Hedging of Financial Derivatives and Portfolio Insurance

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1 Hedging of Financial Derivatives and Portfolio Insurance Gasper Godson Mwanga African Institute for Mathematical Sciences 6, Melrose Road, 7945 Muizenberg, Cape Town South Africa. Supervisor : Prof. J. C. Ndogmo Department of Mathematical University of Western Cape Private Bag X17, 7535 Bellville, Capetown South Africa June 23, 2005

2 Contents List of Figures iii Acknowledgements iv 1 Introduction Background Markets Derivative security Option Forwards and Futures Swaps Important Formulae The Greek Letters Naked and Covered Position Stop-Loss Strategy Delta Hedging Hedging Performance Delta of Forward Contract Delta of European Calls and Puts Delta of Other European Options Theta Gamma Hedging

3 CONTENTS ii Making a Portfolio Gamma-Neutral Calculation of Gamma Relationship between, Θ & Γ Rho (ρ) Vega (V) Calculation of Vega Scenario Analysis Portfolio Insurance Preliminary Using Index Options for Portfolio Insurance Creating Options Synthetically Use of the Trading of Portfolio Use of Index Option Conclusion 28 A Determination of Delta for a European call Option 29 B Determination of Theta for a European call Option 31 Bibliography 32

4 List of Figures 2.1 Stop-Loss strategy Calculation of delta Theta of European call option Gamma of European options Rho of European call option Vega of European options

5 Acknowledgements I am indebted to many people from the early stages to the final write up of this work for their help, ideas and suggestions. My greatest debt is to my supervisor, Prof. J. C. Ndogmo for his assistance and encouragement during the preparation of this work. I would be wrong if I will not acknowledge Dr. M. Pickles, Prof. W. Kotze, L. Wills and my fellow students, for their assistance in correcting and editing this work. Finally, I want to acknowledge all the AIMS staff, in particular Prof. F. Hahne, Prof. N. G. Turok as well as all the sponsors of the AIMS programme for making my stay in AIMS such a wonderful experience. I will not regret my decision of coming to AIMS, because I learnt a lot in the courses offered and also I interacted with world renowned academics.

6 To my parents

7 Abstract Risk management is an important issue in finance because of the considerable impact of the volatility of asset prices on financial holdings. Investment banks, financial corporations and insurance companies around the globe are searching for techniques to enhance their risk management practices. Because of the rapid development of derivative markets, this practice becomes more complex and challenging. This accelerates the development of more advanced techniques in risk management and creates many interesting theoretical and practical problems for researchers. Hedging is the trading strategy which attempts to reduce the degree of risk exposure. In this essay we analyse some common hedging strategies such as naked and covered positions, stop-loss strategies and show how more specific hedging strategies denoted by Greek letters, namely delta, gamma, theta, Vega, and rho can be used to improve the hedging performance. The relationship among these Greek letters and the way in which each affects the change in the portfolio value will also be discussed, as well as scenario analysis and portfolio insurance Mathematics Subject Classification codes 62PXX, 62P20, 91BXX, 91B30

8 Chapter 1 Introduction This chapter will highlight the main concepts we will discuss in the subsequent chapters. 1.1 Background The option pricing theories we are familiar with nowadays has strong roots in stochastic calculus. This concept traces back as far as 1877, when Charles Castelli wrote a book called The Theory of Options in Stock and Shares. This book introduced the concepts of hedging and speculation. Also the financial mathematician Louis Bachelier in 1900 wrote his thesis Théorie de la Spéculation. In this paper he discussed the analysis of the stock and option markets and it also contains some ideas in the theory of Brownian motion. Five years later in 1905 A. Einstein wrote a famous paper on Brownian motion which we use for the mathematical modeling of price movements and the evaluation of contingent claims in financial markets [6]. In 1973, Fisher Black and Myron Scholes published their ground breaking paper The Pricing of Options and Corporate Liabilities in the Journal of Political Economy. This work gained its recognition when in 1997, Robert Merton and Myron Scholes were given the Nobel prize. Part of this work exposes the issue of hedging which we will discuss in this essay [10]. Most scholars continue to criticise the assumptions underlying the Black-Scholes model which led to much research in this area which give rise to more advances the models. The work of H. E. Leland titled Option pricing and replication with Transaction costs published in the Journal of Finance 1985, tries to rectify the trap of the Black-Scholes assumption of no transaction cost; it introduces the type of hedging strategy depending on the value of Leland number A = 2 π. k σ, δt where k is the round-trip transaction cost, σ is the volatility of the underlying asset and δt is the time-lag between transactions. When A < 1 the Black-Scholes delta-hedging is valid [9].

9 1.2 Markets Markets In financial markets the traded item may be an asset (basic equity) such as a stock, bond, or a unit of currency. The item s value may be directly derived from the value of some other traded asset. If so, its future price is tied to the price of another asset. In this case the item is a financial derivative; the asset it refers to is called the underlying asset. A collection of assets all owned by the same individual or organisation is called a portfolio. The person or firm who formulates the contract and offers it for sale is termed as the writer, while a person or firm who purchases the contract is called the holder. The value of a portfolio made up of underlying assets is simply a linear combination of their prices. To see this let the market have d + 1 assets labelled S 0, S 1,... S d, where we assume that the first is riskless, so that its price S 0 (0) determines its price S 0 (t) at future time t with certainty. Then, other assets are risky, so that their prices S i (t), i = 1, 2,..., d are random variables. We usually refer to the risky assets as stocks. Clearly the value at any future time t of a portfolio containing θ i assets is given by d V t (θ) = θ i S i (t). i=0 This linearity of portfolio prices allows us to price other assets in terms of the underlying ones provided we are able to construct a notional portfolio whose value at all times is the same as that of the asset we seek to price. This is the fundamental idea underpinning hedging strategies, which is the key concept in modelling a financial market. In financial markets there are three major types of trading participants. Together they provide important liquidity to facilitate entry into and exit from the market. These traders are as follows: 1. Hedgers These are traders who want to avoid risk exposure due to the price movements of an asset. They do this by taking a position in an option or forward contract and use hedging strategies. 2. Speculators Speculators make profit from predicting directional changes in price in the market. If they are betting that price will go up, they can for example take a long position in a call option because the asset will have a higher price in future and if they are betting that price will go down, they may take a short position in call option (see this in discussion of options). If they are successful they make a profit, if not they incur losses. 3. Arbitrageurs Arbitrageurs take advantage of price discrepancies between the underlying market and the derivatives market with the intention of making a profit, by buying in the cheaper market and selling in the more expensive market. Over time the actions of the arbitrageur usually force

10 1.3 Derivative security 3 the markets back into equilibrium. Arbitrageurs make risk-free profits, although arbitrage opportunities occur infrequently. Summary All these traders (hedgers, speculators and arbitrageurs) are important for the efficient operation of futures and options market. For example if the market provides no economic function for the speculator to assume the hedger s risk, there would be no market. In this essay we focus on the strategies hedgers use to minimise risk. 1.3 Derivative security First what is a derivative? It is a financial instrument whose price depends on, or is derived from, the price of another asset (that is an underlying asset)[3]. Definition 1.1 : A derivative security (also called a continent claim) is a financial contract whose value at its expiry date T is fully determined by the prices at time T (or at a fixed range of times within [0,T]) of the underlying assets. Here are some examples of derivative securities: Option An option is a financial instrument which gives the holder the right, but not the obligation to trade at a specified price, at (or by) a specified date. A call option gives the holder the right to buy an asset, and a put option gives the right to sell an asset. The strike price X is the price at which the future transaction will take place, and is fixed in advance at time 0 (now). The option is called European if the transaction can take place only at the expiry (or exercise) date; while an American options can be exercised at any trading date up to the expiry time. Note that in all of these options it is only the option holder who has the choice to exercise or not. Most of the work in this essay focuses on European options. To short an asset refers to selling of an asset not owned by the seller with the intention of replacing it at a later date. On other hand, a short ( long) position in an option contract refers to the position of the writer ( holder ) of the contract. For a European call option with the stock price S T at expiry date T and with strike price X will be exercised only if S T > X, since otherwise the trader could simply buy the stock from the market for less than X and the option is worthless. Then the value of an option at time T is V T = max{(s T X), 0}. For t [0, T ], if S t > X the call option is said to be in-the-money; when S t = X, the call option is said to be at-the-money; and finally when S t < X, the call option is said to be out-of-the-money. For the put option V T = max{(x S T ), 0} and the inequalities are reversed.

11 1.4 Important Formulae Forwards and Futures A forward contract is a binding agreement to buy or sell an asset S at future date T at a certain future price. This contract must be fulfilled regardless of the future price. Unlike options there are no premiums to be payed to enter into this contract. The price is arranged in such a way that at time t = 0, neither the short nor the long position has a profit. On the other hand in futures the price are determined by the law of supply and demand. The contract is the same as in the case of forward contracts, but the exchange now requires both parties to open margin accounts which will be monitored by the exchange (or clearing house). The clearing house will adjust these marging accounts on a daily basis with some debits or deposits, according to the market price movements Swaps Swaps are exchanges between two partners of future cash flows according to agreed criteria that depend on the value of some underlying assets. The swap market developed because two different investors would find that while one of them had a comparative advantage in borrowing in one market, he was at a disadvantage in the particular market in which he wanted to borrow. They get the best of both worlds through a swap. 1.4 Important Formulae This section provide some of the important formulae we need in the later chapters. The most basic partial differential equation derived by Black-Scholes in 1973 on option pricing is given by V t σ2 S 2 2 V V + rs rv = 0 (1.1) S2 S where V is the value of the option contract at time t (maturing at time T ), σ is the volatility of an asset which is the variable showing how the return of the underlying asset will fluctuate between now and the expiration of the option, S is the stock price and r is the riskless interest rate. By solving the differential equation (1.1) it can be shown that, the value C of a European call option on a non-dividend paying stock is given by [11] C = SN(d 1 ) Xe r(t t) N(d 2 ) with d 1 = ln(s/x) + (r + σ2 /2)(T t) σ T t d 2 = ln(s/x) + (r σ2 /2)(T t) σ = d 1 σ T t T t (1.2)

12 1.4 Important Formulae 5 where N(d 1 ) is the cumulative normal distribution of d 1, S is the stock price at time t, T is expiry time of the option, X is the strike price or exercise price, and σ is the volatility of the underlying stock. The formula for a European put option P on a non-dividend paying stock is given by [11] P = Xe r(t t) N( d 2 ) SN( d 1 ) (1.3) with d 1 and d 2 as in equation (1.2) and other symbols have the usual meaning. Theorem 1.1 Call-Put Parity: Let C(S, t) and P (S, t) be the price at time t of a European call and a European put option respectively, on the same underlying stock and with the same time to the maturity T. Then r(t t) C(S, t) P (S, t) = S Xe where X is the strike price and S is the stock price at time t. Similarly, for stock that pays a continuous dividend yield at rate q, the formula for a European call option C is given by C = Se q(t t) N(d 1 ) Xe r(t t) N(d 2 ) with, d 1 = ln(s/x) + (r q + σ2 /2)(T t) σ T t d 2 = ln(s/x) + (r q σ2 /2)(T t) σ = d 1 σ T t. T t (1.4) The value P of a European put option on dividend paying stock is given by P = Se q(t t) N( d 1 ) + Xe r(t t) N( d 2 ) (1.5) with d 1 and d 2 as in equation (1.4). The formula for a European call option C on currency with risk-free interest rate of foreign currency r f is given by C = Se r f (T t) N(d 1 ) Xe r(t t) N(d 2 ) with, d 1 = ln(s/x) + (r r f + σ 2 /2)(T t) σ T t d 2 = ln(s/x) + (r r f σ 2 /2)(T t) σ = d 1 σ T t. T t (1.6) The formula for a European put option P on a currency is given by with d 1 and d 2 as in equation (1.6). P = Se r f (T t) N( d 1 ) + Xe r(t t) N( d 2 ) (1.7)

13 Chapter 2 The Greek Letters In this chapter we establish the meaning of some Greek letters we use for hedging strategy. Each Greek letter measures a different dimension of the risk in an option position and the aim of a trader is to manage them so that all risks are minimised. We can express the formula for Greek letters by using a binomial model [2] or by the Black-Scholes model (in discrete time [11] or continuous time). In this essay we will use the continuous - time Black-Scholes model. Consider the following example, Example 2.1 Suppose that a financial institution has sold for a European call option on N = shares of a (non-dividend paying) stock, that is C = 1.50 the price of each call. Suppose that at the time the contract interred the stock price is S 0 = 36, and that the strike price is X = 37, the interest rate is r = 5% per annum (continuously compounded), the stock return volatility is σ = 20% per annum. The time to maturity of the contract is T = 3 month (that is, T = years), and the expected return on the stock is µ = 10% per annum. The financial institution sold the call option at the price of C = 1.50 which is higher than the theoretical value of C = 1.10 per each share predicted by the Black-Scholes equation (1.2). Now the financial institution is faced with the problem of hedging its exposure. 2.1 Naked and Covered Position Lets now investigate what kind of strategies the financial institution can adopt in example (2.1). The financial institution can adopt what is called a naked position, which means doing nothing. When the call expires, there are two possible cases:

14 2.2 Stop-Loss Strategy 7 Case 1: The price is below the strike price (S T < X). Then, the call will not be exercised and the financial institution will make the profit of In this case the strategy works. Case 2: The price exceeds the strike price (S T > X) say S T = 40. Then, the call will be exercised, and the financial institution will have to buy shares at 40 in order to deliver the stock at X to fulfil the contract. The financial institution will incur a loss of N (S T X) = 30000, with present value e rt N (S T X) = This loss is higher than the they received. So in this case the strategy did not work. As an alternative to the naked position, the financial institution can adopt a covered position. In this strategy the financial institution buy shares as soon as the option has been sold. For these shares they will have to pay ; the financial institution starts with a debt of Now the two cases above are reduced to: Case 1: If S T > X, then they deliver the shares that they already own. They are going to receive , with present value The financial institution will realize a payoff with present value of The strategy is good in this case [7]. Case 2: If S T < X say S T = 30, the option will not be exercised. The financial institution will lose due to the difference N[S 0 e rt S T ] = 63676, which is present value of the loss. This strategy can bring a big loss, hence is not a good strategy [7]. So neither a naked position nor a covered position provides a satisfactory hedge. If the assumptions underlying the Black-Scholes formula hold, the cost to the financial institution should always be for a perfect hedge using equation (1.2). 2.2 Stop-Loss Strategy Another strategy that a financial institution would employ, is buying the stock as soon as the stock price reaches the strike price (X = 37) and sell it as soon as it drops below X. In other words this strategy would ensure that a financial institution is naked when S t < X and covered when S t > X, for all t [0, T ]. It appears to produce payoffs that are the same as the payoffs on the option. This strategy is more advanced than the naked and covered position since the financial institution will hold the stock only when the option is in-the-money and will be naked when the option is out-of-the-money. Nevertheless the problem of this approach arises from the nature of the Brownian motion. If the price reaches the strike price, say from below (see figure 2.1), a financial institution cannot tell if it will continue to rise (and therefore buy the stock) or if it will decline again (and therefore do nothing). It is obvious that a financial institution has to choose some value ε, and employ the strategy buying at X +ε and selling at X ε. It is clear that this method creates losses equal to 2ε (apart from transaction cost). On the other hand, if a financial institution tries

15 2.3 Delta Hedging 8 S(t) S=SELL B=BUY epsilon X epsilon B S B S B S B S T(Deliver) Time Figure 2.1: Stop-Loss strategy; we buy when price is X + ε from below and sell stock when price is X ε from above to let ε 0, it is easily shown that the number of trades required will tend to infinity (see figure 2.1), making this approach not feasible. Since in none of these strategies do we achieve a satisfactory hedge (see section 2.3.1), then we need more sophisticated schemes than those mentioned so far. These involve calculating measures such as delta, gamma, rho, theta and Vega. 2.3 Delta Hedging The simple way to look at delta hedging is when we have sold a call option. Suppose we observe that when the stock price goes up $1, the call price goes up by $0.50, that is two for one. We could balance out 100 calls with 50 share of stock. Similarly if call price went up $0.20 when the stock price went up $1, this is five for one ratio. To hedge or balance 100 calls, we would only need to sell 20 shares of stock [5]. In mathematical terms we can say this Ratio = change in option price change in stock price Definition 2.1 Delta ( ) is the rate of change of the option price with respect to the price of the underlying asset. It is the slope of the curve that relates the option price to the underlying asset price; thus an increase in stock price leads to an increase in delta. See figure (2.2).

16 2.3 Delta Hedging 9 C = C (2.1) S where, C is the call option price, C is the delta of call option (note we can replace the call with another contract like a put option, futures or a portfolio of options of value (Π)). Suppose that the delta of the call option on a stock is 0.3. This means that when the stock price changes by a small amount, the option price changes by about 30% of that amount. If one manages to create a portfolio that has Π = 0, called delta neutral, then its value will not be affected when the underlying asset price changes (during the next instant). To understand this concept assume in example (2.1) the delta of the call option on a stock to be 0.3. Then the financial institution position could be hedged by buying = 3000 shares. Now the gain (loss) on the call option position would tend to be offset by a loss (gain) on the stock position. For example if after some time the stock price rises by 1, producing a gain of 3000 on the share bought, the call option will go up by = 3000 producing a loss on the call option written. The same argument follows when the stock price will fall to a certain value. Thus the overall delta of a financial institution is zero. It is obvious that the value of will depend on the asset price itself, therefore it will change over time. In order to maintain a delta neutral portfolio, one has to re-balance it in a continuous fashion, a strategy called dynamic delta hedging. If we have two portfolios with values Π 1 and Π 2, then the composite portfolio Π = Π 1 + Π 2 will have delta equal to the sum of the individual deltas Π = Π S = (Π 1 + Π 2 ) = Π1 + Π2 S Now suppose that one starts with a portfolio Π 1, with delta Π1, and wants to take a position in shares, Π 2 = w S S (w S is the number of shares and S is the price of each stock), to make the composite position delta neutral. Clearly, the delta of the position in shares will be Π2 = w S S = w S (since the underlying asset has delta equal to one that is, S = ds = 1). This will imply that ds the composite portfolio has to be constructed by selling Π1 shares, w S = Π1 to make the portfolio delta neutral. Then the delta of the composite portfolio is Π = Π1 + Π2 = 0. We know that the delta of a single derivative in a portfolio is given by = Π S. Then, if a portfolio n Π has w i derivative securities with 1 i n then = w i i where i is the delta of ith derivative. Thus in general if we have N portfolios then the delta of the combined portfolio is given N by = j where j is the delta of jth portfolio. j=1 i=1

17 2.3 Delta Hedging call Option price 1 delta= Stock price, S(t) Figure 2.2: Calculation of delta ( = 0.7) Hedging Performance The question to ask is: why do we use delta hedging rather than a stop-loss strategy? To answer this question we have to measure the performance of these two hedging strategies. The performance measure is the ratio of the standard deviation of the cost of writing the option and hedging it to the Black-Scholes (that is theoretical) price of the option. John Hull [3] did a Monte Carlo simulation based on M = 1, 000 sample paths with the following data; S 0 = 49, X = 50, r = 0.05, σ = 0.02, T = and µ = 0.13 (all symbols have the same meaning as defined in example 2.1). The cost of writing the call option is $ 300, 000 while the theoretical price calculated using Black-Scholes formula (1.2) is $ 240, 000. The result of the simulation gave the following tables (table 2.1 and 2.2). Let the cost caused by applying the mth hedging strategy be κ m m = 1, 2,..., M. Then the sample variance (ϱ 2 ) is given as ϱ 2 = The performance measure is given by 1 (M 1) M κ m 1 M m=1 M j=1 κ j 2. M = ϱ 2 C(S 0, T ) (2.2) where C(S 0, T ) is the Black-Scholes option price (call option in this case).

18 2.4 Delta of European Calls and Puts 11 t (Weeks) M Table 2.1: Performance of the Stop-Loss Strategy t (Weeks) M Table 2.2: Performance of the Delta Hedging They observed that for a stop-loss strategy it is not possible to find a scheme that has performance measure lower than 0.7 regardless of how small t is made (see table 2.1). But for delta hedging the performance measure for five weeks is even better than 0.25 weeks in stop-loss strategy (see table2.2). The performance measure of the Delta hedging (table 2.2) is getting more better when they re-balance the delta of an option frequently (that is in short time interval). Thus Delta hedging provides a better hedge compared to the Stop-loss strategy. Since for a perfect hedge the performance measure (M) must reduce to zero Delta of Forward Contract For any derivative whose price f depends on S, the delta is given by = f S. Consider a long forward contract on a non-dividend-paying stock r(t t) f = S Xe (where r is the riskless interest rate and X the strike price) which has a = 1. Thus the delta of forward contract on one share of non-dividend paying stock is always 1.0. Thus, a short forward contract on one share can be hedged by purchasing one share, whereas a long forward contract on one share can be hedged by shorting one share. This is a hedge and forget scheme (that is, no changes need to be made to the position in the stock during the life of the contract) [3]. 2.4 Delta of European Calls and Puts For a European call option on a non-dividend paying stock (from the Black-Scholes formula ) we have = N(d 1 ) (2.3) where d 1 is defined in equation (1.2) see Appendix A. This means that using delta hedging for a short position in a European call option involves keeping a long position of N(d 1 ) shares at any

19 2.4 Delta of European Calls and Puts 12 given time. Similarly for a long position, it involves maintaining a short position of N(d 1 ) shares at any given time. For a European put option on a non-dividend paying stock (from the Black-Scholes formulae similarly as shown in Appendix A) we can show that = N( d 1 ) = N(d 1 ) 1 (2.4) where d 1 is as given in equation (1.2). The delta of a European put in non-dividend paying stock is negative which implies that, the long position in a put option should be hedged with the long position in the underlying stock. Also the short position in a put option should be hedged with a short position in the underlying stock Delta of Other European Options The following are formulae of deltas of other contracts which are derived in the same way as for the European call option given in appendix A, using Black-Scholes formula corresponding to each case. These equations can be interpreted in a similar way as the two equations above (that is 2.3 and 2.4). For the call option on a stock index paying dividend yield at rate q = e q(t t) N(d 1 ) where d 1 is defined in equation (1.4). For a put options on a stock index For the call option on a currency = e q(t t) [N(d 1 ) 1]. = e r f (T t) N(d 1 ) where r f is the foreign risk-free interest rate and d 1 = ln(s 0/X)+(r r f +σ 2 /2)(T t). For a put options σ (T t) on a currency we have = e r f (T t) [N(d 1 ) 1] For a futures call options = e r(t t) N(d 1 ). this result from differentiation of a future call option C = e r(t t) [F N(d 1 ) XN(d 2 )] with d 1 = ln(f 0/X)+(σ 2 /2)(T t) and F 0 the is spot price. The futures put options for non-dividend σ (T t) paying stock has a delta given by = e r(t t) [N(d 1 ) 1]

20 2.5 Theta Theta Stock price, S(t) Figure 2.3: Theta relation for European call option. We set the strike X = 100, the volatility σ = 20%, T = 0.25 and the interest rate r = 5%. 2.5 Theta Theta is a measure of decay of time value in a portfolio. Mathematically it is given as Thus theta is the sensitivity of the portfolio value with respect to time. Θ = Π t. (2.5) For a European call option on a non-dividend paying stock (see equation 1.2) by differentiation w.r.t t as shown in Appendix B Θ = C t = SN (d 1 )σ 2 T t rxe r(t t) N(d 2 ) (2.6) where d 1 and d 2 is as defined in equation (1.2) and N (d 1 ) = 1 2π e d Theta of the call option is always negative (see figure 2.3), thus as time to maturity decreases with all market variables (such as underliers, implied volatilities, interest rates etc.) remaining constant, the option tends to become less valuable. For a European put option on a non-dividend paying stock, it can be shown (in the same way as Appendix B) that Θ = SN ( d 1 )σ 2 T t + rxe r(t t) N( d 2 ) (2.7)

21 2.6 Gamma Hedging 14 where d 1 and d 2 are defined in equation (1.2) and all other symbols have the usual meaning. For a European call option on a stock index paying dividend at rate q as defined in equation (1.4) by differentiation we can show that Θ = SN (d 1 )σe q(t t) 2 + qsn(d 1 )e q(t t) rxe r(t t) N(d 2 ) (2.8) T t For a European put option on a stock index paying dividend, from equation (1.5) we can show that Θ = SN ( d 1 )σe q(t t) 2 qsn( d 1 )e q(t t) + rxe r(t t) N( d 2 ) (2.9) T t with d 1 and d 2 as defined in equation (1.5). For a European call option on currency with foreign interest rate r f as defined in equation (1.6) by differentiation we can show that Θ = SN (d 1 )σe r f (T t) 2 T t + r f SN(d 1 )e r f (T t) rxe r(t t) N(d 2 ) (2.10) For a European put option on currency, from equation (1.7) we can show that Θ = SN ( d 1 )σe r f (T t) 2 T t r f SN( d 1 )e r f (T t) + rxe r(t t) N( d 2 ) (2.11) with d 1 and d 2 as defined in equation (1.6). Note that all the formulas given above can be derived in the same way as in Appendix B. If theta of a particular contract or portfolio is negative, then its value will decay as time passes and vice-versa. Theta is usually given per days so, because other parameters are measured in years, we have to divide the final result by the number of trading days (252) in a year [3]. The question to answer is, do we need theta hedging? The answer is NO since there is no uncertainty on time since the time is determined in advance from the beginning of the contract. Some traders prefer using theta as just descriptive statistics of the contract or the portfolio they hold. However we can make use of theta as a proxy of gamma. 2.6 Gamma Hedging Delta hedging works well for small stock price movements. For larger movements in stock price the delta does not accurately reflect the option price changes. This leads to another Greek letter called gamma. Gamma (Γ) of a portfolio of options is defined as the rate of change of the portfolio s delta with respect to the price of underlying asset. Mathematically, Γ = S = 2 Π S 2 (2.12)

22 2.6 Gamma Hedging 15 Thus gamma is a measure of the curvature of a graph of option price w.r.t stock price (see figure 2.2). Hence the approximation of option price made by both gamma and delta is better than the one with delta only; since when the delta measures the linear relationship gamma will measure the quadratic changes. If Γ is small, then changes slowly when the asset price changes. Hence the portfolio will be re-balanced infrequently. If Γ is large then delta will change quickly with the change of asset price, hence the portfolio needs to be re-balanced more frequently. Thus gamma tells how often and how much we will have to adjust of a portfolio of derivatives to ensure that we compensate for changes in the underlying. Keeping gamma near 0 helps to reduce the frequency of adjusting the underlying (that is delta hedging), and hence minimises the transaction costs [4] Making a Portfolio Gamma-Neutral If a portfolio is constructed in such a way that the portfolio s gamma is zero, then the portfolio is gamma neutral. We make a portfolio delta neutral by taking a position in the underlying asset, but does not apply for gamma neutrality since the gamma of an asset Γ S = 0 and hence no stock value can contribute to the overall gamma of the portfolio. Thus to make a portfolio gamma neutral we have to take a position in an option because it is a non-linear function of S. Since delta measures the slope of the curve while gamma measures the curvature of the graph of the option price w.r.t the stock price, then when we have a portfolio of option which is both gamma and delta neutral we will have better hedge than using only delta hedging. This strategy is called delta-gamma hedging [2]. Suppose we have a portfolio Π 1 with gamma Γ Π1, and say we use an option with price C. We want to buy w c units of the option. Then the composite portfolio will be Using the linearity of the derivative we get, Π 2 = Π 1 + w c C. Γ Π2 = Γ Π1 + w c Γ c. To achieve gamma neutrality we sell Γ Π 1 Γ c units of option (that is w c = Γ Π 1 Γ c ). Then our portfolio will be Π 2 = Π 1 Γ Π 1 Γ c C. Now to make this portfolio delta neutral, we buy w s shares of the underlying asset (S) (this will not alter the gamma neutrality since Γ s = 0). Then the composite portfolio will be Π 3 = Π 2 + w s S = Π 1 + Γ Π 1 Γ c C + w s S. The delta of this portfolio is Π3 = Π1 Γ Π 1 Γ c C + w s.

23 2.6 Gamma Hedging 16 To make this position delta neutral we sell Π1 Γ Π 1 Γ c C shares (that is w s = Π1 + Γ Π 1 Γ c C ). Hence the value of composite portfolio will be Π = Π 1 Γ Π 1 Γ C C ( Π1 Γ ) Π 1 C S. Γ C Then Π = Γ Π = 0, hence the portfolio is delta - gamma hedged [7] Calculation of Gamma The gamma of a European call option on non-dividend-paying stock is Γ = N (d 1 ) Sσ T t (2.13) and this can be derived simply by differentiating equation (2.3) w.r.t S. By using the call-put parity given in Theorem (1.1), it is easy to show by twice differentiation w.r.t S that, the gamma of a European put (Γ P ) and call (Γ C ) options are equal. That is Γ C = Γ P. Thus the graph of gamma for a European put or call option with respect to stock price (S) is always concave upward and positive. It picks the maximum value when the option is at-the-money (that is when S = X) see figure (2.4). In a similar way it is easy to show that for a European call or put option on a stock index paying a continuous dividend yield at rate q Γ = N (d 1 ) Sσ T t e q(t t) (2.14) where all symbols have the usual meaning. For the European call or put option on a currency with foreign interest rate r f it is Γ = N (d 1 ) Sσ T t e r f (T t) (2.15) where N(d 1 ) = 1 2π e d and d 1 is as defined in equation (1.6). Note the gamma of forward contract is zero(0). All of these formulae are derived in the same way as the delta formula shown in Appendix A, using the Black-Scholes formula corresponding to each case Relationship between, Θ & Γ From the Black-Scholes pricing differential equation (1.1), the value Π of a portfolio of a single derivative dependent on a non-dividend paying stock is given by [3] Π t σ2 S 2 2 Π Π + rs = rπ. (2.16) S2 S

24 2.6 Gamma Hedging Gamma Stock price, S(t) Figure 2.4: Gamma of European call or Put option. We set the strike X = 98, the volatility σ = 20%, T = 0.25 and the interest rate r = 5%. Using the definitions of = Π S, Θ = Π t and Γ = 2 Π S 2 we have Θ + rs σ2 S 2 Γ = rπ. (2.17) It is easy to show that equation ( 2.17) satisfies individual contract (like European call, or put options). For example consider the case of a single European call option on a non-dividend-paying stock we have = N(d 1 ), Θ = SN (d 1 )σ 2 T 1 rxe (T t) N(d 2 ) and Γ = N (d 1 ) Sσ. Thus on substitution of these T t relations in equation (2.17) we have SN (d 1 )σ 2 T t rxe (T t) N(d 2 ) + rsn(d 1 ) σ2 S 2 N (d 1 ) Sσ T t = rc where, C = N(d 1 ) Xe r(t t) N(d 2 ). On simplification we have rsn(d 1 ) rxe r(t t) N(d 2 ) + SσN (d 1 ) 2 (1 1) = rc T t ( ) thus r SN(d 1 ) Xe r(t t) N(d 2 ) = rc. For delta neutral ( = 0), equation (2.17) reduces to Θ σ2 S 2 Γ = rπ. (2.18)

25 2.7 Rho (ρ) 18 This shows that when theta is large and positive, gamma tends to be large and negative and vice-versa; this explains why theta can be regarded as a proxy for gamma. 2.7 Rho (ρ) This is one of the factors used by traders to measure markets risk exposure in derivative portfolios as the result of change of the interest rate. We can define rho as the rate of change of the portfolio value with respect to the interest rate. If Π is the given portfolio value we have in mathematical form ρ = Π r. (2.19) Rho is the measure of how a portfolio is sensitive to change in interest rates. For example, if rho is 14.2, it means that for every percentage point (that is 0.01) increase in the interest rate, the value of the option increases by 14.2%. We can calculate the rho as other Greek measures discussed so far by simply taking differentiation of a value of any contract with respect to interest rates r. It is easy to show that we have the same formula for a European call option on a non-dividend-paying stock, dividend-paying stock and on currency which is given by ρ = X(T t)e r(t t) N(d 2 ) (2.20) where all symbols and the value for d 2 is given in section (1.5) (note that the value of d 2 is considered in the respective contract). Similarly for a European put option on a non-dividend paying stock, dividend-paying-stock and on currency has the same formula, which is given by ρ = X(T t)e r(t t) N( d 2 ) (2.21) where all symbols and the value for d 2 are given in section (1.5) (note that, d 2 is considered in the appropriate contract). Rho is always positive for European call options and negative for European put options (refer to equation 2.20 and 2.21). Therefore as the interest rates increases, call option values will rise while the put option value will fall. Since in a currency option we have two interest rates, one for local and other for foreign, a European Call and put option on currency for local interest rates is the same as in equation (2.20) and (2.21) respectively. To get a European call or put option for foreign interest rate (r f ) replace r with r f in equation (2.20) and (2.21) respectively. Also like theta, rho is not commonly used for hedging. However it gives a statistical account since it shows how sensitive an option is w.r.t the interest rates.

26 2.8 Vega (V) Rho 10 r=0.5 r=0.1 r= Stock price, S(t) Figure 2.5: The behaviour of rho of European call option with different values for r. 2.8 Vega (V) Vega measures the relationship between the stock volatility and the option value. We can define vega (V) of the portfolio as the rate of change of the value of the portfolio with respect to volatility (σ). Volatility is a measure of the uncertainty of the return realized on an asset. Mathematically vega is given as V = Π σ. (2.22) If the value of vega is high, the option s value is very sensitive to small changes in volatility, whereas if the value of vega is low volatility changes have relatively little impact on the option value. Moreover, the higher the volatility the greater is the probability that the option will end up with a higher price. A position in the underlying asset or in a forward contracts has zero vega since they are both independent of volatility. Therefore one has to rely on nonlinear contracts (such as options) to make a portfolio vega neutral. To achieve a gamma-vega hedge, one will have to use two different derivative securities. This is due to the fact that generally speaking, for a given derivative the values of gamma and vega will be different [7]. Say that we hold a portfolio with value Π 1 which has a given delta, Π1, gamma, Γ Π1, and vega V Π1. Our goal is to find the position we have to take in two different derivatives of the underlying

27 2.8 Vega (V) 20 asset, in order to achieve vega-gamma neutrality. Say that we use two options with prices C 1 and C 2, with known deltas, C1 and C2, gammas, Γ C1 and Γ C2, and vegas, V C1 and V C2. We also use the underlying asset, which has price S to achieve delta neutrality (we know S = 1 and Γ S = V S = 0). Let us make this portfolio delta-gamma-vega neutral [7]. Suppose that we buy units of a derivative securities given by Z C1 and Z C2. The composite portfolio will be Π 2 = Π 1 + Z C1 C 1 + Z C2 C 2. (2.23) The gamma of this portfolio is Γ Π2 = Γ Π1 + Z C1 Γ C1 + Z C2 Γ C2. (2.24) To have gamma neutral we need Γ Π2 = 0. The vega of portfolio in equation (2.23) is V Π2 = V Π1 + Z C1 V C1 + Z C2 V C2. (2.25) To have vega neutral we seek V Π2 = 0. Thus we must solve for Z C1 and Z C2 in equations (2.24 and 2.25), such that { V Π1 + Z C1 V C1 + Z C2 V C2 = 0 Γ Π1 + Z C1 Γ C1 + Z C2 Γ C2 = 0 On solving equation (2.26) we have (2.26) Z C1 = Γ Π 1 V C2 V Π1 Γ C2 Γ C1 V C2 V C1 Γ C2 Z C2 = Γ Π 1 V C1 V Π1 Γ C1 Γ C2 V C1 V C2 Γ C1 Now to make the portfolio in equation (2.23) delta neutral we must take a position in the underlying asset (note that the asset do not have an effect on vega or gamma neutrality). Suppose we buy W S shares, the composite portfolio will be The delta of this portfolio is Π 3 = Π 1 + Z C1 C 1 + Z C2 C 2 + W S S. (2.27) Π3 = Π1 + Z C1 C1 + Z C2 C2 + W S. We need Π3 = 0 to achieve delta neutrality. Hence we buy W S = ( Π1 + Z C1 C1 + Z C2 C2 ) shares. Thus equation (2.27) on substitution of the values of Z C1, Z C2 and W S this portfolio Π 3 will be delta-gamma-vega neutral [7].

28 2.8 Vega (V) T=1 year 30 Vega T=6 Months T=3 Months Stock price, S(t) Figure 2.6: The behaviour of vega of a European call or put option with different values for time to maturity (T ) Calculation of Vega We adopt similar techniques to those used in the calculation of the previous Greek letters for calculating the vega of various contracts. Here we differentiate the value of a contract w.r.t the volatility (σ). For a European call or put option on non-dividend paying stock, it can be easily shown from the Black-Scholes formulae that vega is given by where d 1 is defined as in equation (1.2) and N (d 1 ) = 1 2π e d 1 2 /2. V = S T tn (d 1 ) (2.28) For a European call or put option on a stock, or stock index paying a continuous dividend yield at a rate q, V = S T tn (d 1 )e q(t t) (2.29) where d 1 is defined in equation (1.4). If we replace q with a foreign interest rate r f, equation (2.29), we will get vega for a European put or call option on a currency.

29 2.9 Scenario Analysis Scenario Analysis In addition to the use of the Greek letters in measuring the sensitivity of the portfolio or contract, most traders prefer to do scenario analysis. This is an analysis which involves calculating the possible gains or losses on the portfolio over the specified period under a variety of scenarios which may lead to changes in the underlying determinants of portfolio value (such as interest rates, exchange rates, stock prices, commodity prices, etc.) In most cases the time is chosen depending on the liquidity of the instrument (such as derivative, commodity, an index etc.). A scenario can be chosen by management or can be generated by a model. Scenario analysis is an important technique in risk management, because it helps firms and especially financial institutions to forecast possible market trends and therefore to take the most appropriate position in the market. That is, it helps the financial institution to know beforehand what scenario can lead to a loss or a gain, hence providing valuable information for decisions concerning the portfolio.

30 Chapter 3 Portfolio Insurance 3.1 Preliminary Portfolio managers often wish to insure themselves against the value of their portfolios falling below a certain value. One way of achieving this is through portfolio insurance. Portfolio insurance is the use of options and futures theories to guide trading so as to set a floor below which the value of an investment portfolio will not fall [1]. Note that the positive yields are diminished by an insurance premium (price of the option). This eliminates the arbitrage opportunities. Hyne E. Leland and Mark Rubinstein of the University of Carlifonia at Berkeley in 1976 came up with this idea of portfolio insurance. Black, Scholes and Merton also contributed towards the development of portfolio insurance by using the replicating portfolio strategy [1]. In principle the value of portfolio can be insured by buying a put option with strike price equal to the desired floor. As an alternative portfolio manager can insure the portfolio by creating an option synthetically. This concept of synthetic option was invented by Leland, he observed that the desirable put options to provide a certain insurance may not be available. For example, managers with large funds the market does not always have the liquidity to absorb the trades they require. Also the fund managers often require strike prices and exercises dates that are different from those available in the exchange-traded option market [3]. In April 1982, the Chicago Mercantile Exchange launched a futures contract on the Standard and Poor s (S&P) 500 index (S&P 500 index is a market indicator which consists of a basket of 500 stocks, this is the benchmark established to judge overall U.S market performance). This benchmark is most widely used by portfolio managers. The introduction of this index futures provided a simpler way of implementing a portfolio insurance.

31 3.2 Using Index Options for Portfolio Insurance Using Index Options for Portfolio Insurance A portfolio manager can use the index options to limit his downside risk. If a portfolio manager holds a well-diversified portfolio it means that the portfolio she held mirrors the market. portfolio has beta ( β ) equal to one and the dividend yield in the market (or stock index) is equal to that of the portfolio. We can define beta as the factor showing the relationship between the expected return on the portfolio of stocks and the return in the market. When β = 1.0, the return on the portfolio tends to mirror the return on the market. When β = 2.0, the return on the portfolio tends to be twice the return on the market. This implies the portfolio is twice as volatile as the stock underlying the futures contracts and the position in futures should be twice as great [3]. Suppose a fund manager has a well-diversified portfolio, which mirrors the S&P 500 index (β = 1.0). Suppose one futures contract traded on S&P 500 index is on 100 times the index. Given that, the value of index is S I, then the value of the portfolio is protected against the possibility of falling below X if for each $ 100S I in the portfolio, the portfolio manager buys one put option with strike price X. If the portfolio manager holds a portfolio worth $ 400, 000 and the value of the index is S I = $ 1, 000, then the portfolio is worth 400 S I. The portfolio manager is protected from the value of her portfolio falling below $ 384, 000 in three months by buying four put options with strike price $ 960. Suppose in three months the value of the index S I = $ 900, then the portfolio will be worth 400 $ 900 = $ 360, 000. The payoff from the put option will be 4 ($ 960 $ 900) 100 = $ 24, 000, which makes the total value of the portfolio to be $ 360, $ 24, 000 = $ 384, 000 which is the insured value. When the portfolio s beta is not 1.0, we use the Capital Asset Pricing Model (CAPM) which states the expected excess return of a portfolio (r Π ) over the risk-free interest rate (r) equals beta (β) times the excess return of market index (r I ) over the risk-free interest rate (that is, r Π r = β(r I r)) [3]. Consider the case when β = 2.0 and suppose that the current value of the portfolio is $ 1 million with r = 12% per annum (or 3% per three months) and the dividend yield (q) of both index and the portfolio be 4% per annum (or 1% per three months). Further more suppose the current value of the index is $ 1, 000. Let us see what will be the expected value of the portfolio in three months if the value of the index in three months is $ 960. The return from the change in index is 40 or 4% per three months. The total return from 1000 the index will be (r I )= q + ( 4) = 3% per three months. Then the expected return from the portfolio (r Π ) can be obtained by making use of CAPM r Π r = β(r I r) r Π = r + β(r I r) = 9% per three months. But since the dividend yield from the portfolio is 1% per three months, then the increase in the value of the portfolio is 9 1 = 10%. Thus the expected value of the portfolio in three months it will be $ 1, 000, $ 1, 000, 000 = $ 0.90 millions. 100 This

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