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Hedging of Financial Derivatives and Portfolio

Insurance

Gasper Godson Mwanga

African Institute for Mathematical Sciences 6, Melrose Road, 7945 Muizenberg, Cape Town

South Africa.

e-mail: gasper@aims.ac.za, gmlangwe@yahoo.co.uk

Supervisor : Prof. J. C. Ndogmo Department of Mathematical

University of Western Cape

Private Bag X17, 7535 Bellville, Capetown South Africa

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Contents

List of Figures iii

Acknowledgements iv 1 Introduction 1 1.1 Background . . . 1 1.2 Markets . . . 2 1.3 Derivative security . . . 3 1.3.1 Option . . . 3

1.3.2 Forwards and Futures . . . 4

1.3.3 Swaps . . . 4

1.4 Important Formulae . . . 4

2 The Greek Letters 6 2.1 Naked and Covered Position . . . 6

2.2 Stop-Loss Strategy . . . 7

2.3 Delta Hedging . . . 8

2.3.1 Hedging Performance . . . 10

2.3.2 Delta of Forward Contract . . . 11

2.4 Delta of European Calls and Puts . . . 11

2.4.1 Delta of Other European Options . . . 12

2.5 Theta . . . 13

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CONTENTS ii

2.6.1 Making a Portfolio Gamma-Neutral . . . 15

2.6.2 Calculation of Gamma . . . 16

2.6.3 Relationship between ∆, Θ & Γ . . . 16

2.7 Rho (ρ) . . . 18 2.8 Vega (V) . . . 19 2.8.1 Calculation of Vega . . . 21 2.9 Scenario Analysis . . . 22 3 Portfolio Insurance 23 3.1 Preliminary . . . 23

3.2 Using Index Options for Portfolio Insurance . . . 24

3.3 Creating Options Synthetically . . . 25

3.3.1 Use of the Trading of Portfolio . . . 25

3.3.2 Use of Index Option . . . 26

Conclusion 28

A Determination of Delta for a European call Option 29

B Determination of Theta for a European call Option 31

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List of Figures

2.1 Stop-Loss strategy . . . 8

2.2 Calculation of delta . . . 10

2.3 Theta of European call option . . . 13

2.4 Gamma of European options . . . 17

2.5 Rho of European call option . . . 19

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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.

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Abstract

Risk management is an important issue in finance because of the considerable impact of the volatil-ity 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 prac-tices. 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 manage-ment 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.

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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 Th´eorie de la Sp´eculation. 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 =

q

2 π.σ√kδt,

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].

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1.2 Markets 2

1.2

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 S0, S1, . . . Sd, where we assume that the first

is riskless, so that its price S0(0) determines its price S0(t) at future time t with certainty. Then,

other assets are risky, so that their prices Si(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 θi assets is given by Vt(θ) = d X i=0 θiSi(t).

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

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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:

1.3.1 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 ST at expiry date T and with strike price X

will be exercised only if ST > 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 VT = max{(ST − X), 0}. For t ∈ [0, T ], if St > X the call option is said to be in-the-money; when

St = X, the call option is said to be at-the-money; and finally when St < X, the call option is

said to be out-of-the-money. For the put option VT = max{(X − ST), 0} and the inequalities are

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1.4 Important Formulae 4

1.3.2 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.

1.3.3 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 + 1 2σ 2S2∂2V ∂S2 + rS ∂V ∂S − rV = 0 (1.1)

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(d1) − Xe−r(T −t)N(d2) with d1 = ln(S/X) + (r + σ2/2)(T − t) σ√T − t d2= ln(S/X) + (r − σ 2/2)(T − t) σ√T − t = d1− σ √ T − t (1.2)

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1.4 Important Formulae 5 where N(d1) is the cumulative normal distribution of d1, 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(−d1) (1.3)

with d1 and d2 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

C(S, t) − P (S, t) = S − Xe−r(T −t)

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(d2) with, d1= ln(S/X) + (r − q + σ 2/2)(T − t) σ√T − t d2= ln(S/X) + (r − q − σ 2/2)(T − t) σ√T − t = d1− σ √ 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(−d2) (1.5)

with d1 and d2 as in equation (1.4).

The formula for a European call option C on currency with risk-free interest rate of foreign currency rf is given by C = Se−rf(T −t)N(d 1) − Xe−r(T −t)N(d2) with, d1 = ln(S/X) + (r − rf + σ2/2)(T − t) σ√T − t d2 = ln(S/X) + (r − rf − σ 2/2)(T − t) σ√T − t = d1− σ √ T − t. (1.6)

The formula for a European put option P on a currency is given by P = −Se−rf(T −t)N(−d

1) + Xe−r(T −t)N(−d2) (1.7)

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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 £ 15000 a European call option on N = 10000 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 S0= £ 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 = 0.24658 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:

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2.2 Stop-Loss Strategy 7

Case 1: The price is below the strike price (ST < X). Then, the call will not be exercised and the

financial institution will make the profit of £ 15000. In this case the strategy works. Case 2: The price exceeds the strike price (ST > X) say ST = £ 40. Then, the call will be

exercised, and the financial institution will have to buy shares at £ 40 in order to de-liver the stock at X to fulfil the contract. The financial institution will incur a loss of N (ST − X) = £ 30000, with present value e−rTN (ST − X) = £ 29632. This loss is

higher than the £ 15000 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 10000 shares as soon as the option has been sold. For these shares they will have to pay £ 360000; the financial institution starts with a debt of £ 210000. Now the two cases above are reduced to:

Case 1: If ST > X, then they deliver the shares that they already own. They are going to receive

£ 370000, with present value £ 365470.The financial institution will realize a payoff with present value of £ 20470. The strategy is good in this case [7].

Case 2: If ST < X say ST = £ 30, the option will not be exercised. The financial institution will

lose due to the difference N [S0− e−rTST] = £ 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 £ 11000 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 St < X and covered when

St > 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

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2.3 Delta Hedging 8 X epsilon epsilon S=SELL B=BUY B S B S B S B S T(Deliver) Time S(t)

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).

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2.3 Delta Hedging 9

∆C =

∂C

∂S (2.1)

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 0.3 × 10000 = 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 0.3 × 10000 × £ 1 = £ 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 Π1 and Π2, then the composite portfolio Π = Π1+ Π2 will

have delta equal to the sum of the individual deltas ∆Π=

∂Π ∂S =

∂(Π1+ Π2)

∂S = ∆Π1+ ∆Π2

Now suppose that one starts with a portfolio Π1, with delta ∆Π1, and wants to take a position

in shares, Π2 = wSS (wS 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 = wS∆S=

wS (since the underlying asset has delta equal to one that is, ∆S=

dS

dS = 1). This will imply that the composite portfolio has to be constructed by selling ∆Π1 shares, wS = −∆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 Π has wi derivative securities with 1 ≤ i ≤ n then ∆ =

n

X

i=1

wi∆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 by ∆ =

N

X

j=1

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2.3 Delta Hedging 10 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 Option price Stock price, S(t) r=0.05 call delta=0.7

Figure 2.2: Calculation of delta (∆ = 0.7)

2.3.1 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; S0 = 49, X = 50, r = 0.05, σ = 0.02,

T = 0.3846 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 κm m = 1, 2, . . . , M . Then the

sample variance (%2) is given as

%2 = 1 (M − 1) M X m=1  κm− 1 M M X j=1 κj   2 .

The performance measure is given by

M= p%

2

C(S0, T )

(2.2) where C(S0, T ) is the Black-Scholes option price (call option in this case).

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2.4 Delta of European Calls and Puts 11 ∆t (Weeks) 5 4 2 1 0.5 0.25

M 1.02 0.93 0.82 0.77 0.76 0.76 Table 2.1: Performance of the Stop-Loss Strategy ∆t (Weeks) 5 4 2 1 0.5 0.25

M 0.43 0.0.39 0.26 0.19 0.14 0.09 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.

2.3.2 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 f = S − Xe−r(T −t)

(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(d1) (2.3)

where d1 is defined in equation (1.2) see Appendix A. This means that using delta hedging for a

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2.4 Delta of European Calls and Puts 12 given time. Similarly for a long position, it involves maintaining a short position of N(d1) shares

at any given time.

For a European put option on a non-dividend paying stock (from the Black-Scholes formulae simi-larly as shown in Appendix A) we can show that

∆ = −N(−d1) = N(d1) − 1 (2.4)

where d1 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.

2.4.1 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 d1 is defined in equation (1.4). For a put options on a stock index

∆ = e−q(T −t)[N(d 1) − 1].

For the call option on a currency

∆ = e−rf(T −t)N(d

1)

where rf is the foreign risk-free interest rate and d1 = ln(S

0/X)+(r−rf+σ2/2)(T −t)

σ√(T −t) . For a put options

on a currency we have

∆ = e−rf(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(d2)]

with d1 = ln(F

0/X)+(σ2/2)(T −t)

σ√(T −t) and F0 the is spot price. The futures put options for non-dividend

paying stock has a delta given by

∆ = e−r(T −t)[N(d 1) − 1]

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2.5 Theta 13 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 80 90 100 110 120 130 140 150 160 170 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 Θ = ∂Π

∂t . (2.5)

Thus theta is the sensitivity of the portfolio value with respect to time.

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 = − SN0(d 1)σ 2√T − t − rXe −r(T −t)N(d 2) (2.6)

where d1 and d2 is as defined in equation (1.2) and N0(d1) = √1e−

d12

2 . 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 = −SN0(−d1)σ

2√T − t + rXe

−r(T −t)N(−d

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2.6 Gamma Hedging 14 where d1 and d2 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 = −SN0(d1)σe−q(T −t)

2√T − t + qSN(d1)e

−q(T −t)− rXe−r(T −t)N(d

2) (2.8)

For a European put option on a stock index paying dividend, from equation (1.5) we can show that Θ = −SN 0(−d 1)σe−q(T −t) 2√T − t − qSN(−d1)e −q(T −t)+ rXe−r(T −t)N(−d 2) (2.9)

with d1 and d2 as defined in equation (1.5).

For a European call option on currency with foreign interest rate rf as defined in equation (1.6) by

differentiation we can show that

Θ = −SN0(d1)σe−rf(T −t)

2√T − t + rfSN(d1)e

−rf(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

0(−d

1)σe−rf(T −t)

2√T − t − rfSN(−d1)e

−rf(T −t)+ rXe−r(T −t)N(−d2) (2.11) with d1 and d2 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 =

∂2Π

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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].

2.6.1 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 ΓS= 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 Π1 with gamma ΓΠ1, and say we use an option with price C. We want

to buy wc units of the option. Then the composite portfolio will be

Π2= Π1+ wcC.

Using the linearity of the derivative we get,

ΓΠ2 = ΓΠ1 + wcΓc.

To achieve gamma neutrality we sell ΓΠ1

Γc units of option (that is wc = −

ΓΠ1

Γc ). Then our portfolio

will be

Π2= Π1−

ΓΠ1

Γc

C.

Now to make this portfolio delta neutral, we buy ws shares of the underlying asset (S) (this will

not alter the gamma neutrality since Γs= 0). Then the composite portfolio will be

Π3 = Π2+ wsS = Π1+ −ΓΠ1

Γc

C + wsS.

The delta of this portfolio is

∆Π3 = ∆Π1 −

ΓΠ1

Γc

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2.6 Gamma Hedging 16 To make this position delta neutral we sell ∆Π1 −

ΓΠ1

Γc ∆C shares (that is ws = −∆Π1 +

ΓΠ1

Γc ∆C).

Hence the value of composite portfolio will be Π = Π1− ΓΠ1 ΓC C −  ∆Π1− ΓΠ1 ΓC ∆C  S. Then ∆Π= ΓΠ= 0, hence the portfolio is delta - gamma hedged [7].

2.6.2 Calculation of Gamma

The gamma of a European call option on non-dividend-paying stock is Γ = N0(d1)

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 (ΓP) and call (ΓC) options are equal. That is

ΓC = ΓP.

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

0(d 1)

Sσ√T − te

−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 rf it is

Γ = N0(d1) Sσ√T − te −rf(T −t) (2.15) where N(d1) = √1 2πe −d1 2 2 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.

2.6.3 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 + 1 2σ 2S2∂2Π ∂S2 + rS ∂Π ∂S = rΠ. (2.16)

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2.6 Gamma Hedging 17 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 70 80 90 100 110 120 130 140 150 160 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 = ∂∂S2Π2 we have

Θ + rS∆ + 1 2σ

2S2Γ = 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(d1) , Θ = −SN

0(d1

−2√T −1 − rXe−(T −t)N(d2) and Γ =

N0(d1)

Sσ√T −t. Thus on substitution of these

relations in equation (2.17) we have −SN0(d1)σ 2√T − t − rXe −(T −t)N(d 2) + rSN(d1) + 1 2σ 2S2 N0(d1) Sσ√T − t = rC where, C = N(d1) − Xe−r(T −t)N(d2).

On simplification we have rSN(d1) − rXe−r(T −t)N(d2) +

SσN0(d 1) 2√T − t(1 − 1) = rC thus rSN(d1) − Xe−r(T −t)N(d2)  = rC. For delta neutral (∆ = 0), equation (2.17) reduces to

Θ +1 2σ

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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 d2is given in section (1.5) (note that the value of d2is 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(−d2) (2.21) where all symbols and the value for d2 are given in section (1.5) (note that, d2 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 (rf) replace r with rf

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.

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2.8 Vega (V) 19 0 5 10 15 20 25 60 70 80 90 100 110 120 130 140 150 160 Rho Stock price, S(t) r=0.05 r=0.1 r=0.5

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 Π1 which has a given delta, ∆Π1, gamma, ΓΠ1, and vega

VΠ

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2.8 Vega (V) 20 asset, in order to achieve vega-gamma neutrality. Say that we use two options with prices C1 and

C2, with known deltas, ∆C1 and ∆C2, gammas, ΓC1 and ΓC2, and vegas, VC1 and VC2. We also

use the underlying asset, which has price S to achieve delta neutrality (we know ∆S = 1 and

ΓS = VS = 0). Let us make this portfolio delta-gamma-vega neutral [7].

Suppose that we buy units of a derivative securities given by ZC1 and ZC2. The composite portfolio

will be

Π2= Π1+ ZC1C1+ ZC2C2. (2.23)

The gamma of this portfolio is

ΓΠ2 = ΓΠ1 + ZC1ΓC1 + ZC2ΓC2. (2.24)

To have gamma neutral we need ΓΠ2 = 0.

The vega of portfolio in equation (2.23) is VΠ

2 = VΠ1 + ZC1VC1 + ZC2VC2. (2.25)

To have vega neutral we seek VΠ2 = 0. Thus we must solve for ZC1 and ZC2 in equations

(2.24 and 2.25), such that

( VΠ

1 + ZC1VC1 + ZC2VC2 = 0

ΓΠ1+ ZC1ΓC1 + ZC2ΓC2 = 0

(2.26) On solving equation (2.26) we have

ZC1 = − ΓΠ1VC2− VΠ1ΓC2 ΓC1VC2− VC1ΓC2 ZC2 = − ΓΠ1VC1− VΠ1ΓC1 ΓC2VC1− VC2ΓC1

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 WS

shares, the composite portfolio will be

Π3 = Π1+ ZC1C1+ ZC2C2+ WSS. (2.27)

The delta of this portfolio is

∆Π3 = ∆Π1+ ZC1∆C1+ ZC2∆C2 + WS.

We need ∆Π3 = 0 to achieve delta neutrality. Hence we buy WS = −(∆Π1 + ZC1∆C1 + ZC2∆C2)

shares. Thus equation (2.27) on substitution of the values of ZC1, ZC2 and WS this portfolio Π3

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2.8 Vega (V) 21 0 5 10 15 20 25 30 35 40 60 70 80 90 100 110 120 130 140 150 Vega Stock price, S(t) T=1 year T=6 Months T=3 Months

Figure 2.6: The behaviour of vega of a European call or put option with different values for time to maturity (T ).

2.8.1 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

V= ST − tN0(d

1) (2.28)

where d1 is defined as in equation (1.2) and N0(d1) = √1 2πe

−d12/2.

For a European call or put option on a stock, or stock index paying a continuous dividend yield at a rate q,

V= ST − tN0(d

1)e−q(T −t) (2.29)

where d1 is defined in equation (1.4). If we replace q with a foreign interest rate rf, equation (2.29),

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2.9 Scenario Analysis 22

2.9

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 es-pecially 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.

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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. Lelandand 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.

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3.2 Using Index Options for Portfolio Insurance 24

3.2

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. This 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 SI, then the value of the portfolio is protected against the possibility of falling

below X if for each $ 100SI 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 SI = $ 1, 000, then the portfolio is worth 400×SI. 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 SI = $ 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, 000 + $ 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 (rΠ) over the risk-free interest rate (r) equals beta (β) times

the excess return of market index (rI) over the risk-free interest rate (that is, rΠ−r = β(rI−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

1000 or − 4% per three months. The total return from the index will be (rI)= q + (−4) = −3% per three months. Then the expected return from the

portfolio (rΠ) can be obtained by making use of CAPM

rΠ− r = β(rI− r) → rΠ= r + β(rI− 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, 000 + −10

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3.3 Creating Options Synthetically 25 Now in this example, for a portfolio manager to ensure that the value of the portfolio will not fall below $ 0.90 millions in three months, she has to buy a three months put option with strike price X = $ 960. Because the value of beta is 2 she has to hold 2 put option contracts for every $ 100SI

in the portfolio. Since the value of the portfolio is $ 1million and the value of the index is $ 1000, the portfolio manager must buy 20 put option contracts. For illustration we consider if the value of the index falls to $ 900 in three months, then the expected value of the portfolio can be shown to be $ 780, 000. The payoff from the put options will be 20 × ($ 960 − $ 900) × 100 = $ 120, 000, this makes the total value of the portfolio to be $ 120, 000 + $ 780, 000 = $ 0.90 millions which is the insured value.

3.3

Creating Options Synthetically

As an alternative to buying an option is to create it synthetically. This strategy involves taking a varying position in the underlying asset (or futures on the underlying asset) so that the delta of the position is maintained equals the delta of the required option [3]. Creating an option synthetically is the opposite of hedging it (that is we take the position that match the ∆ or Γ or V). The synthetic option can be created either by a trading portfolio or by using index futures.

3.3.1 Use of the Trading of Portfolio

The concept here is a simple strategy of portfolio insurance, since at any given time the fund is divided into two parts: the stock portfolio in which the insurance is required and the riskless asset (a riskless asset is an asset whose future return is known with certainty). We consider the case of a put option. The delta of a European put option on a dividend paying stock at rate q ( see section 2.3.1) is given by

∆ = e−q(T −t)[N(d

1) − 1] (3.1)

where d1 is defined in equation (1.4) and other symbols have the usual meaning.

To create the put option synthetically, the formula of a delta of a put option given in equation (3.1) should be

∆s = e−q(T −t)[1 − N(d1)] (3.2)

and the symbols have the same meaning as in equation (3.1).

Thus, at any given time the portfolio manager must sell N × ∆s shares in the stock and invest

the proceeds in the riskless asset (where N is the number of stocks in the portfolio). Suppose that ∆s= 0.40, then 40% of the portfolio must be sold initially; if after some time the value of original

portfolio decreases, and for example if delta of the portfolio changes to ∆s = 0.45, a further 5%

of the portfolio should be sold and used to purchase riskless assets. When the ∆s becomes less

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3.3 Creating Options Synthetically 26 and the stocks should be purchased. This strategy has a cost, since we buy the stocks when the price is high and sell when the price is low.

3.3.2 Use of Index Option

Using index futures to create portfolio insurance is more preferable than using the underlying stocks, provided that the index futures market is sufficiently liquid to handle the required trade due to the following reasons: it is easier to deliver the futures contracts on a stock than the stock itself since most of the futures contract are normally settled by cash. This makes the futures contract more feasible particularly to those investors with limited capital, as they may find it difficult to come up with the funds to buy the required stocks when a contract is exercised.

Consider a stock with riskless interest rate r, futures price F0 at time zero (now) and the maturity

time T∗ (which is not necessarily equal to the time to maturity (T ) of an option). If S

0 is the spot

price we have the relation

F0 = S0erT

. (3.3)

Thus if the stock price increases by ∆S, the futures price increases by ∆SerT∗

which implies that the delta of the futures contract is erT∗

. This is the same as to say an e−rT∗

increase in one futures contract corresponds to ∆S increase of a stock price. This lead to the relation HF = e−rT

HS

where, HS is the stock position at time zero for delta hedging and HF is the required position for

a futures contract at time zero. If we have the stock index which pays dividend yield at rate q this relation will be

HF = e−(r−q)T

HS (3.4)

and thus the delta of the futures contract is e(r−q)T∗.

Now, in the use of the index futures for portfolio insurance, the portfolio manager keeps the stock portfolio intact and short the index futures contracts. We can see that the futures contract shorted as a proportion of the dollar value of the portfolio is obtained by combining equations (3.2) and (3.4) to give e−qTe−(r−q)T∗ [1 − N(d1)] = eq(T ∗−T ) e−rT∗ [1 − N(d1)] (3.5)

where t = 0 as discussed in definition of the futures position above and HS = e−qT[1 − N(d1)].

If a portfolio is worth K1 times the index and each future contract is on K2 times the index ( in

the example used in section (3.2) K1 = 400 and K2 = 100), then the number of futures contracts

shorted should be eq(T∗−T )e−rT∗ [1 − N(d1)] K1 K2 .

For example: Consider a portfolio manager holding a portfolio worth $ 400 million with beta (β)equals to one who wants to create a three months put option (i.e. T = 0.25 years) with strike price X = $ 384 million. Given that the volatility of the index (σ) is 25% per annum, the risk free

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3.3 Creating Options Synthetically 27 interest rate (r) is 9% and the dividend yield on the index (q) is 3% per annum. Then, when we use trades in the portfolio the delta of the put option will be

∆ = e−qT[N(d

1) − 1] = −0.446.

Thus 44.6% of the portfolio should be sold initially. If we use six months (or T∗ = 0.5) Nikkei 225

Stock Average, then K2 = 225 and K1= 400, 000 for the value of the index equals to $1, 000. Then

the number of futures contract shorted should be eq(T∗−T )e−rT∗

[1 − N(d1)]

K1

K2 ≈ 770.

When Beta is different from one, we use the similar method as discussed in section (3.2) to find the strike price which will be equal to the value to reach the floor (that is the expected level of the market index when the portfolio’s value reaches its insured value).

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Conclusion

This essay tries to exploit the concept of hedging of a financial derivative. We based our discussion on three kinds of derivatives which are: options, futures and forwards. We looked at rudimentary strategies for hedging called the naked and the covered positions which we saw do not give a satisfactory hedge. This lead to another strategy called stop-loss, which seems to be a good idea but we still did not have a satisfactory hedge. Furthermore we looked at the sensitivity of the portfolio w.r.t asset prices, interest rates, volatility and time. Depending on the parameters these sensitivities are named using the Greek letters delta, theta, gamma, rho and vega (or Kappa). We showed how we can use these Greek letters to hedge the financial derivatives. Delta hedging provides the better hedging performance compared to the stop-loss strategy. We showed that if hedging is done dynamically using delta, we get better performance which is closely to the theoretical price. We showed that delta is a slope of an option price w.r.t stock price so, cannot capture the curvature of the curve and this leads to introduction of gamma which measures the curvature. Vega hedging also helps to reduce the randomness of the volatility in a portfolio. We showed that to achieve a better hedge we use combination of more than one hedging parameter like the delta-gamma hedge, or the delta-gamma-vega hedge. We did not use theta or rho for hedging since there is no uncertainty in both time and the interest rates, but they provide useful statistical descriptions of the portfolio.

We looked in brief on what is scenario analysis. We saw scenario analysis is important since it can help the financial institution to know which scenario may lead to loss and which may lead to gain. This information helps the financial institution to establish the right risk management strategy. Also we discussed the concept of portfolio insurance which is widely used by portfolio manager to insure their portfolio from falling below a certain level called the floor. Portfolio managers can insure their portfolio by using index options or by creating an option synthetically.

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Appendix A

Determination of Delta for a

European call Option

From the Black-Scholes differential equation (1.1) one can show that, the European call option is given by C = SN(d1) − Xe−r(T −t)N(d2) with d1 = ln(S/X) + (r + σ2/2)(T − t) σ√T − t and d2 = d1− σ √

T − t and all symbols carry the same meaning as defined in section 1.5. We show that

∆ = ∂C ∂S = N(d1). We have ∂C ∂S = N(d1) + SN 0(d 1) ∂d1 ∂S − Xe −r(T −t)N0(d 2) ∂d2 ∂S with N0(d 2) = 1 √ 2πe −d2 2 2 = √1 2πe −(d1−σ √ T −t)2 /2 = 1 2πe −d1 2 2 e(σ √ T −td1σ 2 2 (T −t)). (A.1)

By substitution of value of d1 we have

N0(d 2) = N0(d1)e(ln(S/X)+(r+σ 2 /2)(T −t)−σ22(T −t)) ⇔ N0(d 2) = N0(d1)e(ln(S/X)+r(T −t)). (A.2) Then equation (A.1) becomes

∆ = N(d1) + SN0(d1) ∂d1 ∂S − Xe −r(T −t)N0(d 1)e(ln(S/X)+r(T −t)) ∂d2 ∂S = N(d1) + N0(d1)  S∂d1 ∂S − Xe (ln(S/X)∂d2 ∂S  = N(d1) + N0(d1)  S∂d1 ∂S − S ∂d2 ∂S 

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Determination of Delta for a European call Option 30 = N(d1) + SN0(d1) ∂ ∂S(d1− d2) = N(d1) + SN0(d1) ∂ ∂S(σ √ T − t). Hence ∆ = N(d1).

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Appendix B

Determination of Theta for a

European call Option

A European call option on a non-dividend paying stock is given by C = SN(d1) − Xe−r(T −t)N(d2)

with d1 =

ln(S/X) + (r + σ2/2)(T − t)

σ√T − t and d2 = d1 − σ √

T − t and all symbols carry the usual meaning as defined in section 1.5. We show that

Θ = ∂C ∂t = − SN0(d 1)σ 2√T − t − rXe −r(T −t)N(d 2) Θ = ∂C ∂t = SN 0(d 1) ∂d1 ∂t − rXe −r(T −t)N(d 2) − Xe−r(T −t)N0(d2) ∂d2 ∂t (B.1) From equation (A.2) we have

N0(d

2) = N0(d1)e(ln(S/X)+r(T −t))

On substitution in equation (B.1) we have Θ = SN0(d 1) ∂d1 ∂t − rXe −r(T −t)N(d 2) −  Xe−r(T −t)N0(d 1)eln(S/X)−r(T −t) ∂d2 ∂t = SN0(d 1) ∂d1 ∂t − rXe −r(T −t)N(d 2) −  Xeln(S/X)N0(d 1) ∂d 2 ∂t = SN0(d 1)∂d1 ∂t − rXe −r(T −t)N(d 2) − SN0(d1)∂d2 ∂t = SN0(d 1) ∂ ∂t(d1− d2) − rXe −r(T −t)N(d 2).

But we know that, d2= d1− σ

√ T − t we have Θ = SN0(d 1) ∂ ∂t  σ√T − t− rXe−r(T −t)N(d 2). Hence Θ = −SσN0(d1) 2√T − t − rXe −r(T −t)N(d 2).

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Bibliography

[1] D. Mackenzie, “The Big, Bad Wolf and Rational Market: Portfolio Insurance, the 1987 crash and the Performativity of Economics”, University of Edinburgh, 2004, pp. 12–23.

[2] H. Harry, “Financial Economics: with applications to investments, Insurance and Pension”, The Actuarial Foundation., 1998, pp. 291–293.

[3] J. C. Hull, “Options, Futures, and other derivatives securities”, Prentice-Hall NJ., 2003, pp. 307–341.

[4] J. E. Robert & P. E. Kopp, “Mathematics of Financial Markets ”, Springer Science + Business Media Inc., 2005, pp. 217–221.

[5] J. Stampfli, & V. Goodman, “The Mathematics Finance: Modeling and Hedging”, Thomson Learning, 2001, pp. 122–135.

[6] K. Rubash, “A Study of Option Pricing Models”, Bradley University, http://bradley.bradley.edu/˜arr/bsm/model.html.

[7] K. Chaurdakis, “Lecture notes for course: Financial derivatives”, University of London , 2003. [8] M. R. Sheldom, “An introduction to Mathematical Finance”, Cambridge University Press.,

1999.

[9] M. Avellaneda & P. Antonio, “Dynamic Hedging Portfolios For Derivative Securities in the Presence of Large Transaction Costs”, Courant Institute of Mathematical Sciences., 1994, pp. 165–193 “manual script”.

[10] Paul Schaafsma, “Financial Engineering News: A Financial Patent History Lesson Part I”, bimonthly print publication, http://www.fenews.com/fen40.

[11] P. Wilmott, “DERIVATIVES the Theory and Practice of Financial Engineering”, John Wiley and Sons, 1999.

[12] P. Wilmott, S. Howison, & J. Dewynne, “The Mathematics of Financial Derivatives”, Cam-bridge University Press., 1995, pp. 33–56.

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BIBLIOGRAPHY 33 [13] S. Joseph & G. Victor, “The Mathematics of Finance: Modelling and Hedging”, Gary Ostedt

Indiana Univ., 2001.

[14] W. David, “Financial Derivatives Hedging with futures, forwards, options and swaps”, Clys Ltd, St Ives plc, 1995, pp. 10–12.

References

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