FINANCIAL RISK THE GREEKS The whole of this presentation is as a result of the collective efforts of both participants. MICHAEL AGYAPONG BOATENG Email: [email protected] MAVIS ASSIBEY-YEBOAH Email: [email protected] DECEMBER 1, 2011 Contents 1. INTRODUCTION 3 2. THE BLACK-SCHOLES MODEL 3 3. THE GREEKS 4 4. HIGHER-ORDER GREEKS 7 5. CONCLUSION 8 6. STUDY GUIDE 8 7. REFERENCES 8 2 1. INTRODUCTION The price of a single option or a position involving multiple options as the market changes is very difficult to predict. This results from the fact that price does not always move in correspondence with the price of the underlying asset. As such, it is of major interest to understand factors that contribute to the movement in price of an option, and what effect they have. Most option traders therefore turn to the Greeks which provide a means in measuring the sensitivity of an option price by quantifying the factors. The Greeks are so named because they are often denoted by Greek letters. The Greeks, as vital tools in risk management measures sensitivity of the value of the portfolio relative to a small change in a given underlying parameter. This treats accompanying risks in isolation to rebalance the portfolio accordingly so as to achieve a desirable exposure. In this project, the derivation and analysis of the Greeks will be based on the Black-Scholes model. This is because, they are easy to calculate, and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. The most commonly used Greeks are the delta, vega, theta, rho and gamma out of which we give an insight into delta hedging and how it works. The first four are first order derivatives whilst gamma is of second-order. Some higher order Greeks will be discussed as well. 2. THE BLACK-SCHOLES MODEL The definition of compound interest satisfies the ordinary differential equation (ODE) dx(t)/x(t)= rdt (r is the interest rate). In the case where rate is uncertain, it is believed to be perturbed by the noise r + ξ(t) which results in the following stochastic differential equation (SDE) dX(t) = (r + σξ(t))X(t). dt This means that (1) dX(t)= rX(t)dt + σX(t)dB(t). When there is no noise, that is σ = 0, we recover the deterministic equation whose solution is obtained by separating variables as x(t) = ert. The solution to (1) is governed by a geometric Brownian motion and it is given by (r σ2/2)t+σB(t) X(t)= e − . Now, let us consider a capital market consisting of a stock, a bond and options on the stock. We assume that the riskless interest rate is a constant r, µ is the mean rate of return and σ is the volatility where σ(S(t)) = σS(t). The stock price process S(t) satisfies the SDE (2) dS(t)= µS(t)dt + σS(t)dB(t). Using Itˆo’s formula with f(x)= ln(x) we find that a strong solution for (2) is obtained as 3 (µ σ2/2)t+σB(t) S(t)= S(0)e − . Furthermore, (B(t))t 0 is the bond price process given by ≥ B(t)= B(0)ert. where r > 0 is a constant. At time 0, the asset prices S(0) and B(0) are known strictly positive real numbers. Through a series of calculations, which we will not bore you with, we arrive at the natural price of a simple European contingent claim with the payoff g(S(T )) at time of maturity T ]0, [ which also happens to be the Black-Scholes equation. The equation is shown below ∈ ∞ 2 rτ (r σ )τ+σ√τG υ(t,s)= e− E[g(se − 2 )] where G N(0, 1), 0 t<T , and τ = T t. ∈ ≤ − Hence, Black and Scholes derived the price at time 0 of a European call with strike price K and termination date T in 1973 as follows S(0) σ2 S(0) σ2 ln K + (r + 2 )T rT ln K + (r 2 )T (3) c(0,S(0),K,T )= S(0)φ( ) Ke− φ( − ). σ√T − σ√T S 2 S 2 ln (0) +(r+ σ )T ln (0) +(r σ )T We make the following denotions; d = ( K 2 ) and d = ( K − 2 ) 1 σ√T 2 σ√T 3. THE GREEKS As mentioned earlier in the introduction, the Greeks of an option measure the sensitivity of the option price from its parameters. In mathematical notations, a simple European option on a stock S with payoff function (S(T ) K)+) at time of maturity T has the price υ(t,S(t) expressed as − 2 rτ (r σ )τ+σ√τG (4) υ(t,s)= e− E[g(se − 2 )] at time t<T . The partial derivatives of (4) with respect to (t,s,r,σ) evaluated at the point (t,S(t)) are called the Greeks of the option. DELTA In options trading, delta is the measure of how the value of an option changes with respect to changes in the value of the underlying asset. It is denoted by the Greek letter ∆. It measures the sensitivity of an option to a change in the price of the underlying parameter. Delta is a number 4 between negative one and one (or in trading jargon, between 100 and +100) which shows the amount of money one losses or gains in the market. In general,− ∂υ ∆= ∂S where υ is the price of the option and S is the stock price. The Black-Scholes model analysis has a close relationship with ∆ in that it constructs a riskless portfolio made up of a position in the option on the stock and a position in the stock. Hence, for a European call option on a non-dividend paying stock, the delta of a call is given by ∂c = φ(d ) ∂S 1 PROOF The call price formula is given by (3). Taking its derivative with respect to the stock price gives ∂c ∂d1 rτ ∂d2 = φ(d )+ S(ϕ) Ke− ϕ(d ) ∂S 1 ∂S − 2 ∂S ∂d ∂d 1 = 2 ∂S ∂S ∂c ∂d1 rτ = φ(d )+ (S(ϕ) Ke− ϕ(d ) ⇒ ∂S 1 ∂S − 2 rτ (S(ϕ) Ke− ϕ(d )=0 − 2 ∂c = φ(d ) ∂S 1 In a call option, an increase in the price of the underlying asset increases the value of the option by an amount equal to the delta of the option. The opposite occurs when there is a decrease in the underlying asset’s price. This scenario exists when the investor takes a long position in the call option as the delta of the call option is a positive number ranging between 0 and 1. Put option on the other hand has a negative delta ranging between 1 and 0. Due to the inverse relationship between the underlying asset and the option price in a− put option, there is a decrease in the option’s price by an amount equal to the delta of the option whenever the underlying asset’s price increases. Also, a decrease in the price of the underlying asset causes the price of the option to increase. The delta for a European put is also given below without proof ∂c = φ( d ) ∂S − − 1 For a portfolio of options with a single asset price as S, the delta of the portfolio is given by ∂π ∂S where π is the value of the portfolio. The delta of the portfolio is the sum of the individual options in the portfolio. A portfolio consisting of a quantity x of option i(1 i n) has a delta i ≤ ≤ n ∆= X xi∆i i=1 where ∆i is the delta of the ith position. A delta neutral portfolio is the position in the price of the underlying asset necessary to make the delta of the portfolio zero. Because delta changes, the ability to make the portfolio delta neutral can exist for a short period of time and the investor has to adjust his position regularly in order to adapt to this changing atmosphere and this is referred to as rebalancing. 5 DELTA AS A HEDGING STRATEGY First of all, hedging can be thought of as an insurance. When individuals decide to hedge, they are insuring themselves against a negative event and this does not necessarily prevent the occurence of a negative event. But when it does happen and one is properly hedged, the impact of the event is essentially reduced. Hedging occurs in our everyday lives. As an example, when one buys a health insurance, unforseen disasters in ones’ health is hedged against. If we assume that we have a portfolio of options with V (t) as the value of the portfolio at time t, then V (t)= n1c + n2S + B where c is the value of a single option, S represents the underlying share’s value, B is the bond invested in a riskless asset and n1 and n2 are the number of options and shares respectively. If n1 is negative, it implies we have a short position in the option whilst a positive value of n1 means we have a long position. The delta of the portfolio will then be ∂V ∂c (5) = n + n ∂S 1 ∂S 2 For this portfolio to be made delta neutral (insensitive to minimal changes in the value of the underlying parameter), we set the left hand side of (5) to zero.