Exotic Options Trading E-Book

Frans de Weert

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 283

Veröffentlichungsjahr: 2011

Beschreibung

Written by an experienced trader and consultant, Frans de Weert's Exotic Options Trading offers a risk-focused approach to the pricing of exotic options. By giving readers the necessary tools to understand exotic options, this book serves as a manual to equip the reader with the skills to price and risk manage the most common and the most complex exotic options. De Weert begins by explaining the risks associated with trading an exotic option before dissecting these risks through a detailed analysis of the actual economics and Greeks rather than solely stating the mathematical formulae. The book limits the use of mathematics to explain exotic options from an economic and risk perspective by means of real life examples leading to a practical interpretation of the mathematical pricing formulae. The book covers conventional options, digital options, barrier options, cliquets, quanto options, outperformance options and variance swaps, and explains difficult concepts in simple terms, with a practical approach that gives the reader a full understanding of every aspect of each exotic option. The book also discusses structured notes with exotic options embedded in them, such as reverse convertibles, callable and puttable reverse convertibles and autocallables and shows the rationale behind these structures and their associated risks. For each exotic option, the author makes clear why there is an investor demand; explains where the risks lie and how this affects the actual pricing; shows how best to hedge any vega or gamma exposure embedded in the exotic option and discusses the skew exposure. By explaining the practical implications for every exotic option and how it affects the price, in addition to the necessary mathematical derivations and tools for pricing exotic options, Exotic Options Trading removes the mystique surrounding exotic options in order to give the reader a full understanding of every aspect of each exotic option, creating a useable tool for dealing with exotic options in practice. "Although exotic options are not a new subject in finance, the coverage traditionally afforded by many texts is either too high level or overly mathematical. De Weert's exceptional text fills this gap superbly. It is a rigorous treatment of a number of exotic structures and includes numerous examples to clearly illustrate the principles. What makes this book unique is that it manages to strike a fantastic balance between the theory and actual trading practice. Although it may be something of an overused phrase to describe this book as compulsory reading, I can assure any reader they will not be disappointed." --Neil Schofield, Training Consultant and author of Commodity Derivatives: Markets and Applications "Exotic Options Trading does an excellent job in providing a succinct and exhaustive overview of exotic options. The real edge of this book is that it explains exotic options from a risk and economical perspective and provides a clear link to the actual profit and pricing formulae. In short, a must read for anyone who wants to get deep insights into exotic options and start trading them profitably." --Arturo Bignardi

Leseprobe

Table of Contents

Title Page

Copyright Page

Preface

Acknowledgements

Chapter 1 - Introduction

Chapter 2 - Conventional Options, Forwards and Greeks

2.1 CALL AND PUT OPTIONS AND FORWARDS

2.2 PRICING CALLS AND PUTS

2.3 IMPLIED VOLATILITY

2.4 DETERMINING THE STRIKE OF THE FORWARD

2.5 PRICING OF STOCK OPTIONS INCLUDING DIVIDENDS

2.6 PRICING OPTIONS IN TERMS OF THE FORWARD

2.7 PUT-CALL PARITY

2.8 DELTA

2.9 DYNAMIC HEDGING

2.10 GAMMA

2.11 VEGA

2.12 THETA

2.13 HIGHER ORDER DERIVATIVES LIKE VANNA AND VOMMA

2.14 OPTIONS’ INTEREST RATE EXPOSURE IN TERMS OF FINANCING THE DELTA HEDGE

Chapter 3 - Profit on Gamma and Relation to Theta

Chapter 4 - Delta Cash and Gamma Cash

4.1 EXAMPLE: DELTA AND GAMMA CASH

Chapter 5 - Skew

5.1 REASONS FOR HIGHER REALISED VOLATILITY IN FALLING MARKETS

5.2 SKEW THROUGH TIME: ‘THE TERM STRUCTURE OF SKEW’

5.3 SKEW AND ITS EFFECT ON DELTA

5.4 SKEW IN FX VERSUS SKEW IN EQUITY: ‘SMILE VERSUS DOWNWARD SLOPING’

5.5 PRICING OPTIONS USING THE SKEW CURVE

Chapter 6 - Simple Option Strategies

6.1 CALL SPREAD

6.2 PUT SPREAD

6.3 COLLAR

6.4 STRADDLE

6.5 STRANGLE

Chapter 7 - Monte Carlo Processes

7.1 MONTE CARLO PROCESS PRINCIPLE

7.2 BINOMIAL TREE VERSUS MONTE CARLO PROCESS

7.3 BINOMIAL TREE EXAMPLE

7.4 THE WORKINGS OF THE MONTE CARLO PROCESS

Chapter 8 - Chooser Option

8.1 PRICING EXAMPLE: SIMPLE CHOOSER OPTION

8.2 RATIONALE BEHIND CHOOSER OPTION STRATEGIES

Chapter 9 - Digital Options

9.1 CHOOSING THE STRIKES

9.2 THE CALL SPREAD AS PROXY FOR THE DIGITAL

9.3 WIDTH OF THE CALL SPREAD VERSUS GEARING

Chapter 10 - Barrier Options

10.1 DOWN-AND-IN PUT OPTION

10.2 DELTA CHANGE OVER THE BARRIER FOR A DOWN-AND-IN PUT OPTION

10.3 FACTORS INFLUENCING THE MAGNITUDE OF THE BARRIER SHIFT

10.4 DELTA IMPACT OF A BARRIER SHIFT

10.5 SITUATIONS TO BUY SHARES IN CASE OF A BARRIER BREACH OF A LONG DOWN-AND-IN PUT

10.6 UP-AND-OUT CALL

10.7 UP-AND-OUT CALL OPTION WITH REBATE

10.8 VEGA EXPOSURE UP-AND-OUT CALL OPTION

10.9 UP-AND-OUT PUT

10.10 BARRIER PARITY

10.11 BARRIER AT MATURITY ONLY

10.12 SKEW AND BARRIER OPTIONS

10.13 DOUBLE BARRIERS

Chapter 11 - Forward Starting Options

11.1 FORWARD STARTING AND REGULAR OPTIONS COMPARED

11.2 HEDGING THE SKEW DELTA OF THE FORWARD START OPTION

11.3 THE FORWARD START OPTION AND THE SKEW TERM STRUCTURE

11.4 ANALYTICALLY SHORT SKEW BUT DYNAMICALLY NO SKEW EXPOSURE

11.5 FORWARD STARTING GREEKS

Chapter 12 - Ladder Options

12.1 EXAMPLE: LADDER OPTION

12.2 PRICING THE LADDER OPTION

Chapter 13 - Lookback Options

13.1 PRICING AND GAMMA PROFILE OF FIXED STRIKE LOOKBACK OPTIONS

13.2 PRICING AND RISK OF A FLOATING STRIKE LOOKBACK OPTION

Chapter 14 - Cliquets

14.1 THE RATCHET OPTION

14.2 RISKS OF A RATCHET OPTION

Chapter 15 - Reverse Convertibles

15.1 EXAMPLE: KNOCK-IN REVERSE CONVERTIBLE

15.2 PRICING THE KNOCK-IN REVERSE CONVERTIBLE

15.3 MARKET CONDITIONS FOR MOST ATTRACTIVE COUPON

15.4 HEDGING THE REVERSE CONVERTIBLE

Chapter 16 - Autocallables

16.1 EXAMPLE: AUTOCALLABLE REVERSE CONVERTIBLE

16.2 PRICING THE AUTOCALLABLE

16.3 AUTOCALLABLE PRICING WITHOUT CONDITIONAL COUPON

16.4 INTEREST/EQUITY CORRELATION WITHIN THE AUTOCALLABLE

Chapter 17 - Callable and Puttable Reverse Convertibles

17.1 PRICING THE CALLABLE REVERSE CONVERTIBLE

17.2 PRICING THE PUTTABLE REVERSE CONVERTIBLE

Chapter 18 - Asian Options

18.1 PRICING THE GEOMETRIC ASIAN OUT OPTION

18.2 PRICING THE ARITHMETIC ASIAN OUT OPTION

18.3 DELTA HEDGING THE ARITHMETIC ASIAN OUT OPTION

18.4 VEGA, GAMMA AND THETA OF THE ARITHMETIC ASIAN OUT OPTION

18.5 DELTA HEDGING THE ASIAN IN OPTION

18.6 ASIAN IN FORWARD

18.7 PRICING THE ASIAN IN FORWARD

18.8 ASIAN IN FORWARD WITH OPTIONAL EARLY TERMINATION

Chapter 19 - Quanto Options

19.1 PRICING AND CORRELATION RISK OF THE QUANTO OPTION

19.2 HEDGING FX EXPOSURE ON THE QUANTO OPTION

Chapter 20 - Composite Options

20.1 AN EXAMPLE OF THE COMPOSITE OPTION

20.2 HEDGING FX EXPOSURE ON THE COMPOSITE OPTION

Chapter 21 - Outperformance Options

21.1 EXAMPLE OF AN OUTPERFORMANCE OPTION

21.2 OUTPERFORMANCE OPTION DESCRIBED AS A COMPOSITE OPTION

21.3 CORRELATION POSITION OF THE OUTPERFORMANCE OPTION

21.4 HEDGING OF OUTPERFORMANCE OPTIONS

Chapter 22 - Best of and Worst of Options

22.1 CORRELATION RISK FOR THE BEST OF OPTION

22.2 CORRELATION RISK FOR THE WORST OF OPTION

22.3 HYBRIDS

Chapter 23 - Variance Swaps

23.1 VARIANCE SWAP PAYOFF EXAMPLE

23.2 REPLICATING THE VARIANCE SWAP WITH OPTIONS

23.3 GREEKS OF THE VARIANCE SWAP

23.4 MYSTERY OF GAMMA WITHOUT DELTA

23.5 REALISED VARIANCE VOLATILITY VERSUS STANDARD DEVIATION

23.6 EVENT RISK OF A VARIANCE SWAP VERSUS A SINGLE OPTION

23.7 RELATION BETWEEN VEGA EXPOSURE AND VARIANCE NOTIONAL

23.8 SKEW DELTA

23.9 VEGA CONVEXITY

Chapter 24 - Dispersion

24.1 PRICING BASKET OPTIONS

24.2 BASKET VOLATILITY DERIVED FROM ITS CONSTITUENTS

24.3 TRADING DISPERSION

24.4 QUOTING DISPERSION IN TERMS OF CORRELATION

24.5 DISPERSION MEANS TRADING A COMBINATION OF VOLATILITY AND CORRELATION

24.6 RATIO’D VEGA DISPERSION

24.7 SKEW DELTA POSITION EMBEDDED IN DISPERSION

Chapter 25 - Engineering Financial Structures

25.1 CAPITAL GUARANTEED PRODUCTS

25.2 ATTRACTIVE MARKET CONDITIONS FOR CAPITAL GUARANTEED PRODUCTS

25.3 EXPOSURE PRODUCTS FOR THE CAUTIOUS EQUITY INVESTOR

25.4 LEVERAGED PRODUCTS FOR THE RISK SEEKING INVESTOR

Appendix A - Variance of a Composite Option and Outperformance Option

Appendix B - Replicating the Variance Swap

Bibliography

Index

For other titles in the Wiley Finance Series please see www.wiley.com/finance

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Preface

This book is appropriate for people who want to get a good overview of exotic options in practice and are interested in the actual pricing of them. When dealing with exotic options it is very important to quantify the risks associated with them and at which stock or interest rate levels the Greeks change sign. Namely, it is usually the case that the Greeks of exotic options show much more erratic behaviour than the Greeks of regular options. This instability of the Greeks forces the trader to choose different hedging strategies than the standard option model would prescribe. Therefore the risk management of exotic options entails much more than just obeying the model, which in turn has an impact on the price. The non-standard risk management of exotic options means that when pricing an exotic option, one first needs to understand where the risks lie that affect the hedging strategy and hence the pricing of the particular exotic option. Once the risks have been mapped and the hedging strategy has been determined, the actual pricing is often nothing more than a Monte Carlo process. Moreover, when knowing the risks, the actual pricing of an exotic option can in some cases even be replicated by a set of standard options. In other words, the starting point for pricing exotic options is to have a full awareness of the risks, which in turn has an impact on how one needs to accurately price an exotic option.

The aim of this book is to give both option practitioners and economics students and interested individuals the necessary tools to understand exotic options and a manual that equips the reader to price and risk manage the most common and complicated exotic options. To achieve this it is imperative to understand the interaction between the different Greeks and how this, in combination with any hedging scheme, translates into a real tangible profit on an exotic option. For that reason, this book is written such that for every exotic option the practical implications are explained and how these affect the price. Knowing this, the necessary mathematical derivations and tools are explained to give the reader a full understanding of every aspect of each exotic option. This balance is incredibly powerful and takes away a lot of the mystique surrounding exotic options, turning it into useable tools for dealing with exotic options in practice.

This book discusses each exotic option from four different angles. First, it makes clear why there is investor demand for a specific exotic option. Secondly, it explains where the risks lie for each exotic option and how this affects the actual pricing of the exotic option. Thirdly, it shows how to best hedge any vega or gamma exposure embedded in the exotic option. Lastly, for each exotic option the skew exposure is discussed separately. This is because any skew exposure is typically harder to quantify, 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.

Acknowledgements

This book is based on knowledge acquired during my work as a trader at Barclays Capital. Therefore I would like to thank my colleagues at Barclays Capital who have been very helpful in teaching me the theory and practice of options. I would like to thank Faisal Khan and Thierry Lucas for giving me all the opportunities to succeed in mastering and practicing option trading; Stuart Barton, Karan Sood and Arturo Bignardi who have all made an indispensable contribution to my knowledge of options; Arturo Bignardi for his many suggestions and corrections when reviewing my work and Alex Boer for his mathematical insights; and special thanks go out to Neil Schofield for his many suggestions and for reviewing the work very thoroughly. I would also like to thank my great friends for all their support and motivation. Special thanks go out to my parents and my two sisters for having been a motivating force throughout my life and for a fantastic upbringing. I also want to thank my father for reviewing this book and being able to get an understanding of this complicated subject and subsequently being able to correct mistakes.

Petra, the light of my life. Finally, we can shine together!

1

Introduction

Exotic 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 of exotic options. Typically, exotic options have a correlation component. Which means that their price depends on the correlation between two or more assets. To understand an exotic option one needs to know above all 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 point during the term of this option does it have the largest Greeks. Secondly, one needs to understand the dynamics of the risks. This means that one needs to know how the risks evolve over time and how these risks behave for a changing stock or basket price. The reason that one needs to understand the risks of an exotic option before actually pricing it is because the risks determine how an exotic option should be priced. Once it is known where the risks lie and the method for pricing it is determined, one finds that the actual pricing is typically nothing more than a Monte Carlo method. In other words, the price of an exotic option is generally based on simulating a large set of paths and subsequently dividing the sum of the payoffs by the total number of paths generated. The method for pricing an exotic option is very important as most exotic options can be priced by using a set of different exotic options and therefore saving a considerable amount of time. Also, sometimes one needs to conclude that the best way to price a specific exotic option is by estimating the price with a series of standard options, as this method better captures the risk involved with this exotic option. The digital option is a good example of that and will be discussed in Chapter 9.

Before any exotic option is discussed it is important to fully understand the interaction between gamma and theta. Although this book assumes an understanding of all the Greeks and how they interact, the following two sections give a brief summary of the Greeks and how the profit of an option depends on one of the Greeks, namely the gamma. A more detailed discussion of the Greeks and the profit related to them can be found in An Introduction to Options Trading, F. de Weert.

2

Conventional Options, Forwards and Greeks1

This section is meant to give a quick run through of all the important aspects of options and to provide a sufficient theoretical grounding in regular options. This grounding enables the reader to enter into the more complex world of exotic options. Readers who already have a good working knowledge of conventional options, Greeks and forwards can skip this chapter. Nonetheless, even for more experienced option practitioners, this section can serve as a useful look-up guide for formulae of the different Greeks and more basic option characteristics.

2.1 CALL AND PUT OPTIONS AND FORWARDS

Call and put options on stocks have been traded on organised exchanges since 1973. However, options have been traded in one form or another for many more years. The most common types of options are the call option and the put option. A call option on a stock gives the buyer the right, but not the obligation, to buy a stock at a pre-specified price and at 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 and at or before a pre-specified date. The pre-specified price at which the option holder can buy in the case of a call and sell in the case of a put is called the strike price. The buyer is said to exercise his option when he uses his right to buy the underlying share in case of a call option and when he sells the underlying share in case of a put option. The date at or up to which the buyer is allowed to exercise his option is called the maturity date or expiration date. There are two different terms regarding the timing of the right to exercise an option. They are identified by a naming convention difference. The first type is the European option where the option can only be exercised at maturity. The second type of option is the American option where the option can be exercised at any time up to and including the expiry date.

Figure 2.1 Payoff profile at maturity for a call option with strike price K

Obviously, the buyer of a European call option would only exercise his right to buy the underlying stock if the share price was higher than the strike price. In this case, the buyer can buy the share for the predetermined strike price by exercising the call and subsequently sell it in the market at the prevailing price in the market, which is higher than the strike price and therefore making a profit. The payoff profile of the call option is shown in Figure 2.1. The buyer of a European put option acts opposite to the buyer of the call option in the sense that the buyer of a put option would only exercise his option right, at maturity, if the share price was below the strike price. In this case the option buyer can first buy the share in the market at the prevailing market price and subsequently sell it at the strike price by exercising his put option, earning a profit as a result. The payoff profile at maturity of a put option is shown in Figure 2.2.

A forward is different to an option in the sense that the buyer of the forward is obliged to buy the stock at a pre-specified price and at a pre-specified date in the future. The pre-specified price of a forward is chosen in such a way that the price of the forward is zero at inception of the contract. Therefore, the expected fair value of the stock at a certain maturity date is often referred to as the forward value of a stock or simply the forward associated with the specific maturity. The payoff profile at maturity of a forward contract is shown in Figure 2.3. Figure 2.3 makes clear that there is a downside in owning a forward. Whereas the owner of an option always has a payout at maturity which is larger than zero and therefore the maximum loss is equal to the premium paid for the option, the maximum loss on one forward is equal to the strike price of the forward, which occurs if the share price goes to zero. Since the definition of a forward prescribes that the contract is worth zero at inception, the strike price of the forward is equal to the forward value, which is discussed more elaborately in sub-section 2.4.

Figure 2.2 Payoff profile at maturity for a put option with strike price K

Figure 2.3 Payoff profile at maturity for a forward with strike price K which is equal to the fair forward value

2.2 PRICING CALLS AND PUTS

In 1973 Black and Scholes introduced their famous Black-Scholes formula. The Black-Scholes formula makes it possible to price a call or a put option in terms of the following inputs:

• The underlying share price, St;

• The strike price, K;

• The time to maturity, T - t;

• The risk free interest rate associated with the specific term of the option, r;

• The dividend yield during the term of the option, d;

• The volatility of the underlying during the term of the option, σ.

• The volatility of the underlying during the term of the option, σ.

The thought Black and Scholes had behind getting to a specific formula to price options is both genius and simple. The basic methodology was to create a risk neutral portfolio consisting of the option one wants to price and, because of its risk neutrality, the value of this portfolio should be yielding the risk free interest rate. Establishing the risk neutral portfolio containing the option one wants to price was again genius but simple, namely for a call option with a price ct and underlying share price St and for a put option with price pt

(2.1)

(2.2)

Respectively, and are nothing more than the derivatives of the call option price with respect to the underlying share price and the put option price with respect to the underlying share price. In other words, if, at any time, one holds number of shares against one call option, this portfolio is immune to share price movements as the speed at which the price of the call option changes with any given share price movement is exactly times this share price movement. The same holds for the put option portfolio. Since both portfolios are immune to share price movements, if one assumes that all other variables remain unchanged, both portfolios should exactly yield the risk free interest rate. With this risk neutral portfolio as a starting point Black and Scholes were able to derive a pricing formula for both the call and the put option. It has to be said that the analysis and probability theory used to get to these pricing formulae are quite heavy. A separate book can be written on the derivations used to determine the actual pricing formulae. Two famous theorems in mathematics are of crucial importance to these derivations, namely Girsanov’s theorem and Ito’s lemma. It is far beyond the scope of this book to get into the mathematical details, but the interested reader could use Lamberton and Lapeyre, 1996 as a reference. Although the derivations of the call and the put price are of no use to working with options in practice, it is very useful to know the actual pricing formulae and be able to look them up when necessary. The prices, at time t, of a European 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 following formulae2:

(2.3)

(2.4)

In these formulae, N(x) is the standard Normal distribution, and d1, d2 are defined as Equations 2.3 and 2.4 are incredibly powerful as all the variables that make up the formulae are known or can be treated as such, except for the volatility. Although it is not known what the interest rate will be over the term of the option, it can be estimated quite easily and on top of that there is a very liquid market for interest rates. Hence, the interest rate can be treated as known. Therefore, the only uncertainty left in the pricing of options is the volatility. The fact that the volatility over the term of the option is not known up front might make it impossible to price the option exactly, but it does give the opportunity to start betting or trading, especially on this variable.

(2.5)

(2.6)

2.3 IMPLIED VOLATILITY

Implied 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 volatility implied by the market place. Since options have a value in the market place, one can derive the volatility implied by this price in the market by equating 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 are known or can be treated as known and hence equating the Black-Scholes formula for an option to the specific price in the market of that option results in an equation in one unknown, namely the volatility. Solving for the volatility value in this equation is called the implied volatility. Although this is an equation in one unknown, one cannot solve this equation analytically but it can easily be solved numerically.

2.4 DETERMINING THE STRIKE OF THE FORWARD

Forwards are agreements to buy or sell shares at a future point in time without having to make a payment up front. Unlike an option, the buyer of a forward does not have an option at expiry. For example, the buyer of a 6 months forward in Royal Dutch/Shell commits himself to buying shares in Royal Dutch/Shell at a pre-agreed price determined by the forward contract. The natural question is, of course, what should this pre-agreed price be? Just like Black and Scholes did for the pricing of an option, the price is determined by how much it will cost to hedge the forward position. To show this, consider the following example. An investment bank sells a 2 year forward on Royal Dutch/Shell to an investor. Suppose that the stock is trading at € 40, the interest rate is 5 % 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 a Royal Dutch/Shell share in 2 years’ time. The bank will hedge itself by buying a Royal Dutch/Shell share today. By buying a Royal Dutch/Shell share the bank pays € 40, over which it will pay interest for the next 2 years. However, since the bank is long a Royal Dutch/Shell share, it will receive a dividend of €1 in 1 year’s time. So over the first year the bank will pay interest over € 40 and over the second year interest over € 39. This means that the price of the forward should be

(2.7)

A more general formula for the forward price is

(2.8)

In the previous example the cost of carry is the interest the bank has to pay to hold the stock minus the dividend it receives for holding the stock.

2.5 PRICING OF STOCK OPTIONS INCLUDING DIVIDENDS

When dividends are known to be paid at specific points in time it is easy to adjust the Black-Scholes formula such that it gives the right option price. The only change one needs to make is to adjust the stock price. The reason for this is that a dividend payment will cause the stock price to go down by exactly the amount of the dividend. So, in order to get the right option price, one needs to subtract the present value of the dividends paid during the term of the option from the current stock price, which can then be plugged into the Black-Scholes formula (see equations 2.3 and 2.4). As an example, consider a 1 year call option on BMW with a strike price of € 40. Suppose BMW is currently trading at € 40, the interest rate is 5 %, the stock price volatility is 20 % per annum and there are two dividends in the next year, one of € 1 after 2 months and another of € 0.5 after 8 months. It is now possible to calculate the present value of the dividends and subtract it from the current stock level.

(2.9)

So, the price of the call option will be

(2.12)

2.6 PRICING OPTIONS IN TERMS OF THE FORWARD

Instead of expressing the option price in terms of the current stock price, interest rate and expected dividend, it is more intuitive to price an option in terms of the forward, which comprises all these three components. The easiest way to rewrite the Black-Scholes formula in terms of the forward is to assume a dividend yield rather than dividends paid out at discrete points in time. This means that a continuous dividend payout is assumed. Although this is not what happens in practice, one can calculate the dividend yield in such a way that the present value of the dividend payments 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, the forward at time t can be expressed as

(2.13)

From the above equation it is clear that dividends lower the price of the forward and interest rates increase it. As shown in the previous sub-section, one can calculate the price of an option by substituting a stock price equal to St × e{-d(T-t)} into the Black-Scholes formula. By doing this one can rearrange the Black-Scholes formula to express the price of an option in terms of the forward. The price of the call can then be expressed as

(2.14)

In the same way the price of the put can be expressed as

(2.15)

where d1 and d2 are

(2.16)

(2.17)

2.7 PUT-CALL PARITY

The put-call parity is a very important formula and gives the relation between the European call and the put price, where the call and the put have the same strike and maturity, in terms of the share price and the strike price. The formula is as follows:

(2.18)

where d is the dividend yield of the share and r is the risk free interest rate. One can prove this put-call parity by assuming it does not hold and show that one can then build a portfolio that leads to a riskless profit at maturity. For example, when

(2.19)

one can prove that the portfolio

• sell a call option with strike price K and maturity T - t(income is ct);

• buy a put option with strike price K and maturity T - t (income is - pt);

• buy a share (income is

leads to a riskless profit at maturity for any level of share price at maturity.

Indeed, if ST > K the call will be exercised and therefore the share within the portfolio is sold at € K and the put expires worthless. The net position at maturity is therefore zero, but since the generated income including the financing plus € K is larger than zero, the strategy makes a riskless profit. Indeed, rewriting equation 2.19 gives

(2.20)

In the same way one can prove that the above portfolio leads to a riskless profit if ST < K.

If one assumes that

(2.21)

one can show that the following portfolio leads to a riskless profit for any level of share price at maturity:

• buy a call option with strike price K and maturity T - t (income

• is ct);

• sell a put option with strike price K and maturity T-t(income is pt);

• sell a share (income is Ste-d(T-t)).

The put-call parity also signals for which strike the call and the put are worth the same. As one would expect, the strike for which the put and the call are worth the same is equal to the forward value of the share. Equation 2.13 established that the forward is equal to

(2.22)

2.8 DELTA

Delta is one of the most important Greeks and instrumental to the Black- Scholes derivation of the price of an option (see Section 2.2). Delta measures the sensitivity of an option price to the stock price. Mathematically, delta, δ, is the derivative of the option price with respect to the stock price. By taking the actual derivative of equations 2.3 and 2.4, the delta of European call and put options on a non-dividend paying stock are as given below. This assumes a call option price of ct, a put option price of pt and a stock price of St.

(2.23)

(2.24)

Equation 2.23 shows that the delta of a call option is between 0 and 1 and of a put option between—1 and 0. Table 2.1 shows for which stock price the delta reaches its extremes. The deltas of call and put options with strike K versus the stock price are shown graphically in Figures 2.4 and 2.5 respectively.

Table 2.1 Extreme delta values

Type of OptionDelta (δ)Far in the money call option1Far out of the money call option0Far in the money put option-1Far out of the money put option0

Figure 2.4 Variation of delta with stock price for a call option

Figure 2.5 Variation of delta with stock price for a put option

2.9 DYNAMIC HEDGING

Black and Scholes showed that a portfolio consisting, at any time, of one option and minus delta shares is risk neutral and should therefore yield the risk free interest rate. For a long call option, delta hedging means selling shares and for a long put put option, minus delta shares means effectively buying shares as the delta of a put option is negative. To buy minus delta shares against one option is called delta hedging. In practice it is not possible to have minus delta shares against one option at any time. This would mean that one would have to continuously adjust the number of shares, which is not possible. Therefore, in practice an option is delta hedged at discrete points in time and is called dynamic hedging. Although this might seem inconvenient, it is exactly the reason that traders make money on volatility. This is discussed in Chapter 3.

2.10 GAMMA

Since delta changes whenever the stock price changes, it is useful to have a measure that captures this relationship. This measure is called gamma and gives the sensitivity of delta to a small change in stock price. Mathematically, gamma is the derivative of delta with respect to the stock price. Since delta is the derivative of the option price with respect to the stock price, gamma is the second order derivative of the option price with respect to the stock price. For European call and put options, gamma is given by the following formula.

(2.25)

(2.26)

where d1 is defined as in equation 2.5 and

(2.27)

The above formulae show that the gamma of a European call option is equal to the gamma of a European put option. Respectively, Figures 2.6 and 2.7 indicate the way in which gamma varies with the stock price and the time to maturity. It is helpful to interpret gamma in terms of how the delta hedge of an option changes for a change in stock price.

Figure 2.6 Variation of gamma with stock price