<s> Exotic Options Trading
Frans de Weert
Exotic Options Trading
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Exotic Options Trading
Frans de Weert
C 2008Copyright 
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British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 978-0-470-51790-1 (HB)Typeset in 11/13pt Times by Aptara Inc., New Delhi, IndiaPrinted and bound in Great Britain by TJ International Ltd, Padstow, Cornwall, UKThis 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.
ContentsPreface
xi
Acknowledgements
xiii
1 Introduction
1
2 Conventional Options, Forwards and Greeks2.1 Call and Put Options and Forwards2.2 Pricing Calls and Puts2.3 Implied Volatility2.4 Determining the Strike of the Forward2.5 Pricing of Stock Options Including Dividends2.6 Pricing Options in Terms of the Forward2.7 Put-Call Parity2.8 Delta2.9 Dynamic Hedging2.10 Gamma2.11 Vega2.12 Theta2.13 Higher Order Derivatives Like Vanna and Vomma2.14 Options’ Interest Rate Exposure in Terms ofFinancing the Delta Hedge
3368891011121414161819
3 Profit on Gamma and Relation to Theta
23
4 Delta Cash and Gamma Cash4.1 Example: Delta and Gamma Cash
2526
21
vi
Contents
5 Skew5.1 Reasons for Higher Realised Volatility inFalling Markets5.2 Skew Through Time: ‘The Term Structure of Skew’5.3 Skew and its Effect on Delta5.4 Skew in FX versus Skew in Equity: ‘Smile versusDownward Sloping’5.5 Pricing Options Using the Skew Curve
27
3234
6 Simple Option Strategies6.1 Call Spread6.2 Put Spread6.3 Collar6.4 Straddle6.5 Strangle
353537394042
7 Monte Carlo Processes7.1 Monte Carlo Process Principle7.2 Binomial Tree versus Monte Carlo Process7.3 Binomial Tree Example7.4 The Workings of the Monte Carlo Process
4545464648
8 Chooser Option8.1 Pricing Example: Simple Chooser Option8.2 Rationale Behind Chooser Option Strategies
494951
9 Digital Options9.1 Choosing the Strikes9.2 The Call Spread as Proxy for the Digital9.3 Width of the Call Spread versus Gearing
53555555
10 Barrier Options10.1 Down-and-In Put Option10.2 Delta Change over the Barrier for a Down-and-InPut Option10.3 Factors Influencing the Magnitude of the BarrierShift10.4 Delta Impact of a Barrier Shift10.5 Situations to Buy Shares in Case of a BarrierBreach of a Long Down-and-In Put
5758
272829
58606363
Contents
10.610.710.810.910.1010.1110.1210.13
vii
Up-and-Out CallUp-and-Out Call Option with RebateVega Exposure Up-and-Out Call OptionUp-and-Out PutBarrier ParityBarrier at Maturity OnlySkew and Barrier OptionsDouble Barriers
6364646565656668
11 Forward Starting Options11.1 Forward Starting and Regular Options Compared11.2 Hedging the Skew Delta of the ForwardStart Option11.3 The Forward Start Option and the Skew TermStructure11.4 Analytically Short Skew but Dynamically NoSkew Exposure11.5 Forward Starting Greeks
7171
12 Ladder Options12.1 Example: Ladder Option12.2 Pricing the Ladder Option
777778
13 Lookback Options13.1 Pricing and Gamma Profile of Fixed StrikeLookback Options13.2 Pricing and Risk of a Floating Strike LookbackOption
79
14 Cliquets14.1 The Ratchet Option14.2 Risks of a Ratchet Option
838385
15 Reverse Convertibles15.1 Example: Knock-in Reverse Convertible15.2 Pricing the Knock-in Reverse Convertible15.3 Market Conditions for Most Attractive Coupon15.4 Hedging the Reverse Convertible
8787898990
16 Autocallables16.1 Example: Autocallable Reverse Convertible
9393
72737475
7980
viii
Contents
16.216.316.4
Pricing the AutocallableAutocallable Pricing without Conditional CouponInterest/Equity Correlation withinthe Autocallable
959798
17 Callable and Puttable Reverse Convertibles17.1 Pricing the Callable Reverse Convertible17.2 Pricing the Puttable Reverse Convertible
9999102
18 Asian Options18.1 Pricing the Geometric Asian Out Option18.2 Pricing the Arithmetic Asian Out Option18.3 Delta Hedging the Arithmetic Asian Out Option18.4 Vega, Gamma and Theta of the Arithmetic AsianOut Option18.5 Delta Hedging the Asian in Option18.6 Asian in Forward18.7 Pricing the Asian in Forward18.8 Asian in Forward with Optional Early Termination
105105107109110110112114116
19 Quanto Options19.1 Pricing and Correlation Risk of the Option19.2 Hedging FX Exposure on the Quanto Option
119119122
20 Composite Options20.1 An Example of the Composite Option20.2 Hedging FX Exposure on the Composite Option
125125126
21 Outperformance Options21.1 Example of an Outperformance Option21.2 Outperformance Option Described as a CompositeOption21.3 Correlation Position of the OutperformanceOption21.4 Hedging of Outperformance Options
129129
131132
22 Best of and Worst of Options22.1 Correlation Risk for the Best of Option22.2 Correlation Risk for the Worst of Option22.3 Hybrids
135135137138
130
Contents
ix
23 Variance Swaps23.1 Variance Swap Payoff Example23.2 Replicating the Variance Swap with Options23.3 Greeks of the Variance Swap23.4 Mystery of Gamma Without Delta23.5 Realised Variance Volatility versus StandardDeviation23.6 Event Risk of a Variance Swap versus a SingleOption23.7 Relation Between Vega Exposure and VarianceNotional23.8 Skew Delta23.9 Vega Convexity
139140140142144145146147147148
24 Dispersion24.1 Pricing Basket Options24.2 Basket Volatility Derived From its Constituents24.3 Trading Dispersion24.4 Quoting Dispersion in Terms of Correlation24.5 Dispersion Means Trading a Combination ofVolatility and Correlation24.6 Ratio’d Vega Dispersion24.7 Skew Delta Position Embedded in Dispersion
151151152153153
25 Engineering Financial Structures25.1 Capital Guaranteed Products25.2 Attractive Market Conditions for CapitalGuaranteed Products25.3 Exposure Products for the Cautious EquityInvestor25.4 Leveraged Products for the Risk Seeking Investor
157157
153155156
158160163
Appendix A Variance of a Composite Optionand Outperformance Option
167
Appendix B
169
Replicating the Variance Swap
Bibliography
175
Index
177
PrefaceThis book is appropriate for people who want to get a good overviewof exotic options in practice and are interested in the actual pricing ofthem. When dealing with exotic options it is very important to quantifythe risks associated with them and at which stock or interest rate levelsthe Greeks change sign. Namely, it is usually the case that the Greeksof exotic options show much more erratic behaviour than the Greeks ofregular options. This instability of the Greeks forces the trader to choosedifferent hedging strategies than the standard option model would prescribe. Therefore the risk management of exotic options entails muchmore than just obeying the model, which in turn has an impact on theprice. The non-standard risk management of exotic options means thatwhen pricing an exotic option, one first needs to understand where therisks lie that affect the hedging strategy and hence the pricing of the particular exotic option. Once the risks have been mapped and the hedgingstrategy has been determined, the actual pricing is often nothing morethan a Monte Carlo process. Moreover, when knowing the risks, theactual pricing of an exotic option can in some cases even be replicatedby a set of standard options. In other words, the starting point for pricingexotic options is to have a full awareness of the risks, which in turn hasan impact on how one needs to accurately price an exotic option.The aim of this book is to give both option practitioners and economicsstudents and interested individuals the necessary tools to understandexotic options and a manual that equips the reader to price and riskmanage the most common and complicated exotic options. To achievethis it is imperative to understand the interaction between the differentGreeks and how this, in combination with any hedging scheme, translatesinto a real tangible profit on an exotic option. For that reason, this book
xii
Preface
is written such that for every exotic option the practical implications areexplained and how these affect the price. Knowing this, the necessarymathematical derivations and tools are explained to give the reader afull understanding of every aspect of each exotic option. This balance isincredibly powerful and takes away a lot of the mystique surroundingexotic options, turning it into useable tools for dealing with exotic optionsin practice.This book discusses each exotic option from four different angles.First, it makes clear why there is investor demand for a specific exoticoption. Secondly, it explains where the risks lie for each exotic option andhow this affects the actual pricing of the exotic option. Thirdly, it showshow to best hedge any vega or gamma exposure embedded in the exoticoption. Lastly, for each exotic option the skew exposure is discussedseparately. This is because any skew exposure is typically harder toquantify, but it has a tremendous impact on almost every exotic option.For that reason, this book devotes a separate section to skew, Chapter 5,which explains skew and the reasons for it in depth.
AcknowledgementsThis book is based on knowledge acquired during my work as a trader atBarclays Capital. Therefore I would like to thank my colleagues at Barclays Capital who have been very helpful in teaching me the theory andpractice of options. I would like to thank Faisal Khan and Thierry Lucasfor giving me all the opportunities to succeed in mastering and practicing option trading; Stuart Barton, Karan Sood and Arturo Bignardiwho have all made an indispensable contribution to my knowledge ofoptions; Arturo Bignardi for his many suggestions and corrections whenreviewing my work and Alex Boer for his mathematical insights; andspecial thanks go out to Neil Schofield for his many suggestions and forreviewing the work very thoroughly. I would also like to thank my greatfriends for all their support and motivation. Special thanks go out to myparents and my two sisters for having been a motivating force throughout my life and for a fantastic upbringing. I also want to thank my fatherfor reviewing this book and being able to get an understanding of thiscomplicated subject and subsequently being able to correct mistakes.Petra, the light of my life. Finally, we can shine together!
1IntroductionExotic options are options for which payoffs at maturity cannot be replicated by a set of standard options. This is obviously a very broad definition and does not do justice to the full spectrum and complexity ofexotic options. Typically, exotic options have a correlation component.Which means that their price depends on the correlation between two ormore assets. To understand an exotic option one needs to know aboveall where the risks of this particular exotic option lie. In other words,for which spot price are the gamma and vega largest and at which pointduring the term of this option does it have the largest Greeks. Secondly,one needs to understand the dynamics of the risks. This means that oneneeds to know how the risks evolve over time and how these risks behavefor a changing stock or basket price. The reason that one needs to understand the risks of an exotic option before actually pricing it is becausethe risks determine how an exotic option should be priced. Once it isknown where the risks lie and the method for pricing it is determined,one finds that the actual pricing is typically nothing more than a MonteCarlo method. In other words, the price of an exotic option is generallybased on simulating a large set of paths and subsequently dividing thesum of the payoffs by the total number of paths generated. The methodfor pricing an exotic option is very important as most exotic options canbe priced by using a set of different exotic options and therefore savinga considerable amount of time. Also, sometimes one needs to concludethat the best way to price a specific exotic option is by estimating theprice with a series of standard options, as this method better capturesthe risk involved with this exotic option. The digital option is a goodexample of that and will be discussed in Chapter 9.Before any exotic option is discussed it is important to fully understandthe interaction between gamma and theta. Although this book assumesan understanding of all the Greeks and how they interact, the followingtwo sections give a brief summary of the Greeks and how the profit ofan option depends on one of the Greeks, namely the gamma. A moredetailed discussion of the Greeks and the profit related to them can befound in An Introduction to Options Trading, F. de Weert.
2Conventional Options, Forwardsand Greeks1This section is meant to give a quick run through of all the importantaspects of options and to provide a sufficient theoretical grounding inregular options. This grounding enables the reader to enter into the morecomplex world of exotic options. Readers who already have a good working knowledge of conventional options, Greeks and forwards can skipthis chapter. Nonetheless, even for more experienced option practitioners, this section can serve as a useful look-up guide for formulae of thedifferent Greeks and more basic option characteristics.
2.1 CALL AND PUT OPTIONS AND FORWARDSCall and put options on stocks have been traded on organised exchangessince 1973. However, options have been traded in one form or anotherfor many more years. The most common types of options are the calloption and the put option. A call option on a stock gives the buyer theright, but not the obligation, to buy a stock at a pre-specified price andat or before a pre-specified date. A put option gives the buyer the right,but not the obligation, to sell the stock at a pre-specified price andat or before a pre-specified date. The pre-specified price at which theoption holder can buy in the case of a call and sell in the case of a putis called the strike price. The buyer is said to exercise his option whenhe uses his right to buy the underlying share in case of a call option andwhen he sells the underlying share in case of a put option. The date ator up to which the buyer is allowed to exercise his option is called thematurity date or expiration date. There are two different terms regardingthe timing of the right to exercise an option. They are identified by anaming convention difference. The first type is the European optionwhere the option can only be exercised at maturity. The second type of
1Parts of this chapter have been previously published in de Weert, F. (2006) An Introduction to OptionsTrading, John Wiley & Sons Ltd, Chichester. Reproduced with permission.
Exotic Options Trading
Payoff at maturity
4
K0
|
Share price at maturity
Figure 2.1 Payoff profile at maturity for a call option with strike price K
option is the American option where the option can be exercised at anytime up to and including the expiry date.Obviously, the buyer of a European call option would only exercisehis right to buy the underlying stock if the share price was higher thanthe strike price. In this case, the buyer can buy the share for the predetermined strike price by exercising the call and subsequently sell it inthe market at the prevailing price in the market, which is higher than thestrike price and therefore making a profit. The payoff profile of the calloption is shown in Figure 2.1. The buyer of a European put option actsopposite to the buyer of the call option in the sense that the buyer of a putoption would only exercise his option right, at maturity, if the share pricewas below the strike price. In this case the option buyer can first buythe share in the market at the prevailing market price and subsequentlysell it at the strike price by exercising his put option, earning a profitas a result. The payoff profile at maturity of a put option is shown inFigure 2.2.A forward is different to an option in the sense that the buyer of theforward is obliged to buy the stock at a pre-specified price and at apre-specified date in the future. The pre-specified price of a forward ischosen in such a way that the price of the forward is zero at inception ofthe contract. Therefore, the expected fair value of the stock at a certain
5
Payoff at maturity
Conventional Options, Forwards and Greeks
K0
|
Share price at maturity
Figure 2.2 Payoff profile at maturity for a put option with strike price K
Payoff at maturity
maturity date is often referred to as the forward value of a stock or simplythe forward associated with the specific maturity. The payoff profileat maturity of a forward contract is shown in Figure 2.3. Figure 2.3makes clear that there is a downside in owning a forward. Whereasthe owner of an option always has a payout at maturity which is larger
K0
|
–KShare price at maturity
Figure 2.3 Payoff profile at maturity for a forward with strike price K which is equalto the fair forward value
6
Exotic Options Trading
than zero and therefore the maximum loss is equal to the premium paidfor the option, the maximum loss on one forward is equal to the strikeprice of the forward, which occurs if the share price goes to zero. Sincethe definition of a forward prescribes that the contract is worth zero atinception, the strike price of the forward is equal to the forward value,which is discussed more elaborately in sub-section 2.4.
2.2 PRICING CALLS AND PUTSIn 1973 Black and Scholes introduced their famous Black–Scholes formula. The Black–Scholes formula makes it possible to price a call or aput option in terms of the following inputs:
r The underlying share price, St ;r The strike price, K ;r The time to maturity, T − t;r The risk free interest rate associated with the specific term of theoption, r ;
r The dividend yield during the term of the option, d;r The volatility of the underlying during the term of the option, σ .The thought Black and Scholes had behind getting to a specific formulato price options is both genius and simple. The basic methodology was tocreate a risk neutral portfolio consisting of the option one wants to priceand, because of its risk neutrality, the value of this portfolio should beyielding the risk free interest rate. Establishing the risk neutral portfoliocontaining the option one wants to price was again genius but simple,namely for a call option with a price ct and underlying share price Stct −
∂ct· St∂ St
(2.1)
∂ pt· St .∂ St
(2.2)
and for a put option with price ptpt −
Respectively, ∂∂cStt and ∂∂ pStt are nothing more than the derivatives of thecall option price with respect to the underlying share price and the putoption price with respect to the underlying share price. In other words,if, at any time, one holds ∂∂cStt number of shares against one call option,this portfolio is immune to share price movements as the speed at whichthe price of the call option changes with any given share price movement
Conventional Options, Forwards and Greeks
7
is exactly ∂∂cStt times this share price movement. The same holds for theput option portfolio. Since both portfolios are immune to share pricemovements, if one assumes that all other variables remain unchanged,both portfolios should exactly yield the risk free interest rate. With thisrisk neutral portfolio as a starting point Black and Scholes were ableto derive a pricing formula for both the call and the put option. It hasto be said that the analysis and probability theory used to get to thesepricing formulae are quite heavy. A separate book can be written on thederivations used to determine the actual pricing formulae. Two famoustheorems in mathematics are of crucial importance to these derivations,namely Girsanov’s theorem and Ito’s lemma. It is far beyond the scopeof this book to get into the mathematical details, but the interested readercould use Lamberton and Lapeyre, 1996 as a reference. Although thederivations of the call and the put price are of no use to working withoptions in practice, it is very useful to know the actual pricing formulaeand be able to look them up when necessary. The prices, at time t, of aEuropean call and put option with strike price K , time to maturity T − t,stock price St , interest rate r and volatility σ are given by the followingformulae2 :ct = St N (d1 ) − K e−r (T −t) N (d2 ),
(2.3)
pt = K e−r (T −t) N (−d2 ) − St N (−d1 ).
(2.4)
In these formulae, N (x) is the standard Normal distribution, and d1 , d2are defined as  ln SKt + r + 12 σ 2 (T − t)d1 =,(2.5)√σ T −t√d2 = d1 − σ T − t.(2.6)Equations 2.3 and 2.4 are incredibly powerful as all the variables thatmake up the formulae are known or can be treated as such, except forthe volatility. Although it is not known what the interest rate will be overthe term of the option, it can be estimated quite easily and on top ofthat there is a very liquid market for interest rates. Hence, the interestrate can be treated as known. Therefore, the only uncertainty left in thepricing of options is the volatility. The fact that the volatility over theterm of the option is not known up front might make it impossible to2
The dividend yield is assumed to be zero.
8
Exotic Options Trading
price the option exactly, but it does give the opportunity to start bettingor trading, especially on this variable.
2.3 IMPLIED VOLATILITYImplied volatility is one of the main concepts in options trading. At first,the principle of implied volatility might seem quite difficult. However,it is in fact quite intuitive and simple. Implied volatility is the volatilityimplied by the market place. Since options have a value in the marketplace, one can derive the volatility implied by this price in the market byequating the Black–Scholes formula to the price in the market and subsequently solving for the volatility in this equation with one unknown.Except for the volatility, all variables in the Black–Scholes formula areknown or can be treated as known and hence equating the Black–Scholesformula for an option to the specific price in the market of that optionresults in an equation in one unknown, namely the volatility. Solvingfor the volatility value in this equation is called the implied volatility.Although this is an equation in one unknown, one cannot solve thisequation analytically but it can easily be solved numerically.
2.4 DETERMINING THE STRIKE OF THE FORWARDForwards are agreements to buy or sell shares at a future point in timewithout having to make a payment up front. Unlike an option, the buyerof a forward does not have an option at expiry. For example, the buyerof a 6 months forward in Royal Dutch/Shell commits himself to buyingshares in Royal Dutch/Shell at a pre-agreed price determined by theforward contract. The natural question is, of course, what should thispre-agreed price be? Just like Black and Scholes did for the pricing ofan option, the price is determined by how much it will cost to hedgethe forward position. To show this, consider the following example.An investment bank sells a 2 year forward on Royal Dutch/Shell to aninvestor. Suppose that the stock is trading at € 40, the interest rate is5 % per year and after 1 year Royal Dutch/Shell will pay a dividend of€ 1. Because the bank sells the forward it commits itself to selling aRoyal Dutch/Shell share in 2 years’ time. The bank will hedge itself bybuying a Royal Dutch/Shell share today. By buying a Royal Dutch/Shellshare the bank pays € 40, over which it will pay interest for the next2 years. However, since the bank is long a Royal Dutch/Shell share, itwill receive a dividend of € 1 in 1 year’s time. So over the first year thebank will pay interest over € 40 and over the second year interest over
Conventional Options, Forwards and Greeks
9
€ 39. This means that the price of the forward should beF = 40 + 40 × 0.05 + 39 × 0.05 − 1 = 42.95.
(2.7)
A more general formula for the forward price isF = Price of underlying + Cost of carry.
(2.8)
In the previous example the cost of carry is the interest the bank has topay to hold the stock minus the dividend it receives for holding the stock.
2.5 PRICING OF STOCK OPTIONS INCLUDINGDIVIDENDSWhen dividends are known to be paid at specific points in time it is easyto adjust the Black–Scholes formula such that it gives the right optionprice. The only change one needs to make is to adjust the stock price. Thereason for this is that a dividend payment will cause the stock price to godown by exactly the amount of the dividend. So, in order to get the rightoption price, one needs to subtract the present value of the dividendspaid during the term of the option from the current stock price, whichcan then be plugged into the Black–Scholes formula (see equations 2.3and 2.4). As an example, consider a 1 year call option on BMW witha strike price of € 40. Suppose BMW is currently trading at € 40, theinterest rate is 5 %, the stock price volatility is 20 % per annum and thereare two dividends in the next year, one of € 1 after 2 months and anotherof € 0.5 after 8 months. It is now possible to calculate the present valueof the dividends and subtract it from the current stock level.PV of Dividends = e{− 12 ×0.05} × 1 + e{− 12 ×0.05} × 0.5= 1.4753.2
8
(2.9)
Now the option price can be calculated by plugging into the Black–Scholes formula (see equation 2.3) a stock price of St = 40 − 1.4753 =38.5247 and using K = 40, r = 0.05, σ = 0.2 and T − t = 1. + 0.05 + 12 × 0.22 × 1ln 38.524740= 0.1621 (2.10)d1 =√0.2 × 1√d2 = d1 − 0.2 × 1 = −0.0379.(2.11)So, the price of the call option will bect = 38.5247 × N (0.1621) − 40 × e−0.05 N (−0.0379)= 3.2934.(2.12)
10
Exotic Options Trading
2.6 PRICING OPTIONS IN TERMS OF THE FORWARDInstead of expressing the option price in terms of the current stock price,interest rate and expected dividend, it is more intuitive to price an optionin terms of the forward, which comprises all these three components.The easiest way to rewrite the Black–Scholes formula in terms of theforward is to assume a dividend yield rather than dividends paid out atdiscrete points in time. This means that a continuous dividend payout isassumed. Although this is not what happens in practice, one can calculatethe dividend yield in such a way that the present value of the dividendpayments is equal to St × ed(T −t) − 1 , where d is the dividend yield.So, if the dividend yield is assumed to be d and the interest rate is r , theforward at time t can be expressed asFt = St × e{r (T −t)} × e{−d(T −t)}= St × e{(r −d)(T −t)} .
(