<s> ANINTRODUCTIONTO OPTIONSTRADINGFrans de Weert
AN INTRODUCTION TOOPTIONS TRADING
The Securities & Investment InstituteMission Statement:To set standards of professional excellence andintegrity for the investment and securities industry,providing qualifications and promoting the highestlevel of competence to our members, other individualsand firms.The Securities and Investment Institute is the UK’s leadingprofessional and membership body for practitioners in thesecurities and investment industry, with more than 16,000members with an increasing number working outside theUK. It is also the major examining body for the industry,with a full range of qualifications aimed at people enteringand working in it. More than 30,000 examinations are takenannually in more than 30 countries.You can contact us through our website www.sii.org.ukOur membership believes that keeping up to date is central toprofessional development. We are delighted to endorse theWiley/SII publishing partnership and recommend this seriesof books to our members and all those who work in theindustry.As part of the SII CPD Scheme, reading relevant financialpublications earns members of the Securities & InvestmentInstitute the appropriate number of CPD hours under the SelfDirected learning category. For further information, pleasevisit www.sii.org.uk/cpdschemeRuth Martin
Managing Director
ANINTRODUCTIONTO OPTIONSTRADINGFrans de Weert
Copyright # 2006
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,West Sussex PO19 8SQ, EnglandTelephone (þ44) 1243 779777
Email (for orders and customer service enquiries): cs-books@wiley.co.ukVisit our Home Page on www.wiley.comAll Rights Reserved. No part of this publication may be reproduced, stored in a retrievalsystem or transmitted in any form or by any means, electronic, mechanical, photocopying,recording, scanning or otherwise, except under the terms of the Copyright, Designs andPatents Act 1988 or under the terms of a licence issued by the Copyright Licensing AgencyLtd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing ofthe Publisher. Requests to the Publisher should be addressed to the Permissions Department,John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ,England, or emailed to permreq@wiley.co.uk, or faxed to (þ44) 1243 770620Designations used by companies to distinguish their products are often claimed as trademarks.All brand names and product names used in this book are trade names, trademarks or registeredtrademarks of their respective owners. The Publisher is not associated with any product orvendor mentioned in this book.This publication is designed to provide accurate and authoritative information in regard tothe subject matter covered. It is sold on the understanding that the Publisher is not engagedin rendering professional services. If professional advice or other expert assistance isrequired, the services of a competent professional should be sought.Other Wiley Editorial OfﬁcesJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USAWiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, GermanyJohn Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, AustraliaJohn Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, Ontario, Canada L5R 4J3Wiley also publishes its books in a variety of electronic formats. Some content that appearsin print may not be available in electronic books.
British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN-13 978-0-470-02970-1 (PB)ISBN-10 0-470-02970-6 (PB)Project management by Originator, Gt Yarmouth, Norfolk (typeset in 12/16pt Trump Mediaeval).Printed and bound in Great Britain by T.J. International Ltd, Padstow, Cornwall.This book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.
To Jan and Annelies
CONTENTSPreface
xiii
Acknowledgements
xv
Introduction
xvii
1 OPTIONS
1
1.1 Examples
3
1.2 American versus European options
7
1.3 Terminology
8
1.4 Early exercise of American options
13
1.5 Payoffs
15
1.6 Put–call parity
16
2 THE BLACK–SCHOLES FORMULA2.1 Volatility and the Black–Scholes formula
2128
2.2 Interest rate and the Black–Scholes formula 292.3 Pricing American options
31
viii
CONTENTS
3 DIVIDENDS AND THEIR EFFECT ONOPTIONS
33
3.1 Forwards
34
3.2 Pricing of stock options includingdividends
35
3.3 Pricing options in terms of the forward
36
3.4 Dividend risk for options
38
3.5 Synthetics
39
4 IMPLIED VOLATILITY
41
4.1 Example
44
4.2 Strategy and implied volatility
45
5 DELTA
47
5.1 Delta-hedging
52
5.2 The most dividend-sensitive options
57
5.3 Exercise-ready American calls on dividendpaying stocks
57
6 THREE OTHER GREEKS
61
6.1 Gamma
62
6.2 Theta
65
6.3 Vega
69
CONTENTS
ix
7 THE PROFIT OF OPTION TRADERS
73
7.1 Dynamic hedging of a long call option
74
7.1.1
Hedging dynamically every $1
75
7.1.2
Hedging dynamically every $2
76
7.2 Dynamic hedging of a short call option
77
7.2.1
Hedging dynamically every $1
78
7.2.2
Hedging dynamically every $2
79
7.3 Profit formula for dynamic hedging
80
7.3.1
Long call option
81
7.3.2
Short call option
83
7.4 The relationship between dynamichedging and 
86
7.5 The relationship between dynamichedging and  when the interest rate isstrictly positive
88
7.6 Conclusion
91
8 OPTION GREEKS IN PRACTICE
93
8.1 Interaction between gamma and vega
94
8.2 The importance of the direction of theunderlying share to the option Greeks
97
8.3 Pin risk for short-dated options
98
8.4 The riskiest options to go short
99
x
CONTENTS
9 SKEW
101
9.1 What is skew?
102
9.2 Reasons for skew
103
9.3 Reasons for higher volatilities in fallingmarkets
104
10 SEVERAL OPTION STRATEGIES
105
10.1 Call spread
106
10.2 Put spread
107
10.3 Collar
109
10.4 Straddle
111
10.5 Strangle
112
11 DIFFERENT OPTION STRATEGIES ANDWHY INVESTORS EXECUTE THEM
117
11.1 The portfolio manager’s approach tooptions
118
11.2 Options and corporates withcross-holdings
119
11.3 Options in the event of a takeover
120
11.4 Risk reversals for insurance companies
122
11.5 Pre-paid forwards
123
11.6 Employee incentive schemes
126
11.7 Share buy-backs
126
CONTENTS
xi
12 TWO EXOTIC OPTIONS
129
12.1 The quanto option
130
12.2 The composite option
135
13 REPO
137
13.1 A repo example
138
13.2 Repo in case of a takeover
139
13.3 Repo and its effect on options
140
13.4 Takeover in cash and its effect on theforward
141
AppendicesA PROBABILITY THAT AN OPTION EXPIRESIN THE MONEY
143
B VARIANCE OF A COMPOSITE OPTION
145
Bibliography
149
Index
151
PREFACEThis book is appropriate for people who want to get a goodoverview of options in practice. It especially deals withhedging of options and how option traders earn money bydoing so. To point out where the profit of option traderscomes from, common terms in option theory will be explained, and it is shown how they relate to this profit. Theuse of mathematics is restricted to a minimum. However,since mathematics makes it possible to lift analyses to anon-superficial level, mathematics is used to clarify andgeneralize certain phenomena.The aim of this book is to give both option practitionersas well as interested individuals the necessary tools todeal with options in practice. Throughout this book reallife examples will illustrate why investors use optionstructures to satisfy their needs. Although understandingthe contents of this book is a prerequisite for becoming agood option practitioner, a book can never produce a goodtrader. Ninety percent of a trader’s job is about dealingwith severe losses and still being able to make the rightdecisions if such a loss occurs. The only way to become agood trader is to accept that, when helping clients toexecute their option strategies, the trader will inevitably
xiv
PREFACE
end up with positions where the risk reward is againsthim but the odds are in his favour. For that reason a oneoff loss will almost always be larger than a one-off gain.But, if a trader executes many deals he should be able tomake money on the small margin he collects on everydeal even if he gets a few blow-ups.
ACKNOWLEDGEMENTSThis book is based on knowledge acquired during mywork as a trader at Barclays Capital. Therefore, I wouldlike to thank my colleagues at Barclays Capital who havebeen very helpful in teaching me the theory and practiceof options. At Barclays Capital I would especially like tothank Faisal Khan and Thierry Lucas for giving me all theopportunities to succeed in mastering and practisingoption trading. I would also like to thank Thierry Lucasfor his many suggestions and corrections when reviewingmy work and Alex Boer for his mathematical insights.I would especially like to thank Karma Dajani for firstgiving me a great foundation in probability theory, thensupervising my graduate dissertation in mathematicsand, lastly, reviewing this book. Special thanks goesout to my father, Jan de Weert, for having been a motivating and helpful force in mastering mathematics throughout my life, and for checking the book very thoroughlyand giving me many suggestions in rephrasing sentencesand formulae more clearly.
INTRODUCTIONOver the years derivative securities have become increasingly important. Examples of these are options, futures,forwards and swaps. Although every derivative has itsown purpose, they all have in common that their valuesdepend on more basic variables like stocks and interestrates. This book is only concerned with options, but oncethe theory behind options is known, knowledge can easilybe expanded to other derivatives.This book has three objectives. The first is to introduceterms commonly used in option theory and explain theirpractical interpretation. The second is to show whereoption traders get their profit and how these commonlyused terms relate to this profit. The last objective is toshow why companies and investors use options to satisfytheir financial needs.
Chapter
1OPTIONS
2
AN INTRODUCTION TO OPTIONS TRADING
O
ptions on stocks were first traded on an organizedexchange in 1973. That very same year Black andScholes introduced their famous Black–Scholes formula.This formula gives the price of an option in terms of itsparameters, like the underlying asset, time to maturityand interest rate. The formula and the variables it dependson will be discussed in more detail in the next chapters.Since 1973 option markets have grown rapidly, not onlyin volumes but also in the range of option products to betraded. Nowadays, options can be traded on many different exchanges throughout the world and on many different underlying assets. These underlying assets includestocks, stock indices, currencies and commodities.There are two general kinds of options, the call optionand the put option. A call option gives the holder theright, but not the obligation, to buy the underlying assetfor a pre-specified price and at a pre-specified date. A putoption gives the holder the right, but not the obligation,to sell the underlying asset for a pre-specified price and ata pre-specified date. This pre-specified price is called the‘strike price’; the date is known as the ‘expiration date’, or‘maturity’. The underlying asset in the definition of anoption can be virtually anything, like potatoes, theweather, or stocks. Throughout this book the underlyingasset will be taken to be a stock.When the owner of a call option chooses to buy the stock,it is said that he exercises his option right. Of course, thesame holds for the owner of a put option, only in this casethe owner chooses to sell the stock, but it is still referredto as ‘exercising’ the option. If the option can only be
OPTIONS
3
exercised on the expiration date itself, the option is saidto be a ‘European’ option. If it can be exercised at any timeup to expiration, the option is an ‘American’ option.Although there are many other option types, the mostimportant ones have already been covered: American calloption, American put option, European call option andthe European put option. The vast majority of the optionsthat are traded on exchanges are American. However, it isuseful to analyze European options, because properties ofAmerican options can very often be deduced from itsEuropean counterpart.Before an example is given, it is worthwhile to look atthe boldface sentence in the definition of an option. Theholder of an option is not obligated to exercise the option.This means that at maturity the holder can decide not todo anything. This is exactly why it is called an ‘option’,the holder has a choice of doing something. Because ofthis choice, the largest loss an owner of an option can faceis the price paid for the option.
1.1
EXAMPLES
Consider a holder of a European call option on the stockRoyal Dutch/Shell with a strike price of $42. Supposethat the current stock price is $40, the expiration dateis in 1 year and the option price is $5. Since the option isEuropean, it can only be exercised on the expiration date.What are the possible payoffs for this option? If on theexpiration date the stock price is less than $42, the holderof this option will clearly not exercise his option right.
4
AN INTRODUCTION TO OPTIONS TRADING
For if he did, he would buy the stock for $42 (exercisingthe option), and would only be able to sell the stock onthe market for less than $42. So, the holder would incur aloss if he exercised his right, whereas nothing wouldhappen if he did not exercise this right. In conclusion,if on the expiration date the stock price is less than $42,the holder does not exercise the option. In these circumstances the holder’s loss is the price paid for the option, inthis case $5. If on the expiration date the stock price isbetween $42 and $47, the holder will exercise his optionright. Suppose that the stock price on the expiration dateis $45, then, by exercising his option right, he buys thestock for $42 and immediately sells this stock on themarket for $45, making a profit of $3. However, takinginto account that he paid $5 for the option, he still makesa loss of $2. It is clear that up to $47 the holder losesmoney on the option. If on the expiration date the stockprice is more than $47, the holder will again exercise hisoption right. The difference between this case and theprevious one is that the holder not only makes a profitby exercising his option right, he also makes an overallprofit. Suppose that the stock price on the expiration dateis $49, then his profit is $2, $7 from exercising the optionand $5 from the price paid for the option. Figure 1.1shows the way in which the profit of the holder of a calloption on Royal Dutch/Shell varies with the stock priceat maturity.This example points out that the profit of the holder of acall option increases as the stock price increases. Thus,the holder of a call option is speculating on an increasingstock price.
OPTIONS
5
50
40
Profit ($)
30
20
10
0
10|
20|
30|
40
50
|
|
|
70|
80|
90|
Stock price at maturity ($)
−5−10
60
(42,−5)
47
Figure 1.1 Profit from buying a European call option on RoyalDutch/Shell. Option price ¼ $5, Strike price ¼ $42.
In contrast to a call option, the owner of a put optionis hoping that the stock price will decrease. Consider aholder of a European put option on Unilever with a strikeprice of $50. Suppose that the current stock price is $50,the expiration date is in 6 months and the option price is$3. Again, because the put option is European it can onlybe exercised on the expiration date. What are the possiblepayoffs for the put option? Suppose that the stock price atmaturity (expiration date) is more than $50. In this casethe holder would not exercise his option right. Because hewould have to buy the stock for more than $50, and,under the conditions of the put options, sell the samestock for $50. This would mean he would always incur aloss. So, if the stock price is more than $50, the holderdoes not exercise the option and faces a loss of $3, the
6
AN INTRODUCTION TO OPTIONS TRADING50
40
Profit ($)
30
20
10
0−3−10
Stock price at maturity ($)10|
20|
30|
40
50
|
|
60|
70|
80|
90|
(50,−3)
47
Figure 1.2 Profit from buying a European put option on Unilever.Option price ¼ $3, Strike price ¼ $50.
price paid for the put option. If the stock price at maturityis between $50 and $47, the holder will exercise theoption, but still make an overall loss. He will earn moneyby exercising the option, but it will not be enough tomake up for the initial $3 paid for the option. If thestock price is less than $47 at maturity, the holder willexercise the option, and make an overall profit. He willmake a profit of more than $3 by buying the stock on themarket and immediately selling this stock for $50 underthe conditions of the put option. Since the price paid forthe put option was $3 he will also make an overall profit.Figure 1.2 shows the way in which the profit of the holderof the put option on Unilever varies with the stock priceat maturity.
OPTIONS
1.2
7
AMERICAN VERSUSEUROPEAN OPTIONS
If in above examples the options were American ratherthan European, the holders of the options would not haveto wait until the expiration date before exercising theoptions. In the example of the put option on Unileverwith strike price $50 this would mean that at any timeduring the 6 months, the holder is allowed to exercise theoption. So, suppose that after 3 months the stock price is$40, the holder could decide to exercise his option right.By exercising he would make a profit of $7, $10 fromexercising the option and $3 from the price paid forthe option.
Everything that can be done with a European option canbe done with an American option, but the Americanoption has the additional property that it can be exercisedat any time up to the expiration date. This means thatits price is always at least as much as its Europeancounterpart.
Although most options that are traded on exchanges areAmerican, this book mainly focuses on European options.Since American options are based on the same principlesas European ones, and their properties can, very often, beeasily derived from the European counterparts this isperfectly arguable. Throughout this book, an option isassumed to be European, unless specifically statedotherwise.
8
1.3
AN INTRODUCTION TO OPTIONS TRADING
TERMINOLOGY
An option contract is an agreement between two parties.The buyer of the option is said to ‘have taken a longposition’ in the option. The other party is said to havetaken a ‘short position’ in the option, or is said to have‘written’ the option. So, taking a long position essentiallymeans ‘buying’ and taking a short position means‘selling’. For that reason there are four general optionpositions:1. A short position in a call option.2. A short position in a put option.3. A long position in a call option.4. A long position in a put option.Although it is tempting to say that positions 1 and 4 arebasically the same whenever the options have the samestrike price, and also positions 2 and 3, there is one bigdifference. The short position always bears a greater risk,because losses can be unlimited and profits are bounded.For that reason the party holding a short position getsmoney from the party holding a long position, which is ofcourse equal to the price of the option. For example, theloss of an investor with a short position in a call optioncan be unlimited, whereas the loss of an investor with along position in a call option is bounded by the price ofthe option. The profit of an investor with a short positionin a call option is bounded by the price of the option, andthe profit of an investor with a long position in a call option is unlimited. In fact, it is not totally true that lossescan be unlimited for investors holding a short position. In
OPTIONS
9
10
(42,5)
50
Stock price at maturity ($)
|
|
|
|
|
|
|
|
|
10
20
30
40
50
60
70
80
90
Profit ($)
−10−20−30−40−50
47
Figure 1.3 Profit from writing a European call option on RoyalDutch/Shell. Option price ¼ $5, Strike price ¼ $42:
the case of a short position in a put option, the losses arebounded by the strike price, since the stock price cannotdrop below 0. Respectively, Figures 1.3 and 1.4 show thevariation of the profit and loss with the stock price atmaturity of an investor holding a short position in aEuropean call option on Royal Dutch/Shell, and an investor holding a short position in a European put optionon Unilever. Respectively, when Figures 1.3 and 1.4 arecompared with Figures 1.1 and 1.2, it also becomes clearthat a short position always bears a greater risk than along position.An option has three states, in the money, at the moneyand out of the money. Every state refers to what wouldhappen were the option exercised immediately. For
10
AN INTRODUCTION TO OPTIONS TRADING
(50,3)
30
|
|
|
|
|
10
20
30
40
50
|
|
|
|
60708090Stock price at maturity ($)
Profit ($)
−10−20−30−40−50
47
Figure 1.4 Profit from writing a European put option on Unilever.Option price ¼ $3, Strike price ¼ $50.
example, consider the call option on Royal Dutch/Shellwith a strike price of $42. If the stock price of RoyalDutch/Shell is more than $42, the call option is in themoney, because a profit would be made by exercising theoption immediately. The call option is at the money if thestock price is $42, since neither a profit nor a loss wouldbe made by exercising the option immediately. The calloption is out of the money if the stock price were lessthan $42, since by exercising the option immediately theholder would incur a loss. For the put option the reverseholds. A put option is in the money if the strike price ismore than the stock price, at the money if the strike priceis equal to the stock price and out of the money if the
OPTIONS
11
Table 1.1
States of an option.
Ratio
State call option
State put option
Stx>1¼1þ100K
x% in the money
x% out of the money
St¼1K
At the money
At the money
Stx<1¼1100K
x% out of the money
x% in the money
strike price is less than the stock price. Table 1.1 summarizes the three states of an option.1The intrinsic value of an option is the payoff were theoption exercised immediately. Consider a call option,with strike price $40, on a stock with a price of $50.For this particular option the intrinsic value is $10.This definition implies that in-the-money options havepositive intrinsic values, whereas at-the-money and outof-the-money options have intrinsic values equal to 0.Furthermore, it is good to discuss the economic principleof discounting here. The easiest way to illustrate this isby means of an example. Consider an option trader whoowns a European option with an expiration date in 1 yearand an expected payoff at maturity of $4. One could betempted to say that the price of this option should be $4 aswell, since in this case both sides of the option contractwill have expected profit of 0. However, this is not quite1
St stands for the stock price at time t and K stands for the strike of theoption.
12
AN INTRODUCTION TO OPTIONS TRADING
true, because the seller of the option gets $4 a year earlierthan the buyer of the option. So, in fact the fair price ofthe option should be that amount of money such thatwhen it is put in a savings account for 1 year it has grownto $4. This means that if the interest rate is 6% per yearthe fair price is 4=1:06 ¼ $3:77. The foregoing is referredto as ‘discounting’, and in this case $4 is discounted at6%. In the above calculation it is assumed that interest ispaid out only once a year. In practice, it is actually paidout continuously, which implies that money on a savingsaccount grows faster, because interest is paid out on theinitial amount plus the interest earned up to that time. Inthis case the discounting formula is 4  e0:061 . Ingeneral, if an amount x has to be discounted at an interestrate of r (expressed as a percentage) per year over a periodof T  t years, the discounting formula is x  erðTtÞ .Lastly, it is worth mentioning that it is possible to short(sell) a stock without actually owning it. It basicallycomes down to borrowing a stock from a third partyafter which the stock is sold immediately. However,this stock has to be returned sometime. For example,an investor shorts the stock Unilever for $50. Becausethe investor sells the stock Unilever he gets the $50. Ifafter 1 year the stock price of Unilever is $40 he candecide to buy the stock on the market for $40 and returnit to the party he borrowed it from. With this strategy theinvestor makes a profit of $10 if the interest rate is 0%,and if the interest rate is r (expressed as a percentage) itgives a profit of 50  ð1 þ rÞ  40, or if interest is paidcontinuously 50  er1  40.
OPTIONS
1.4
13
EARLY EXERCISE OFAMERICAN OPTIONS
For a call option on a non-dividend paying stock, itappears that the prices for both European and Americantypes are exactly the same. In other words, it is neveroptimal to exercise a call option on a non-dividend payingstock early. Because the price of a stock is an expectationof the future earnings there are always investors whowant to hold the stock longer than the term of the option.Even if such investors did not care about the option part,they would still be prepared to pay more for an Americancall option than its intrinsic value, since there is aninterest rate advantage. This is because the price of acall option is always less than the price of its underlyingstock, otherwise money could be made by buying thestock and shorting the call option. So, for an investorplanning to hold the underlying non-dividend payingstock longer than the term of the option, it would becheaper to buy the call option for its intrinsic valuethan to actually buy the stock. For that reason this particular investor, which must exist, would be prepared topay more than just the intrinsic value. Knowing this, it iseasy to see why it is never optimal to exercise an American call option on a non-dividend paying stock early.Consider an American call option on Royal Dutch/Shell with a strike price of $40 and an expiration datein 1 year. Suppose that after half a year the stock price is$50. The holder of this call option could be tempted toexercise his option, getting a payoff of $10.
14
AN INTRODUCTION TO OPTIONS TRADING
However, it would be better not to exercise the calloption, but to sell the call option instead. Since the priceof the option is more than its intrinsic value this wouldlead to a higher payoff than $10. An alternative strategy2is to keep the option and short the underlying stock. Thisguarantees a payoff of at least $10. On top of that, interestis earned over $50, since this was received by shorting thestock, which is more than the interest earned on the $10were the American option exercised immediately. Thismeans that the discounted expected payoff is more than$10.
However, for an American put option it can be profitableto exercise the option early. This can be shown by lookingat an extreme but simple example.3 Consider an investorwith a put option on Royal Dutch/Shell with a strikeprice of $40 and an expiration date in 1 year. Now supposethat after half a year Royal Dutch/Shell is virtually worthnothing (very extreme situation). By exercising the putoption, the investor makes an immediate gain of $40. Ifthe investor waits, the gain could be less than $40 but itcould never be more than $40, since stock prices cannotdrop below 0. This already shows that there can be a pointin exercising an American put option on a non-dividendpaying stock early. Furthermore, receiving $40 now ispreferable to receiving $40 in the future (interest rateadvantage). It follows that in this case the American putoption should be exercised immediately.23
Argument on p. 158 of Hull (1993).The following argument is based on pp. 160=161 of Hull (1993).
OPTIONS
1.5
15
PAYOFFS
The examples show that the payoff of an option can neverbe negative, since the holder can decide not to exercisehis option right. This does not mean that the profit of theowner of an option cannot be negative. This is of coursebecause an initial price has been paid for the option. So, adistinction has to be made between the profit of an optionand the payoff. The payoff only consists of the cash flowat the time the option is exercised, whereas the profit