (ii) (iii) (iv)

The current price of the index is 1650. The3-month forward price of the index is 1670. The continuously compounded risk-free interest rate is 0.06. The price of an at-the-money 3-month European call option on the index is 78.

Calculate the price of a 3-month European straddle on the index. (A) 116

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(B) 126

(Cl 1.36

(D) 146

(E) 156

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PRACTICE EXAMS

29. You are considering opening bowling alleys. You can open two bowling alleys in two different towns. Both towns have similar populations, so if the business is succes.sful in one town, it will be succes.sful in the other town.

For each bowling alley, (i) (ii)

Jt will cost 1 million to open. Free cash flows each year will be 150,000 if successful, 40,000 if not.

The cost of capital is 10%. The probability of succes.s is 0.5. You may also open one bowling alley now and see if it is successful. If it is successful, after one year, you open the other bowling alley for 1,000,000.

If you decide to close a bowling alley, you can sell it for the present value of its cash flows. Calculate the value of the option to wait one year before opening the second bowling alley. (A) 45,000

(B) 91,000

(Cl 132.000

30. For two sta.:ks with time-t prices Q(t) and R(t),youare

(D) 159,000

(E) 177,000

given:

(i) Neither stock pays dividends. (ii) The continuously compounded expected rates of return are0.10 for Q and 0.15 for R. (iii) The volatilities are0.20 for Q and 0.30 for R. (iv) The correlation coefficient of 21(1) and Z2(t) is 0.6. (v) Q(0) =5. (vi) R(O) = 20. Consider S(t) = Q(t )R(t ). CalculatePr(S(4) < 150). (A) 0.20

(B) 0.23

(Cl 0.26

(D) 0.32

(E) 0.35

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Practice Exam 9 1. Determine which of the following strategies are appropriate for an investor expecting the price of astocktoincrea,e. I.

Naked collar

II. In-the-money written cm·ered call Ill.

Bull spread with an in-the-money and out-of-the-money

put.

(A) I only (B) II only (C) Ill only (E) The correct answer is not given by (A), (B), (C), or (D).

(D) 1,11,andlll

2 Which of the following ob,ervations would one expect if the weak form of the efficient market hypothesis is true7 I.

Ina scatter plot of the stock return on date I+ 1 against the stock return on day I, there should bean equal number of points in each of the four quadrants.

II. The autocorrelation coefficient of returns should be clo,e to 0. Ill. The variance of returns should not vary by length of period. (D) 1,11,andlll

(A) I and JI only (B) J and Ill only (C) JJ and Ill only (E) The correct answer is not given by (A), (B), (C), or (D).

3. The prices of European call and put option;; on the same stock with the same strike price and hme to expiry vary with r, the continuously compounded annual risk-free interest rate. You are given the following prices

0.04 0.08 0.12

Call 3.11 3.45 3.82

Put 12.24 11.79 11.46

Determine hme to expiry for these option;;. (A) 1

(B) 2

(D) 4

(Cl 3

(E) 5

4. For a large equally weighted portfolio of stocks, the correlation of pail'; of stocks is 0.5 and the volatility of each stock is 0.2. Calculate the limit of the volatility of the portfolio as the number of stocks goes to infinity. (A) 0.02

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(B) 0.05

(D) 0.11

(Cl 0.08

599

(E) 0.14

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PRACTICE EXAMS

5. For a 4-year European call option on a nondividend (i) Thestrikepriceis40e 4'.

paying stock

(ii) The stock price is 40. (iii) The stock price follows the Black-Scholes framework. (iv) The price of the option is 7.28. Determine a, the volahlity of the stock. (A) 0.20 6.

(B) 0.23

(Cl 0.25

(E) 0.30

(D) 0.27

Fora stock

(i) The current price of the stock is 80. (ii) The stock pays dividends of 1 per quarter. (iii) The continuously compounded annual risk-free rate is 4%. Calculate the 2-year forward price of the stock.

(A) 78.38

(B) 78.66

(Cl 78.95

(D) 79.32

(E) 79.57

7. A company has free cash flows of 100,000 for the current year, growing 3% per year. The company's debt-equity ratio is 1. The cost of equity capital is 15%. The cost of debt capital is 6%. The company maintains a constant debt-equity ratio. Thecorporatetaxrateis25%. Calculate the pre.ent value of the tax shield. (A) 133,000

(B) 140,000

(Cl 148.000

(D) 155,000

(E) 163,000

8. For a 1-year off-market forward contract on a stock providing for the purchase of a stock after 1 yearatapriceof91:

(i) (ii) (iii) (iv) (v) (vi)

The continuously compounded annual dividend rate of the stock is 0.02. The 1-year forward price of the stock is 90. The price of a 6-month European call ophon on the forward contract is 2.25. The continuously compounded annual return on the stock is 0.13. The continuously compounded annual risk-free interest rate is 0.06. If an ophon is exercised on the contract, the option purcha,er will own the forward contract but will not get an immediate mark-to-market payment.

Determine the price of a 6-month European put ophon on the forward contract. (A) 3.01

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(B) 3.07

(Cl 3.13

(D) 3.19

(E) 3.25

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PRACTICE EXAM 9

601

9. For two European call ophons on the same stock, C1 and C2 (i) The stock price is 45. (ii) The price of C1 is 10.86. (iii) The delta of C1 is0.6252. (iv) Eta buy,; one call of each type. Eta's portfolio has elasticity 2.~. (v) Z.eta buys one C1 and hvo Ci's. Zeta's portfolio has elasticity 3.03. (vi) Iota buys one C1 and sells one C2. Determine the elasticity of Iota's portfolio. (A) 0.13

(B) 0.35

(D) 0.70

(Cl 0.52

(E) 0.90

10. The time-I priceofastockisS(I). A forward start option will, after a year, give the pun::haser a 2-year gap put ophon on the stock with strike price 0.95(1) and trigger price 1.1$(1). You are given: (i) (ii) (iii) (iv) (v)

The price of the stock follows the Black-Scholes framev.Tirk. The 1-year forward price of the stock is 75. The volatility of the stock is 0.3. The stock pays no dividends. The continuously compounded risk-free interest rate is 0.1.

Calculate the price of the forward-start option. (A) 2.38

(B) 2.50

(Cl 3.02

(D) 3.29

(E) 5.23

11. For an insurance policy, claim counts have a Poisson distribution with mean 2. Claim sizes are lognormal with /J a: 5 and,, a: 2. Reinsurance covers aggregate losses subject to an aggregate deductible of 100. Three years of reinsurance claims are simulated using the inversion methOO. U.e the following uniform random numbers on LO,1) to simulate claim counts: 0.8

0.2

0.4

U.e the following uniform random numbers on LO,1) in the order given, as many as needed, to simulate claim sizes:

Calculate the total simulated reinsurance payments for three years. (A) (B)

(C) (D) (E)

Less than 240 At least 240, but less than 2~ At least 290, but less than 340 At least 340, but less than 3~ Atleast390

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PRACTICE EXAMS

'"' 12

For a 6-month dollar-denominated

(i) (ii) (iii)

American put option on eums:

A 2-period binomial tree based on forward prices is used to model it.

Currently, the exchange rate is €1/$1.50. The strike price of the put option is $1.50. (iv) The continuously compounded risk-free interest rate for dolla15 is 0.04. (v) The continuously compounded risk-free interest rate for eums is 0.05 (vi) The variance of the logarithm of theeum to dollar exchange rate after I years is 0.041.

Calculate the price of the option. (A) 0.079

(B) 0.081

(Cl 0.083

(D) 0.085

(E) 0.087

13. The Sharpe ratio of the market portfolio is 0.4. A stock has 60% correlation with the market and has a beta of 1.2. Calculate the Sharpe ratio for the stock. (A) 0.16

(B) 0.24

(Cl 0.32

(D) 0.40

(E) 0.48

14. The time-/ price of a stock is S(t). You are given (i) (ii) (iii)

For all I, S(t)/S(0) follows a lognormal distribution with parameters /JI and S(0) = 1.2 Pr(S(1) > 1.2) = 0.60642

(iv)

Pr(S(2) > 1.44) =0.34827

aVI.

DetermineVar(S(1)). (A) 0.03

(B) 0.04

(Cl 0.05

(D) 0.06

(E) 0.07

15. Consider the risk measure g(X) = 10 VaR,,(X), 10 times the value-at-risk of X at 95%. Which of the following coherence properties does g(X) satisfy 7 I. Translabon invariance II. Positive homogeneity Ill.

Monotonicity (D) 1,ll,andlll

(A) I and JI only (B) I and JJJonly (C) JJand JJJonly (E) The correct answer is not given by (A) , (B) , (C) , or (D) .

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PRACTICE EXAM 9

16. The price of a stock at time I is 5(t). An option on the stock has the following provisions: (i) There is no upfront payment for the option. (ii) If 5(1) > 50, then the ,eller pays the buyer 5(1) -50. (iii) If 5(1) < 50, then the buyer pays the .eller k at time 1. You are given: (i) (ii) (iii) (iv) (v) (vi)

The price of the stock follows the Black-Scholes framework. 5(0)=.:(). The continuously compounded dividend rate for the stock is 0.02. The annual continuously compounded expected return on the stock is 0.10. The continuously compounded risk-free interest rate is 0.06. The volatility of the stock is 0.40.

Determine k if the option is sold at time I = 0. (A) 9.15 17.

(B) 15.96

(D) 18.72

(Cl 16.95

(E) 19.87

A stock's price after 3 months is modeled with the following one-periOO binomial tree. 66

~ 49.50

55 -------------

You are given (i) The continuously compounded dividend rate on the stock is 8%. (ii) The continuously compounded risk-free interest rate is 4%. A bull spread consists of a long 60-strike put and a short 70--strikeput, each put expiring in3 months. Calculate the number of shares of stock in the replicahng portfolio for the bull spread. (A) 0.356

(B) 0.360

(Cl 0.36.3

(D) 0.367

(E) 0.371

18. A project is expected to generate 5 million in revenue for 10 years, and no revenue after that. Expenses are 6millioninihally; • •

1 million each year; 10% of revenue each year. The annual effective risk-free interest rate is 0.04 and the risk premium is 0.Cll. Beta for the project is 1. Calculate the NPV of the project.

(A) 11.3 million

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(B) 13.8 million

(C) 16.3 million

(D) 19.4 million

(E) 22.4 million

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PRACTICE EXAMS

19. For a }-month futures contract on 250 units of an index: (i) (ii) (iii) (iv) (v) (vi) (vii)

The initial price of the index is 1280. The futures price equals the forward price. The initial margin is 10% of notional value. Maintenance margin is 70% of initial margin. The index pays dividends at a continuously compounded rate of 2%. The continuously compounded risk-free interest rate is 5%. The margin account pays no interest.

A margin call for 11000 is made on the next day. Calculate the price of the index on that day. (A) 1236 20.

(B) 1241

(Cl 1251

(D) 1256

(E) 1263

Which of the following statements are true?

I. Small investors trade too often.

II. lnvesto15 sell winners and hold on to losers due to tax considerahons. JI!. The average mutual fund manager is able to idenhfy profitable trading opportunities. (A) I and JI only (B) I and Ill only (C) JJ and Ill only (E) The correct answer is not given by (A), (B), (C), or (D).

21.

(D) 1,11,andlll

For a stock index and European options on it, you are given:

(i) (ii) (iii) (iv)

The The The The

price of the index is 1000. index pays continuous dividends at a compounded annual rate of0.015. purcha,er of a 950-1050 bull spread of puts on the index expiring in 6 months receives 27.02. continuously compounded risk-free interest rate is 0.04.

Calculate the price of a 9~-1050 European collar on the stock expiring in 6 months. (A) 7.47

22

(B) 9.66

(Cl 11.18

(D) 14.68

(E) 19.55

You are given

(i) (ii) (iii) (iv) (v) (vi) (vii)

The price of a stock follows the assumphons of the Black-Scholes formula. The stock's current price is 40. The stock pays a continuously compounded dividend rate of0.02. The continuously compounded rate of increa,e in the stock price is 0.15. The annual volatility of the stock's price is 0.25. The continuously compounded risk-free interest rate is 0.07. A European call ophonon the stock expiring in 4 years has strike price 50.

Calculate the price of the call option. (A) 5.84

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(B) 6.95

(Cl 7.53

(D) 12.15

(E) 13.55

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PRACTICE EXAM 9

605

23. For an American put option on a futures contract (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

The option expires in 1 year. The futures contract is a 1-yearcontract on a stock index. The stock index has a conhnuously compounded dividend yield of0.02. The continuously compounded risk-free interest rate is 0.05. The 1-year prepaid forward price of the stock is 6.5. The option is modeled using a 2-periOO binomial tree based on forward prices. The volatility of the futures contract is 0.3. Thestrikepriceis6.5.

Determine the price of the put option. (A) 5.90

(B) 6.16

(Cl 6.53

(D) 6.96

(E) 7.25

24. The time-/ price of a nondividend paying stock, S(t ), follows the Black-Scholes framework. You are given: (i) The initial price,S(0),is40. (ii) The continuously compounded expected rate of return of the stock is0.15. (iii) The volatility of the stock is 0.20. CalculateE[S(2)2]. (A) 2293

(B) 2484

(Cl 2691

(D) 2915

(E) 3158

25. You are given: (i) A stock has price 45. (ii) A market-maker writes put ophon I on the stock with price 1.53, delta -0.39, and gamma 0.072. (iii) The market-maker delta-gamma-hedges the option with the stock and with put ophon JJ having price 2.00, delta -0.31, and gamma 0.038. Determine the number of shares of stock to buy to implement the hedge. (A) 0.19

(B) 0.20

(Cl 0.21

(D) 0.22

(E) 0.23

26. You sell 100 shares of a nondividend-paying stock short at 38 per share. You buy a 6-month call option on 100 shares of the stock with strike price 40. At the end of 6 months you close the position. You are given: (i) The cost of the call ophon is 120. (ii) The annual effechve risk-free interest rate convertible semi-annually is 4%. Ignore transaction costs and costs of maintaining a margin. State the minimum and maximum profit for this strategy, and the name of this strategy. (A) (B)

(Cl (D) (E)

Minimum Profit -246.4 -246.4 -246.4 -122.4 -122.4

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Maximum Profit 3753.6 3753.6 Unlimited 3753.6 Unlimited

Name Cap Cm·eredcall Cap Cm·eredcall Covered call

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PRACTICE EXAMS

27. An asymmetric butterfly spread is constructed using put options. The strike prices are 40, 50, and 65. The spread contains 30 puts with strike price 65. Determine the number of puts with strike price 50. (A) 20

(B) 45

(Cl 50

(D) 60

(E) 75

28. Which of the following are not examples of corporate debt? (A)

Mortgage bonds

(B)

Revenue bonds Private placements Yankee bonds Debentures

(C) (D) (E)

29.

For a dollar-denominated

Asian call option on 100 yen:

(i) (ii) (iii) (iv)

The continuously compounded risk-free rate for dollars is 0.05. The continuously compounded risk-free rate for yen is0.01. The current exchange rate is Y100=$1.10. The option will pay,at the end of three years, the excess of the arithmetic average dollar value of 100 yen at theendsof each of the three years m·er $1.12. (v) An otherwise similar Asian put ophon costs $0.23.

Determine the value of the call ophon. (A) 0.27

(B) 0.29

(Cl 0.31

(D) 0.33

(E) 0.34

30. You are given the following covariance matrix for three stocks, A, B, and C: 100 25 0) 25 81 12 ( 0 12 64 Portfolio I consists or 4000 invested in A and 6(XJ(J invested in B. Portfolio JI consists of 4(KJ(J invested in C and 6000 im·ested in B. Calculate the correlation behveen the hvo portfolios. (A) 0.70

(B) 0.72

(Cl 0.75

(D) 0.77

(E) 0.80

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Practice Exam 10 1. An investor buys a collared stock. The current price of the stock is 200. The collar has strike prices 200 and 210 and is zem- K2 (the trigger price) you collect 5 1 and pay K1,and if 5 1 < K2 you pay K1 and collect Sr which is the same as collecting Sr and paying K1, so

In this problem,

22 [Section 14.4] A synthehc forward is a long stock plus a short bond. So a short bond is a long forward plus a short stock. (C) 23. [Section 21.3] The6-month forward rate of eums in pounds is e{llll6-ll04)(1l5) = e001 = 1.01005. Up and down mm·ements, and the risk-neutral probability of an up mm·ement, are II:

tOOl+llJ,/iis: 1.084(}6

d :eOOl-llJ,/iis =0.94110 p·

1-p·

=

~:~°:=~::~~~

= 1-0.4823

0.4823

=0.5177

The binomial tree is shown in Figure A.1. At the upper nOOe of the second column, the put value is calculated as P" = e--003(0.5177)(0.08384) = 0.04212 At the lower nOOe of the second column, the put value is calculated as p~enta~w:

IFMStudyManual-i"edilion CopynghtC2!118ASM

e-llOl((0.4823)(0.08384) + (0.5177)(0.19147)) = 0.13543

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PRACTICE EXAM 1, SOLUTlONS TO QUESTIONS 24-26

6'3

0.94014 0

0.86725 -------0.04212

~0.81616

08 0.J ------------~

--------

0.08384

0.75288 0.14712

~0.70853 0.19147 Figure A.1: E)(change rates and option values for put option of question 23

but the exerci,e value0.9- 0.75288 = 0.14712 is higher so it is optimal to exerci,e. At the initial node, the calculated value of the ophon is ptenta~w= e-lllll((0.4823)(0.04212) + (0.5177)(0.14712)) = 0.09363 Since 0.9 -0.8 = 0.1 > 0.09363, it is ophmal to exerci,e the option immediately, so its value is 0.10 (which means that such an ophon would never exist), and the price of an option for€100 is 100(0.10) = [!Q}. (E) 24. [Section 2.2] Let r be the cost of capital. The NPV in millions is NPV=-16+-

2 -=0 r-0.03

-'-=• r-0.03 8r-0.24=

1

r=¥=~

(E)

25. [Section 23.2] The frachon X(2)/X(0) follows a lognormal distribution with parameters m = 2(0.1 - 0.5(0.22)) = 0.16 and v = 0.2..fi.. Cubing does not affect inequalities, so the requested probability is the same as Pr(lnX(2)- lnX(0) > 0), which is

!)= N(0.56569) = ~

01 1 - N(- ·_ 0.2v2

26. [Section 26.2] By formula (26.3) with

E

(E)

= 1 and h = 1/365,

Market-Maker Profit= -0.5ft 2 - eh - rh(St..-C(S)) = -0.5(0.08)(1 2) +0.02-

~-:l(l00)(0.76)

= -0.04 + 0.02 - 0.00986 = I -0.0298b

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I

- 4] (B)

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PRACTICE EXAM 1, SOLUTlONS TO QUESTIONS 27-30

634

=

27. [Lesson 30] For the bottom two nodes, expected profit is 0.4(1200) +0.6(900) 1020. The expected profits at the four information mx:les in the fourth column, one period before the end, are, in order from top to bottom:

0.7(3000) + 0.3(1400) 0.6(2500) + 0.4(1200) 0.5(2000) + 0.5(1000) 0.4(2000) + 0.6(1000)

=2520 =19&1 =1500 =1400

Pulling back to the third column, these become

2520-1500 =1020

= = =

1980-800 11&1 1500- 800 700 1400 -1000 400 At the upper decision node in the second column, 11&1 is ,elected. At the lower decision nOOe in the .e(:ond column, 700 is .elected. At the informahon mx:le in the first column, expected profit is 0.5(1180 + 2000) +0.5(700 +200) 1940. After subtrachng 900, this yields~ at the initial decision nOOe,which is greater than 1020, so it is the optimal profit. (C)

=

28. [Lesson 20] The risk-neutral probability is e{r-6)h-d e{006-1l0l)(025)_d 0.5 =p· = ~ = 11-d

ellOl-d 11-d

but 11=de 2 ",r,;=de 2(03X1/ll =de 03,so ellOl- d : 0.5 (ell3d - d) eOOl=d (0.5(e113-1)+1)

=1.17493d

ellO\ d = 1.17493 =0.85967 II

= 0.85967eOJ : 1.16(}43

The option only pays at the upper node. The price of the ophon is C

= e-'hp•(S11 - K) = e-llll6{02S)(0.5)(40(1.16043)-40) = ~

(8)

29. [Lesson 5] The variance of the portfolio is 0.16 (0.3 +0.84 (0.2 + 2(-0.5)(0.16)(0.84)(0.2)(0.3) = 0.022464 2

2

2

)

2

)

Let p be the proportion invested in Stock A in the more efficient portfolio. Then 2

0.09p +0.04(1-

p)2

-

0.06p(1 - p) = 0.022464

0.19p -0.14p+0.017536 2

p

=O

0.14+0~~

(C)

30. [Subsection 25.1.7] Delta for a portfolio of options on a single stock is the sum of the individual deltas of the ophons. 100(0.6262) + 100(0.6517) + 200(0.9852) = ~ (E) IFMStudyManual-i"edilion CopynghtC2!118ASM

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PRACTICE EXAM 2, SOLUTlONS

TO QUESTIONS

635

1--4

Answer Key for Practice Exam 2 11 12

D

21

C

13

B

22 23

C

14

C

D D D

15

B A B A A A

1 2 3

C

4 5 6 7 B 9 10

D A

C

16 17 1B

E

19

D

20

27

E A E A A B B

2B

C

29

D A

24 25 26

30

Practice Exam 2

ore trading

1. [Lesson lJ houses. (C) 2.

is direct behveendealer

18.ll By put-call parity,

[Subsection

Ke-006/3_

K = 4~~::1

3.

and pun::haser and does not go through clearing-

4 (k--003/3

=O

=~

(D)

[Lesson 5] Use quotients of consecutive prices to compute annual yields.

32 15 • -1 =0.146168 28.05 ~:~!-1

=0.051944

38 11 • -1 =0.126848 3.3.82 37 00 • -1 = -0.029126 38.11 The average yield is 0.073958. The variance, with division by 3, is 4

3

(0.1461682 + 0.0519442 + 0.1268482 + 0.0291262 4

2)

0.073958

=0.006374

The variance of the mean is this number divided by the size of the sample, which is 4, and the standard error is the square mot of the variance of the mean, so the standard error is ✓o.006374/4 = ~(A)

4. [Section 25.1] Only call theta is usually negative. (C) Vega measures the increase in option price due to increase in volatility. Higher volatility makes the option more likely to pay off and therefore more valuable, making vega positive. IFMStudyManual-i"edilion CopynghtC2!118ASM

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PRACTICE EXAM 2, SOLUflONS TO QUESTIONS 5-9

636

Rho measures the increase in option price due to increa,e in risk-free rate. A higher risk-free rate makes the payment for the stoi::kat expiry have a lower present value, making a call option more valuable and rho positive. Gamma measures the change in delta as the stock price increa.es. Delta increa.es as a function of stock price, making gamma posihve. Theta measures the increase in ophon value as time to expiry decreases. The value of a call ophon usually declines as time to expiry decreases, making theta negative. 5. [Lessons 18 and 21] A European call with the ~me strike price and expiry date would be worthless based on this tree. Therefore by put-call parity the price of the put option is Ke-' 1 - Se-M = 150e--002_100e-ll05=~.(D)

6. [Section 23.1] lnX(5)/X(0) is normally distributed with parameters (a - 0.5a 2)(t) = (0.1 0.5(0.0s2))(5) = 0.49375 and (a 2 )(t) = (0.052 )(5) = 0.0125. Then r,(1n(X(5)/X(O)) > In 1.5) = 1- N(lnl.5-0.49375)

v'o.ims

= 1- N(-0.78964) = 1-0.21487=10.78513] 7. [Section 26.2] By formula (26.3) with

E

(D)

= 0, the profit is

Market-Maker Profit= -eh - rh(St.- C(S)) = 0.Ql -

~~; (25(-0.32)-2.50)

=0.Ql+0.001438=10.011438]

(D)

8. [Lesson 3] We use a, not r, since we are calculating the stock price and not discounting it. The lognormal parameters are /JI= 2(a -0-0.5,1 2) = 2 (0.15-0.03-0.5(0.3)2) =0.15 ,1 VI=

o.3...fi.= 0.424264

The following table shows the results of the runs. Actually, to save time, rather than multiplying each x; separately by 55, we could first average the x ;sand then mulhply by 55. Z;

11;=0.15+0.424264%;

-1.6 -0.3 0.8 1.2

-0.5288 0.0227 0.4894 0.6591

X;=e"'

S;(2)=55x;

0.5893 1.0230 1.6314 1.9331

32.41 56.26 89.72 106.32

The average of the four generated values for $(2) is (32.41 + 56.26 + 89.72 + 106.32)/4 = ~ (C) 9. [Section 24.2] By the Garman-Kohlhagen formula, for 1 Swiss franc, ln(0.25/0.26) + 0.05- 0.04 +0.5(0.1 2 ) 0.1 d2 = -0.24221 - 0.1 = -0.34221 d

1

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0.24221

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PRACTICE EXAM 2, SOLUTlONS TO QUESTIONS 10--14

637

N(d1) = N(-0.24221) =0.40431 N(d2) = N(-0.34221) = 0.36610

c = o.2Se--004 (0.40431) -o.26eThe call options cost 10,000(0.006571) = ~-

110

5(o.36610) = o.006571

(E)

10. [Lesson 9] There is 5,000,000(32) = 160,000,000 of equity. The original debt to equity ratio is 1/2. So the unlevered cost of capital is 2/3(0.1) + 1/3(0.04) = 0.(ll. After borrowing 40,000,000, the debt to equity ratio is 3/4. The cost of equity capital is now rE = ru +

%(ru - ro) = 0.08 + ¾(0.08-

0.04) = ~

(D)

11. [Section 28.2] The volatility of the difference between rates of return of Assets X and Y is a= ✓o.3 2 + 0.22 - 2(0.3)(0.2)(0.6) = 0.24083. The prepaid forward price of X, the strike as.et, is

100 - PV(Dividends) = 100 - e--OOl- e- 1102 = 98.0298 The option as.et, 2 shares of Y, has price 100. In the following, nohce that the risk-free rate is not used; the appropriate discounting rate is the rate of dividends on X which is the strike asset, and we've already calculated the prepaid forward price of X. di=

ln(l00/98.0298) + (-0.02 + 0.5(0.240832))(0.5) _ (1:1 ,/(fs =0.14327 0 24 3

d2 = 0.14327- 0.24083-v'o.s = -0.02702 N(di)=N(0.14327)=0.55696 N(d2) = N(-0.02702) = 0.48922 C = 100e--0ol(o5 J(0.55696)-98.0298(0.48922) =

II!EJ

(D)

12. [Lesson 6] Sharpe raho for a stock equals Sharpe ratio for the market times the correlation with the market, or 0.75(0.4) = @:II.(C) Beta is not needed. 13. [Section 17.3] The raho spread has one long 50-;;trike call and three short 60-strike calls. Profit is higher for the raho spread when S ~ 60 becau,e they both pay the same and the ratio spread is cheaper to buy. For S > 60, we want to equate payoffs minus accumulated costs. The left side of the following equation is profit for I, and the right side is profit for II.

-(12 - 4)(1.04) + 10 = (S - 50) -3(5 - 60) = 130- 25 1.68 =130-25 $:130~1.68=~

(B)

14. [Section 121] I. The pre-money valuation is based on the price of shares in this round, which is 2.50; it is 3,000,000(2.5) = 7,soo,ooo. X

II. The post-money valuahon is 5,000,000(2.5) = 12,500,000. ✓ Ill. The new purcha.ers have 2 million out of 5 million shares, or 40% ownership. ✓ (C) IFMStudyManual-i"edilion CopynghtC2!118ASM

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PRACTICE EXAM 2, SOLUTlONS TO QUESTIONS 15-19

638

15. [Section 17.1] A naked call option has this graph. A floor has this graph as well; the floor limits losses on a stock if the price decreases too much. A cap, however, has a graph which decreases on the left up to a level at which it stays; the cap limits the amount paid to purchase a shorted asset. (B) 16. [Lesson31] The MDBG isa put option, with probability 0.1 of expiring in 10 years and probability of 0.1 of expiring in 20 years. 95,000 is im·ested in the account, so S 95,000, and the option allows selling the account value for 100,000, so K 100,000. Let's value the hvo options.

=

=

di(10)

2 ))(10)

= ln(95,000/100,000~-:~+0.S(0.2 =0.86758 =0.19281

=

O.B6?SB

0.2 Y1Q 0.23513 N(-J2(10)) 0.40705 4(0.40705)-95,000(0.19281) :8968.43 P(10): 100,()()()e-ll ln(95,000/100,000) + (0.04 + 0.5(0.22))(20) 1.28429 d1(20) = _ ,fjjj 02

d2(10) N(-d1(10))

=

d2(20) = 0.86758 - 0.2 V2Q= 0.38987 N(-d1(20)) = 0.09952 N(-d2(20)) = 0.34832 114 P(10) = 100,()()()e- (0.34832)- 95,000(0.09952) = 6196.63

The value of the MDBG is 0.1(8968.43 + 6196.6.3)= ~17.

(A)

[Section 25.2] We calculate L\ and the put premium C so that we can calculate O = di= ln(53.25/54.75)+~~-0.01

+0.5(0.25 2)

¥-

0.25388

di = 0.2538S - 0.25 = 0.00388 N(-d1) = N(-0.25388) = 0.39979 N(-d2) = N(-0.00388) = 0.49845 L\ = -e- 6N(-d 1) = -e- 110 1(0.39979) = -0.39582 C = 54.75e-lllJ7(0.49845)- 53.2.se-110 1(0.39979) = 4.36817 Q = (53.2::~~;3;582)

B]II

(B)

18. [Lesson 5] Let P be the portfolio, which is 1:AJ%in A and -:A)% in B. Cov(B,P) = Cov(B,1.5A -0.5B) = 1.5Cov(A,B)-0.5Var(B) = 1.5(0.6)(0.4)(0.1)-0.5(0.1 2) =0.031

Var(P) = 0.42(1.52) + 0.12(0.s2) - 2(0.6)(1.5)(0.5)(0.1)(0.4) = 0.3265 Corr(B,P)=

19.

(O.l~~=[0.542525]

(A)

[Section 23.2] The parameters for the lognormal distribuhon of growth m·er two years are m =2(0.15-0.5(0.3 2)) =0.21

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PRACTICE EXAM 2, SOLUTlONS TO QUESTIONS 20---25

639

0.3Y2: 0.4243

V:

So the prediction interval for the growth in the stock price over two years is 0.21 ± 1.96(0.4243) = (-0.6216, 1.0416) and the predichon interval for the stock price is (40e--06216 ,4Qe10416 ) = [ (21.48, 113.35) 1(A)

20. [Section 26.3] Let c be the amount of option B to buy. Since the gammas must add up to 0, 0.0174- 0.0369c = 0, from which it follows c = 0.0174/0.0369 = ~(A) 21. [Lesson 4] I is evidence for the weak form but not for the ,emi-strong form. JJ is not evidence of the EMH. lll is evidence of the strong form and therefore also of the semi--strong form. (E) 22. [Section 17.3] The cost of the strategy is 6.80 - 3.30 = 3.50. Accumulated with interest, it costs 3.50(1.02) = 3.57. The expected payout is5(0.4) + 10(0.2) = 4. Expected profit is4-3.57 =~ (A) 23. [Lesson 6] The variance of the portfolio is Var(P) = 0.42 (0.32 +0.2s2 + 2(0.8)(0.3)(0.25)) = 0.0436 The covariance of Stock A with the portfolio is (R is the risk-free asset) Cov(P,A) = Cov(0.4A +0.4B +0.2R,A) = 0.4 Var(A) + 0.4Cov(A, B) +0.2Cov(A, R) =0.4(0.3 2)+0.4(0.8)(0.3)(0.25) =0.06

Then~~ = 0.06/0.0436 = 11.376151. (E) 24. [Subsection 27.4.1] By Black-Scholes, the price of a put option is 2 di = O.Q75- 0.0~~; 0.5(0.24 )

0.3283.3

d2 =0.32833-0.24=0.0883.3 N(-d1) = N(-0.328.33) =0.37133 N(-di) = N(-0.088.33) =0.46481

Put:

4(Je-llll75(0.46481)- 4(Je-llll25(0.3713.J): 2.7625

By compound option parity, CallOnPut = PutOnPut-xe-rr,

+Put

: 2.3179- 5e-ll1J7.'i(Ol.5) + 2.7625: ~ 25. [Lesson 20] By put-call parity, P-C

(A)

--

= Ke-rt -Se-M

Thus Ke-rt is the amount lent for a portfolio of a long put and a short call; Ke-rt is Bput- Bcaa,where B is the amount lent. We are given that the replicahng portfolio for the call borrows 22, or Bca.11 = -22. This means that Bputcan be derived from Ke-rt = Bput- Braa = Bput+ 22 IFMStudyManual-i"edilion CopynghtC2!118ASM

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PRACTICE EXAM 2, SOLUTlONS TO QUESTIONS 26-29

&10

4Qe-llll5{0.5):

Bput+22

Bput = 4Qe-llll2S- 22

I

= 39.0124 - 22 = 17.01241

(A)

26. [Section 7.2] Use formula (7.2). The los.s L is40%, with probability p = 0.05. 0.04+0.0S(0.4)=~

(B)

27. [Lesson 22] For a put option, the early-exercise boundary is the highest value of the stock for which you would exercise the option. If you're more likely to exercise the op hon, the early-exercise boundary is higher. (A) Making r higher makes exercising the option more appealing since you get cash from selling the stock which earns higher interest, so you'd accept less payoff on the ~le, meaning the early-exercise boundary is higher. ✓ (B) Making a higher makes the ophon more volahle and more likely to pay off even if it is more out-of-the-money, so you'd be less likely to exercise the option, making the early-exercise boundary lower.X (C) Making fi higher means the stock pay,; higher dividends so you have less reason to sell it and less inclination towards exercising the option, making the early-exen::ise boundary lower. ✓ (D) As the option gets closer to expiry, it is less likely to change in value much, making it more likely that it should be exercised, so the early-exercise boundary is higher. ✓ (B)

28. [Lesson 18] By put-call parity, 40e-r - se-

6

= 1.12-8.25

50e-r -se-

6

=6.47-4.15=

= -7.13 2.32

Subtracting the first from the second 10e-r = 7.13+2.32 =9.45

By put-call parity at 45, 45e-r - Se- 6 = P2 - 5.40 5e-, + (40e-' - Se- 6 ) = P2 - 5.40

9

·;5+(-7.13)=Pi-5.40 Pi=~-7.13+5.40=~

29.

(C)

[Section 23] We integrate f(x) to obtain F(x), the distribution function. For x < 5, Ix - 51 = 5-x.

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PRACTICE EXAM 2, SOLUTlON TO QUESTlON 30

641

This is for x < 5, but there is no need to go further, since the 1' 1 percenhle occurs in this range. e21'.x-.5)

~=O.Ql

x-5=0.5ln0.02

x=~

(D)

30. [Section 27.2] The first option is worthless because the final price (65) is greater than the strike price(60). The second option is worth something because the final price (65) is less than the strike price (70) and it didn't hit the barrier. It pay,; 70 -65 = IT]. The third option is worthless because it didn't hit the barrier. The fourth option hit the barrier of 70 but is worthless because the final price (65) is greater than the strikeprice(50). (A)

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PRACTICE EXAM 3, SOLUflONS TO QUESTIONS 1-3

Answer Key for Practice Exam 3 1 2 3 4

5 6 7 B 9 10

E A A A D

11 12

D

13 14 15

C C

21 22 23 24 25

C

16 17 1B

D

26 2B

19

E E E

29

A

20

D

30

C

D E B

D

B E

E B

D A A D

C C

27

Practice Exam 3 1. [Section 15.2] You need to know that the futures contract has notional value of 2.:(1times the index. The maintenance margin is 0.8(0.1)(250)(1400) = ~(E) 2 [Lesson 6] ~ is the covariance of ZYX with the market divided by the variance of the market. In calculahng these, we'd ordinary divide the sum of products or squares by 4, but the 4s will cancel, so we won't do the division. We'll u,e subscripts of i for ZYX and Mkt for market. Much of the following calculation can be done with the stahstical funchons of a calculator.

R;= 0.08+-~·+0.15 R,\W

=0. 078

= 0.12+-~·+0.20

=0.086

_L(R; - R.;)(R,\nt - R,\nt) = (0.Cll)(0.12)+ • • • + (0.15)(0.20)-5(0.078)(0.086) _L(R,\nr -

=0.052460

R,\u1 )2-5R~w = 0.122 + •· • +o.2a2 -5(0. 2.s0e1102

46.2253-S > 2.5505 S 50, profit on the first strategy is -3.70e 1102 . Profit on the ,econd strategy is 2.SOe002 - (S - .:()). -3.7Qe002

>

2.:()e 1102-$+5Q

-3.7747 > 52.5505- S S > 56.3252 (E)

You can arrive at the correct answer choice without any calculahons.

The first strategy pays more for

low prices and high prices by its very nature, and choice Eis the only choice with that structure. 9. [Lesson31] Option I pays Sr - m 8. (B)

pays 60 - 52

=

=55-45 =10. Option

II pays M -Sr=

60-55 =5. Opbon Ill

10. [Section 8.1] All three statements are true. (D) 11. [Section 17.4] All 3 of these strategies gain by increa,ed volatility. For a short butterfly spread, the gain is limited. (D) 12.

[Section 24.2] Yen are the domestic currency and dollars are the foreign currency, so r = 0.02 and

r1 =0.08. di

ln(l00/105) + 0-~\- 0.03 + 0.5(0.32)

-0.21263

di = -0.21263- 0.3 = -0.51263

N(di) = N(-0.21263) =0.41581 N(di) = N(-0.51253) = 0.30410

C(100, 105, 1) = 100e-lHWl(0.41581) - 105e-llll2(0.30410)= 7.09 Thecostof100optionsisY709,or~.

(B)

13. [Lesson 31] The earnings-enhanced death benefit is a call ophon. The stock price is 50,000 and the strike price is also :A.1,000. The benefit is 35% of the excess of stock price m·er strike price. The call may expirein5yea15or10years. d1(5) = (0.04 \~;55(~s2))(5)

0.76399

d2(5) = 0.76399 - 0.15../s = 0.66589

N(d1(5)) = 0.77756 C(5)

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N(d2(5)) = 0.66589

=:A.1,000(0.77756)-:A),()()()e-lll(0.66589)=11,619

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PRACTICE EXAM 3, 50LUTl0N5

d1(10) =

TO QUE5Tl0N5 14-17

(0.04+0.5(0.1~))(10) o.lsv'io

645

1.0&!444

d2(10) = 1.08044 - o.1sv'io = 0.60610 N(d1(10)) = 0.860(11

N(d2(10)) = 0.72778 ~,()()()e- 02(0.72778) = 18,609

C(10) = ~.000(0.860(11) -

The value of the earnings-enhanced death benefit is 0.05(0.35)(11,619 + 18,609) = ~ (C) 14. [Section 23.3] The probability that the option pays off is N(d2) and the parhal expectation of the stock price is 5oe"t N(j1)- Let's calculate jl and jl·

J1 =

2 ln(5o/ K): a+ 0.5,1

ln(40/45)+0.05+0.5(0.J2) 0.3 N(d1) = N(-0.07594) =0.46973

d2 = d1 -

,,

0.07594

= -0.01594 - o.3 = -0.37594

N(d2) = N(-0.37594) = 0.35348 Therefore the expected price of the stock given that it is greater than45 is Pr(51 > 45) =0.35348 PEl51 I 51 > 45] = Soe"N(d1) = 40e00·\0.46973) = 19.753 1 3 El51 I 51 > 45] = = 55.88 0 8

_~;!

The average payoff fmm the option is 55.8S - 45 = 10.88. (E) 15. [Section 23.2] When logging a lognormal variable, we subtract O.S.12 fmm the rate of return a.

For2 units of hme (11-9), isgreaterthanOis

µt = -1.5(2) = -3and

a= 3..fi.. The probability that the lognormal variable

0 3 1- ,\1( ~~ )) = 1- N(0.70711) =I0.23975]

(C)

[Subsection 14.3.1] Fo,1= 52e 0 1ll- 0.5e 0015-0.5 = 52.57608 Premium=

=

1 In (52.57608) 1[i - - =~ 52

(D)

17. [Section 6.1] The Sharpe raho of the stock is (0.12-0.04)/0.25 = 0.32. This has toexceed0.4 times the mrrelation, so the mrrelation must be at least 0.32/0.4 = @11. (E)

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PRACTICE EXAM 3, SOLUTlONS TO QUESTIONS 18-23

18. [Lesson 18] Convexity is not violated, so (A) and (B) are out. Checking put-call parity for the :()-strike options (since the other three choices involve them)

which is greater than to 4.95 - 7.65, so the put is underpriced and/or the call is m·erpriced. So to effect an arbitrage, sell the call and buy the put. (E) works. 19. [Section 17.3] A bull spread with 42-strike and 48--strike calls consists of buying the 42-strike call and selling the 48--strikecall. A bull spread with 42-strikeand 48-strike put consists of buying the 42--strike put and ,elling the 48-strike put. By put-call parity, (C(S,42,2)-

C(S,48,2))-

(P(S,42,2)-

P(S,48,2))

=(45-42e-i(oll,))-(45-48e-

2(0 1l\l)

=5.321523 Therefore, P(S,42,2)-

P(S,48,2)

=C(S,42,2)-

The pun::haser of the bull spread receives

=1.70-

C(S,48,2)-5.321523

5.321523 = -3.621523

rrm. (E)

20. [Lesson 10] By formula (10.3), the difference in WACCs is the debt-to-value ratio of 1.5/2.5 = 0.6 hmes ro = 0.04 times the tax rate. The tax rate is therefore 0.006/0.024 = @;ill.(D) 21. [Lesson 25] We deduce that d1 = N- 1(0.6591) = 0.41 and d2 = N- 1(0.3409) = -0.41. Since they are negatives of each other, it follows that

ln(S/ K) + r - 0 +0.5,1 2

ln(S/ K) + r - 0 - 0.5a 2

ln(S/K) + r -0 = -(ln(S/K) + r -0) ln(S/K)+r-0=0 so d1 = (0.5,12)/a = .5 47.5. TVaRo_95(X)= 0.6 ElX I 47.5 5: X 5: 49] + 0.4 ElX IX > 49] = 0.6(48.25) +0.4(59) =~

(C)

28. [Subsection 25.1.4] By put-call parity,

=0.06(38)e- 006112 -o.02(40)e--0 112112 = 2.269-0.799 = 1.47 In the following line, we divide this difference by 365 to express it per day. 0caa : 0put - ( 0put - 0call)

= -0.0169 -~

= t-0.0209 j

(C)

29. [Section 2.2] Let x be the number of units sold in millions. The NPV in millions is 4 (1-(1.02/1.15) -25+0.8(5)x_L--' ( 1.02l )' =-25+-----x=-25+18.9841x bO 1.}s,l+l 1.15 1-1.02/1.15 Setting this equal to 0, we need to sell I 1.317 million

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I units

8

)

annually. (A)

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PRACTICE EXAM 3, SOLUTlON TO QUESTlON 30

30.

[Subsection

649

18.ll By put-call parity, Ke-rt - Se- 61 = P(K) - C(K) Ke-ll1J?/4 -42e- 110214= 6.43 - 4.00 Ke- 00175= 2.43+42e--0 005=44.22

K=44.22e-0017s=~

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(C)

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650

PRACTICE EXAM 4, SOLUffONS TO QUESTIONS 1--4

Answer Key for Practice Exam 4 1 2 3 4 5 6 7 B 9 10

A

11

C C

D D

12 13 14 15 16 17 1B 19

B

20

A D D

C

B E

D D D D A E B E

21 22 23 24 25 26 27 2B

C

29

A

30

C

B

D

C C E B E

Practice Exam 4 1.

[Section 15.1]

2000e-llll2e-61

:200(Je-ll05

The price of the index is 2000e-rt = 2000e- 1102 . The prepaid forward price is =I1902.4&j.

(A)

2. [Section 19.4] (A) is true since the higher the price you can sell the stock for, the more valuable the put. ✓ (B) is true since the lower the price you can buy the stock for, the more valuable the call. ✓ (C) is false since puts with higher strike prices are worth more making the left side positive, while the right side is negative. X (D) is true since the left side is negative and the right side is posihve, but much stronger statements can be made. ✓ (E) correctly expresses convexity for put ophons. ✓ (C)

3. [Section 23.2] The lognormal distribution of the price of the stock after one year has parameters /J = a - 0 - 0.5a 2 = 0.13 - 0.03 - 0.5(0.32) = 0.055 and a = 0.3. The prediction interval for In 5(1), the logarithm of the stock price after one year, is (/J -1.645 60. If the stock goes up and down, it return;; to 50. Thus to hit the barrier, either. The stock goes up hvice immediately (irn); probability 1/4. The stock goes ud or du and then goes up twice; probability (1/2)(1/4) 1/8. The stock goes ud or du followed by another ud or du, and then goes up hvice; probability (1/2)(1/2)(1/4) 1/16. The stock goes down hviceand up 4 times, orddirnuu, probability 1/64.

=

=

The sum of these probabilities is ~- The value of the option is

22. [Lesson 11] J and Ill are true. Highly leveraged company invest in positive NPV projects only if they benefit equity holders, which mean;; the NPV must meet a certain threshold greater than 0. (B) 23. [Section 17.4]

24. [Lesson 4] The weak form of EMH says past stock prices do not provide any information about future stock prices, so Jeffrey cannot outperform the market. The semi-strong form of EMH leaves open the possibility that hard-to-get information may be used to predict future stock prices, so Sarah may outperfom the market even under the semi-strong form of EMH. (C) 25. [Section25.l]

y-r=O(a-r) 0.01-r= -0.6(0.090.064= 1.6r r=[IQ!]

r) (C)

=0.06 and 'I =0.04. =xe-'1 Ke-rt 0.0304 =0.8e--0o4 - 0.Se--006 C(0.8) =0.CD04+ 0.7686 -0.7534 =~ (E)

26. [Subsection 18.4] The domeshc currency is euros, so r 1

C(0.8)- P(0.B) C(0.8) -

27. [Lesson 6] Risk premium is0.13-0.03

-

=0.1.

5 ~ = -o~~~~) =-0.2 r

=0.03-0.2(0.1) =@:ill

(B)

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PRACTICE EXAM 4, SOLUTlONS TO QUESTIONS 28-30

28. [Lesson 20] The value of the option at the end of a year is 4011- 45 at the upper node. The value of the replicating portfolio at the upper node is 0.2625(4011)-8.715. Equahng the two, 4011-45 =0.2625(4011)-8.715 29.511=36.285 11

=[Iill

(E)

29. [Lesson 5] Let fl be the correlation between Iota and Rho. Let P be the portfolio, I be Iota, R be Rho. Cov(J, P)

=Cov(J,0.51

+0.SR)

=0.SVar(I) + 0.5Cov(I,

R)

2)+0.Sp(0.1)(0.3) 0.90&1(0.1)~=0.S(0.1 =0.005+0.015p Var(P) =0.5 2(0.12 +0.2 2 +2fJ(0.1)(0.2)) =0.0125 +0.0lp 0.09080-Jo.0125 +0.01/J

=0.005 + 0.01(!

18.16-Jo.0125+0.01/J =1 +2p

=

4.12232+3.29786p 1+4{! +4e 2 4/J2 +0.70214e-J.122J2=0 /J

=-0.8,@1]

(A)

=

30. [Section 26.2] Market-maker profit is zero when the stock prices mm·es up or down by S,1Vh ':IJ,1,/1!36s. We can back out,,. Since6 =0.6554 and 6 = e-MN(d1)and O = 0, N(d1) = 6 = 0.6554 di= N- 1(0.6554) = 0.4 In(:()/:())+ 0.Q35 +0.Sa 2 " 0.4

o.s.?- 0.4,, + a.ms= o " = 0.4 - ✓o.41 - 4(0.S)(0.03S) = 0.1 The other solution,,= 0.4 + 0.3 = 0.7 is rejected since it is greater than 0.5. So the stock's increa.ed price is:()+:()(0.1)/v.365=~(C)

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PRACTICE EXAM 5, SOLUTlONS TO QUESTIONS 1-3

657

Answer Key for Practice Exam 5 B E B E E

11 12 13 14 15

6 7 B

A A

16 17 1B

9

D A

1

2 3

4 5

B

10

D A D

C C

21 22 23 24 25

B

26

C

A D

27

B

2B

C

29

D

30

C

B

C B

B E

19 20

B

D

Practice Exam 5 1. [Lesson 16] The underlying as.et's increa.e in price is potentially unlimited, so a long forward contract has an unlimited potenhal payoff. Shorting an option can never earn more profit than the premium accumulated with interest. A put option cannot pay off more than the strike price. (B) 2. [Lesson 20] By put-call parity, the replicating portfolio for buying a put and selling a call is P(S,50,0.25)-C(S,50,0.25)

=50e--0is, -Se--0is.i

or lending :A)e--0i5 , and ,elling e- 11256 shares of stock. On the other hand, we are given that the replicahng portfolio for buying a put and ,el ling a call is lending 34.05- (-15.Cll) = 49.13 and selling 0.9925 shares of stock. The replicahng portfolio is unique, so the loan and stock components of the two replicating portfolios we've just described must be equal. Equating the loan components of the two replicahng portfolios, 5Qe-ll2.5r=49.13 r= -4ln

("·")= ~

(E)

=~

3. [Subsection 27.3] The payment equals €13 + max(0,£10 -€13). The first term has present value €13e--OIO,and the derivahve of this with respect to eums is 13e--O10. The second term is 13 put options on cums with pounds. Delta for each option is -e-'f N(-d1)- The current price and the strike price are the same. di

2 0.04 -0.1~_; O.S(0.2 )

N(-d1) = N(0.2)

_ 0 _2

=0.579266

Thus the number of eums you should buy is 13e--O10-13e- 0 10(0.57926) = ~ (B) Alternatively (and easier), the payment could be repre.ented as £10 + max(O, €13 - £10). Thus you put aside £10, who,e delta is O (the derivahve of the pound rate with respect to cums is 0), and buy 13 calls on euros. Delta for one call on eums is e- 01 N(d1). However, d1 = -0.2 as calculated above, so the numberofeuros to buy is 13e- 111N(-0.2) 13e- 111(0.42074) =~-

=

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658

PRACTICE EXAM 5, SOLUflONS TO QUESTIONS 4---fi

4. [Subsection 25.1.lJ .il = N(d1) for a nondividend N-1(0.52392) = 0.06. From the formula for d1, we have

lnS-ln:(J+(r-fi+O.S.1

paying stock.

N-1(0.64802) = 0.38 and

2)1 =0.38aVI

(0 )

=0.06aVI

(.. )

lnS-ln55+(r-0+0.5