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 tstrike 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

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. We will therefore have ∂C (6) 0= n + n 1 ∂S 2

Since n1 is usually known, we can always solve (6) to obtain the optimal number of shares needed to make the portfolio self-financing at time t = 0. This process is what is commonly referred to as ∆ hedging.

VEGA (ν)

Vega measures how sensitive the volatility (σ) is in an option. It is determined by finding the derivative of the option with respect to volatility. The vega of equation (4) is given as ∂υ = Sφ(d ) (τ) ∂σ 1 p In practical situations, vega is expressed as the quantity of money relative to the underlying parameter that the option value will gain or lose when there is a 1% rise or fall in volatility. Vega is an important tool in trading since there is no such thing as constant volatility in the real world. A high absolute value of vega implies that a portfolio’s real value is highly sensitive to minimal changes in σ whilst a low absolute value means changes in σ will have little effect in the asset price with the passage of time.

RHO (ρ)

Rho is a measure of how sensitive the interest rate can be in the price of an option. It is obtained when the derivative of the option price is taken with respect to the rate r and the relation from equation (4) is presented below

∂υ rτ ρ = = Kτe− φ(d ) ∂ρ 2

The risk-free interest rarely affects the value of the option except under extreme circumstances. Hence it is not very common to find traders using rho to monitor options. Also an increase in interest rates will decrease the value of an option by increasing carrying costs.

THETA (Θ) 6 The Greek theta is a sensitivity measure of the option as it grows with time. It is found by taking the derivative of the option with respect to time whilst holding all other parameters constant. In practice, it’s theoretical relation according to (4) is given as

∂υ Sϕ(d1)σ rτ Θ= = rKe Φ(d2) ∂t − 2p(τ) −

Theta is also known as the “time decay factor” normally written as Θ= V (t + x) V (t) − where x is referred to as the day count parameter usually equal to one. This implies that, the value of the option tomorrow (V (t + x)) is calculated and the value for today (V (t)) is deducted from it thereby obtaining the amount of money lost or gained each day. Summing individual theta values of assets in a portfolio gives the portfolio’s theta value. Theta is usually negative for an option.

GAMMA (Γ)

Because of the unstable nature of delta which can lead to huge losses for investors especially on short positions, investors resort to another ’Greek’ called gamma to estimate how delta position changes. It measures the rate of change in the delta with respect to changes in the underlying asset’s price. It is the second partial derivative of the portfolio with respect to asset price. It is mathematically denoted by ∂∆ ∂2υ Γ= = ∂S ∂S2 Long options have positive gamma and short options have negative gamma. If gamma is small, delta changes slowly and adjustments to keep the portfolio delta-neutral need only be made rela- tively infrequently. However, if gamma is large, changes should be made regularly since the delta becomes highly sensitive to the price of the underlying asset. Moreover, when gamma is positive, theta tends to be negative whereas when gamma is negative, theta tends to be positive. In order for an investor to obtain an effective delta-neutral portfolio, it is important for him to neutralize the portfolio’s gamma to enable him have an effective hedge over a wider range of underlying price movements which tends to reduce the returns in the excess of the risk-free rate.

4. HIGHER-ORDER GREEKS

These are the Greeks that involves taking the second or higher order derivatives of the parameters in an option. Notable amongst them are gamma (which we have already talked about), charm (known as delta decay which is a measure of the rate of change of delta over time), speed (measures the rate of change of gamma relative to the price of the underlying asset), colour (measures the rate of change of gamma with respect to the passage of time), etc... ∂∆ ∂2υ Charm = = − ∂t −∂S∂t

∂Γ ∂3∂υ Speed = = ∂S ∂S3

∂Γ ∂3∂υ Colour = = ∂t ∂S2∂t 7 5. CONCLUSION

The calculation of the Greeks should usually be based on a specific model and thier precision is as good as the model used in computing them. It is in view of this that we chose the Black-Scholes model which is widely used by investors in their daily tradings. However, a couple of issues have been raised against this model and as a result has a corresponding impact on the Greeks discussed in this project. For instance, volatility is random and will have to be estimated frequently. Also, interest rate is not constant and the lognormality of the model is not definitive of the real world. In this case, building a risk neutral portfolio would not be possible if the volatility of the asset varies stochastically. One way to cater for the problems of volatility and lognormality in the Black-Scholes model is to employ another model known as the Heston’s model given as 1 dSt = µStdt + p(VtStdWt )

dV = κ(θ V )dt + σ (V dW 2) t − t p t t 1 2 ddWt Wt = ρdt However, with this model, one has to deal with so many parameters and if they are handled well, the problem of stochastic volatility will not be a bother to the investor. The Cox-Ingersol-Ross (CIR) model is another adopted model on the market. This model has the ability to model short term fluctuations in the interest rate. The CIR model is given as dr = a(b r )dt + σ (r )dW t − t p t t

6. STUDY GUIDE

Theory of finance and the workings of the financial markets have been of great interest over the past forty years especially after the awardance of the Nobel prizes to Harry Markowitz and Black- Scholes and Merton. As can be understood from this presentation, the Greeks play an important role in the financial markets as long as we require models for analysis. It is therefore imperative for all in this field to gain a deeper understanding of this subject. There are a lot of good books on the market that give explicit explanations on the subject of the Greeks and the models which are used in computing them. Readers who are keen on knowing more can consult books such as Stochastic Calculus for Finance II, Continuous- Time models by Steve E. Shreve, Christer Borel’s Introductory Compendium to the Black-Scholes Theory, Introduction to Stochastic Calculus With Applications by Fima C. Klebaner and Options, Futures and Other Derivatives by John Hull. Extensive information can also be found on the web.

7. REFERENCES

1. Shreve E. Steven.,2004.Stochastic Calculus for Finance II, Continuous Time Models. New York: Springer Science + Business Media, Inc.

2. Borell Christer.Introductory Compendium to the Black-Scholes Theory. Gothenburg: Chalmers University.

3. Fima C. Klebaner. 2005.Introduction to Stochastic Calculus With Applications. 2nd edition. London: Imperial College Press.

4. Hull C. John.,2009.Options, Futures and Other Derivatives. 7th edition. New Jersey: Pearson Education, Inc. 8 5. Quantonline.2011.Foreign Exchange Derivatives[Online]. Available from: www.quantoline.co.za/documents/FxDerivativesAdvFinT1.pdf

6. Wikipedia.2011.Greeks(finance)[Online]. Available from: http://en.wikipedia.org/wiki/Greeks (finance)

7. Investopedia.2011.Using the Greeks To Understand Options[Online]. Available from: http://www.investopedia.com/articles/optioninvestor/04/121604.asp#axzz1cx7gP5

8. Math.nyu.edu. 2011.The Heston Model:A Practical Approach[Online]. Available from: http://math.nyu.edu/ atm262/fall06/compmethods/a1/nimalinmoodley.pdf

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