Advanced Derivatives Course

Week 9: The Black-Scholes Solution And The “Greeks” (see also Wilmott, Chapter 6,7)

Lecture VIII.1 Plain Vanilla The goal of the next two lectures is to obtain the Black-Scholes solutions for European options, which belong to the type of basic contingent claims called ‘vanilla options’. These lectures may seem a bit too technical. However, I think, it is important to have at least some idea about how the BS equation is solved for various financial instruments. I will try my best to keep things as simple as possible. Let us look at the BS equation. ∂ V 1 2 2 ∂ 2V ∂V + 2σ S + rS − rV = 0. 2 ∂t ∂S ∂S It has two variables, share price S and time t. However, there is a second derivative only with respect to the share price and only a first derivative with respect to time. In finance, these type equations have been around since the early seventies, thanks to Fischer Black and Myron Scholes. However, equations of this form are very common in physics. Physicists refer to them as heat or diffusion equations. These equations have been known in physics for almost two centuries and, naturally, scientists have learnt a great deal about them. Among numerous applications of these equations in natural sciences, the classic examples are the models of • •

Diffusion of one material within another, like smoke particles in air, or water pollutions; Flow of heat from one part of an object to another.

This is about as much I wanted to go into physics of the BS equation. Now let us concentrate on finance.

What Is The Boundary Condition? As I have already mentioned, the BS equation does not say which financial instrument it describes. Therefore, the equation alone is not sufficient for valuing derivatives. There must be some additional information provided. This additional information is called the boundary conditions. Boundary conditions determine initial or final values of some financial product that evolves over time according to the PDE. Usually, they represent some contractual clauses of various derivative securities. Depending on the product and the problem at hand, boundary conditions would change. When we are dealing with derivative contracts, which have a termination date, the most natural boundary conditions are terminal values of the contracts. For example, the boundary

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condition for a European call is the payoff function V(ST,T) = max( ST-E,0) at expiration. In financial problems, it is also usual to specify the behaviour of the solution at S=0 and as S . For example, it is clear that when the share value S , the value of a put option should go to zero. To summarise, equipped with the right boundary conditions, it is possible using some techniques to solve the BS equation for various financial instruments. There are a number of different solution methods, one of which I now would like to describe to you. Transformation To Constant Coefficient Diffusion Equation1 Physics students may find this subsection interesting. Sometimes it can be useful to transform the basic BS equation into something a little bit simpler by a change of variables. For example, instead of the function V(S,t), we can introduce a new function U(x,ô) according to the following rule V(S,t) = eáx + âôU(x, ô) where S = ex , 2τ , σ2  2r  α = − 12  2 − 1, σ  t =T −

2

 2r  β = −  2 + 1 . σ  1 4

Then U(x, ô) satisfies the basic diffusion equation ∂ U ∂ 2U = 2 . ∂τ ∂x It is a good exercise to check (using your week 8) that the above change of variables equation. This equation looks much simpler that can be important, for example when simple numerical schemes.

previous ‘partial derivative exercises’ from indeed gives rise to the standard diffusion than the original BS equation. Sometimes seeking closed-form solutions, or in some

Green’s Functions One solution of the BS equation, which plays a significant role in option pricing, is 1

You can also read about this transformation in the original paper by Black and Scholes, a copy of which you can get from me.

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[

]

 ln( S / S ' ) + (r − σ )(T − t ) 2  2 G( S , t ) = ⋅ exp  −  2 2 σ ( T − t) σS ' 2π (T − t )   e − r (T − t )

2

for any S’. (Exercise: verify this by substituting back into the BS equation.) This solution behaves in an unusual way as time t approaches expiration T. You can see that in this limit, the exponent goes to zero everywhere, except at S=S’, when the solution explodes. This limit is known as a Dirac delta function: lim G (S , t ) → δ ( S , S '). t →T

(Don not confuse this delta function with the delta of delta hedging!) Think of this as a function that is zero everywhere except at one point, S=S’, where it is infinite. One of the properties of ä(S,S’) is that its integral is equal to one: +∞

∫ δ( S, S ' )dS ' = 1.

−∞

Another very important property of the delta-function is +∞

∫ f (S ' )δ ( S , S ' )dS' =

f (S ),

−∞

where f(S) is an arbitrary function. Thus, the delta-function ‘picks up’ the value of f at the point, where the delta-function is singular, i.e. at S’=S. How all of this can help us to value financial derivatives? You will see it in a moment. The expression G(S,t) is a solution of the BS equation for any S’. Because of the linearity of the BS equation, we can multiply G(S,t) by any constant, and we get another solution. But then we can also get another solution by adding together expressions of the form G(S,t) but with different values for S’. Putting this together, and taking an integral as just a way of adding together many solutions, we find that ∞

V ( S ,t ) = ∫ f ( S ')G (S , t )dS ' 0

is also a solution of the BS equation for arbitrary function f(S’).

Now if we choose the arbitrary function f(S’) to be the payoff function of a given derivative problem, then V(S,t) becomes the value of the option. The function G(S,t) is called the Green’s function. The formula above gives the exact solution for

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the option value in terms of the arbitrary payoff function. For example, the value of a European call is given by the following integral ∞

c( S , t ) = ∫ max( S '− E ,0 )G ( S , t ) dS '. 0

Let us check that as t approaches T the above call option gives the correct payoff. As we mentioned this before, in the limit when t goes to T, the Green’s function becomes a delta-function. Therefore, taking the limit we get ∞

c( S T , T ) = ∫ max( S '− E ,0)δ (S T , S ' ) dS ' = max( ST − E ,0). 0

Here we used the property of the delta-function. Thus, the proposed solution for the call option does satisfy the required boundary condition. Formula For A Call Normally, in financial literature you see a formula for European options written in terms of cumulative normal distribution functions. You may therefore wonder how the exact result given above in terms of the Green’s function is related to the ones in the literature. Now I’d like to explain how these two results are related. Let us first focus on a European call. Let us look at the formula for a call ∞

c( S , t ) = ∫ max( S '− E ,0 )G ( S , t ) dS '. 0

We integrate from 0 to infinity. But it is clear that when S’<E, the payoff function vanishes. Therefore, we can integrate from E to infinity and write ∞

c( S , t ) = ∫ (S '− E )G( S ,t )dS '. E

Now if we substitute the explicit expression for the Green’s function into the above integral, then after some manipulations with variables it can be presented in the more familiar form c( S , t ) = SN (d 1 ) − Ee −r (T −t ) N ( d 2 ), where we used the following notations N (d ) =

d

1 2π

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∫e

−∞

−

x2 2

dx

and ln( S / E ) + ( r + σ2 )(T − t ) 2

d1 =

, σ T −t ln( S / E ) + (r − σ2 )(T − t ) d2 = . σ T −t 2

The formula in the box is the famous Black-Scholes formula for a European call written in terms of the cumulative normal distribution function, N(d). The latter is among the standard functions in Excel, NORMDIST(argument,0,1,TRUE). Therefore, it is very easy to use the above formula in practice. A typical plot of the option value as a function of the underlying asset at a fixed time to expiry is shown on the picture below.

call 60 50 40 30 20 10 0 -10 0

50

100

150

200

share price

Options As A Portfolio If you look at the BS formula for a European call, you can note that an option can be duplicated by a portfolio of shares and bonds. Indeed, Ee -r(T-t) can be replaced by the present value of a zero-coupon bond maturing at time T with a face value equal to E. Then N(d1) is the number of shares and N(d2) is the number of borrowed bonds in the duplicating portfolio. The BS formula gives you the optimal portfolio weights at a particular moment in time. The portfolio weights are optimal only over an infinitesimal period of time. Therefore, they have to be readjusted continuously. Formula For A Put The European put option has payoff Payoff(S) = max(E - ST, 0). The value of a put option can be found in the same way as above, or much quicker using put-call parity: 90

c(S,t) + Ee -r(T-t) = p(S,t) + S. This gives p(S,t) = c(S,t) + Ee -r(T-t) – S = S(N(d1) – 1) + Ee-r(T-t)(1 – N(d 2)). Note that 1 – N(d) = N(-d). Thus, we find for the value of a European put p(S,t) = - SN(-d1) + Ee -r(T-t)N(-d2). Is this not beautiful! Formulas For Vanilla Options With Dividends In the previous lecture we derived a formula for options on dividend paying assets: Vt + (1/2)ó 2S 2VSS + (r – D)SVS – rV = 0. In order to find the explicit expressions for the vanilla options, let us perform the following change of variables V(S,t) = e –D(T – t)U(S,t). Then you can check that the function U(S,t) satisfies the following equation Ut + (1/2)ó 2S 2USS + (r – D)SUS – (r – D)U = 0. This is exactly the BS equation but only with the risk-free rate r replaced by r – D. The boundary condition for U(s,t) is U(S T,t) = max(ST – E,0), that is, exactly as before. Thus, we can use the solutions found for the options on assets without dividends by simply substituting r – D instead of r. The results of the calculations are cD (S , t ) = Se− D (T −t ) N (d 1 ) − Ee −r (T −t ) N (d 2 ), p D ( S , t ) = − Se −D (T −t ) N ( −d 1 ) + Ee −r (T −t ) N (− d 2 ), where ln( S / E ) + ( r − D + σ2 )(T − t ) 2

d1 =

, σ T −t ln( S / E ) + (r − D − σ2 )(T − t ) d2 = . σ T −t 2

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Note that the value of a put option again can be found using put-call parity modified for continuously paid dividends: c(S,t) + Ee -r(T-t) = p(S,t) + Se-D(T-t). Example Let us value a European style call currency option traded on the Philadelphia Exchange on November 12, 1999. The contract gives the holder the right to purchase €62,500 for US $63,750 in January, 2000. That is, the strike price of one Euro is US $1.020. The option price is for the purchase of one Euro with US dollars. The formula for the value of the option is

cEuro/ $ ( S , t ) = Se− r f (T −t ) N (d1 ) − Ee− r (T −t ) N (d 2 ), where ln( S / E ) + ( r − rf + σ2 )(T − t ) 2

d1 =

σ T −t ln( S / E ) + (r − r f − σ2 )(T − t )

,

2

d2 =

σ T −t

.

From the FT, we can deduct information about the spot price S, the US one year rate r and the Euro one year rate rf and the time to expiration. Namely, S=€/$=$1.032 (the option is in the money), r=0.061, rf=0.036 and T-t=3 months=0.25. What we cannot find in the newspaper is the volatility of the €/$ exchange rate, σ. So, we have got to use some other sources of information. We can for example call Olsen and Associates a Zurich based company which specialises in estimating volatilities for currencies. Let us assume that the volatility is equal to 0.07. When we substitute all the data into the BS formula, we obtain the price of the call c€/$=$0.0252 or 2.52 cents. The quote in the FT is 2.58 for the January contract. Thus, our estimate is pretty good. Our price of the contract is 62,500 times $0.0252 or $1575. The error in the value of the contract is 62,500(0.0252 – 0.0258) = -$37.5 or about 2% of the deal. This is the amount we would have lost, if we had sold this contract. But we could have bought it too and earned $37.5. If you feel like everything up to this point has been Greek to you, it is the right time to talk about ‘the Greeks’.

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Lecture VIII.2 The Greeks In the previous lecture we have derived the most spectacular results of the BlackScholes model – the values for European options. The fact that this model has become so popular reflects its remarkable success in pricing financial derivatives. Some people say that the BS model is one of the most successful economic theories ever. Now I would like to analyse some of the properties of the BS solutions. The BS solutions are used to define a number of very important characteristics commonly known as the ‘Greeks’. One of them we have already met on many occasions. This is Delta By now we know that the delta of an option or a portfolio of options is the sensitivity of the option or portfolio to the underlying asset. It is the rate of change of the option value with respect to the asset: ∆=

∂V . ∂S

Notice the use of partial derivatives in this definition, which means that as we differentiate with respect to S, all other variables are kept fixed. In this definition, V can be the value of a single derivative contract or of a whole portfolio of contracts. The delta of a portfolio of options is just the sum of the deltas of all the individual positions. When the function V(S,t) is known, we can compute the delta explicitly. However, even without knowing V(S,t) we can conclude that for call options 0 ≤ ∆ ≤ 1. Whereas for puts -1 ≤ ∆ ≤ 0. Indeed, during our second week (which was long time ago!) we derived the upper bound for call options. Namely, we proved that the call can never cost more than the share, c ≤ S. Now if we take partial derivative of both sides, we get ∆ ≤ 1.

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Since we also know that the call value increases when the share price increases, we conclude that the delta must be positive. Thus, we arrive at 0 ≤ ∆call ≤ 1. For puts, we can use the put-call parity and the above result for calls to prove that -1 ≤ ∆put ≤ 0. We know that the delta is used in delta hedging, a procedure allowing us (theoretically and almost practically) to eliminate all risk. Delta hedging means holding one of the options and shorting a quantity Ä of the underlying asset. In general, delta is a function of S and t. For example for the European call and put options, the deltas are respectively given as Ä call = e-D(T-t)N(d 1), Äput = - e-D(T-t)N(-d 1).

Delta (Call) 1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

Spot

These functions vary as S and t vary. This means that the number of shares held must be continuously changed to maintain a delta-neutral position. The delta of a deltaneutral portfolio is equal to zero: ∆Π =

∂Π = 0. ∂S

This procedure is called dynamic hedging. Changing the number of shares held requires the continual purchase and/or sale of the stock. This is called rehedging or rebalancing of the portfolio.

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In practice, the frequency of delta hedging depends on the market liquidity. In highly liquid markets, the BS assumption of the possibility of continuous hedging may be quite accurate. However, in less liquid markets you may not even be able to buy or sell in the quantities you want. Some contracts have a delta that becomes very large at special times or share values. It may even become greater than the whole existing stock. Or like in August 1998, after the Russian government had defaulted on its domestic debt, liquidity simply rapidly dried out from many financial markets, causing asset prices to plunge. That has almost brought down Long-Term Capital Management hedge fund. LTCM tried to stave off disaster by selling assets as the value of its portfolio fell – but the lack of liquidity in the markets prevented it from selling enough. Clearly, in such a situation the basic assumptions of the BS world are inappropriate and all we can do is to look for a way beyond Black-Scholes. However, this goes beyond this course. Gamma The gamma, Ã, of an option or a portfolio of options is the second derivative of the option with respect to the underlying asset: Γ=

∂ 2V ∂ ∆ = . ∂S 2 ∂S

According to the definition, gamma is the sensitivity of the delta to the underlying security. Therefore, it measures how much and how often a position must be rehedged in order to maintain a delta-neutral portfolio. For example, Γcall = Γput =

e − D (T − t ) N ' ( d 1 ) σS T − t − D (T − t ) e N ' (d 1 ) σS T − t

where N '( d ) =

95

1 2π

e

−

d2 2

.

, ,

Gamma (Call/Put) 12 10 8 6 4 2 0 -2 0

0.5

1

1.5

2

If you remember, when we discussed how a change in the share price affects the value of the option, we had to expand V to a second order in dS, because the square of the random part is not negligible. Since the gamma is the second derivative of V , it is dominated by the Brownian nature of the share value movement. We shall see in the last lecture that the gamma plays an important role in discrete hedging. Because of the cost of frequent hedging and because one wants to reduce exposure to model error (meaning that in practice (dW)2 ≠ dt), it is natural to try to minimise the need to rebalance the portfolio too frequently. The corresponding hedging procedure is called a gamma-neutral strategy. To achieve this objective, we have to buy or sell more options, not just the share. By simple differentiation, you can check that a position in the underlying asset or in a futures contract (=Ser(T-t)) on the underlying asset has zero gamma. ∂2S = 0. ∂S 2 Thus, we cannot change the gamma of our position by adding the underlying. However, we can add another option in quantity, which will make the portfolio gamma-neutral. By holding two different options we can make the portfolio both delta- and gamma-neutral. Making A Portfolio Gamma-Neutral Suppose that a delta-neutral portfolio (∆Π = 0) has gamma equal to Γ and a traded option has a gamma equal to ΓO. If the number of traded options added to the portfolio is wO, the gamma of the portfolio is ΓΠ = Γ + wOΓO.

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Hence, the portfolio becomes gamma-neutral, if our position in the traded option is equal to - Γ/ΓO. Of course, as we add the traded option, the delta of the portfolio changes. So the position in the underlying asset then has to be changed to maintain delta-neutrality: ∆S = ∆ + wO ∆O = ∆ - (Γ/ΓO)∆O. Note that the portfolio remains gamma-neutral only instantaneously. As time passes, gamma-neutrality can be maintained only if the position in the traded option is adjusted. So that it is always equal to - Γ/ΓO. Making a portfolio gamma-neutral can be regarded as a first correction for the fact that the position in the underlying asset cannot be changed continuously when delta hedging is used. Theta Theta, Θ, is the rate of change of the option price with respect to time: Θ=

∂V . ∂t

The theta is related to the option value, the delta and the gamma by the BS equation: Θ = r (V − ∆ ⋅ S ) − 12 σ 2 S 2 Γ + D ⋅ ∆ ⋅ S . For vanilla contracts: Θ call = − Θ put = −

σSe− D (T −t ) N ' ( d1 ) 2 T −t σSe N ' ( d1 ) − D (T − t )

2 T −t

+ DSe−D (T −t ) N ( d 1 ) − rEe − r (T −t ) N (d 2 ), − DSe− D (T −t ) N ( −d 1 ) + rEe −r (T −t ) N ( −d 2 ).

For a delta-neutral portfolio, ∆ = 0 and Θ + 12 σ 2 S 2 Γ = rV , where V now is a delta-neutral portfolio of options. The last equation shows that when Θ is large and positive, gamma tends to be large and negative, and vice versa. In a delta-neutral portfolio, theta can be regarded as a proxy for gamma.

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Theta 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05

Vega Vega is a very important but confusing quantity. It is not even denoted by a Greek character2, since it is not a Greek letter although it belongs to the “Greeks”. It is the sensitivity of the option price to volatility

V =

∂V . ∂σ

This is completely different from the other Greeks, since it is a derivative with respect to a parameter and not a variable. You have already heard in our seminars that, in practice, the volatility of the underlying is not known with certainty. Not only is it very difficult to measure at any time, it is even harder to predict what it will do in the future. Suppose that we put a volatility of 30% into an option pricing formula; how sensitive is the price to that number? That is the vega. As with gamma hedging, one can vega hedge to reduce sensitivity to the volatility. This is a major step towards eliminating some model risk, since it reduces dependence on a quantity that is not even known very accurately. Unfortunately, a portfolio that is vega-neutral will not in general be gamma-neutral, and vice versa. If a trader requires a portfolio to be both gamma and vega neutral, at

2

I couldn’t find the right character for vega, so I had to invent my own.

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least two traded derivatives dependent on the same underlying asset must usually be used. For vanilla contracts Vcall = S T − t e− D (T −t ) N ' (d 1 ), V put = S T − t e −D (T −t ) N ' (d 1 ). Vega 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 0

0.5

1

1.5

2

Rho Rho, ρ, is the sensitivity of the option value to the risk-free interest rate : ρ=

∂V . ∂r

In practice, one often uses a whole term structure (remember the talk on interest rate derivatives?) of interest rates, meaning a time-dependent rate r(t). Rho would then be the sensitivity to the level of the rates assuming a parallel shift in rates at all times. For European options: ρ call = E (T − t )e − r (T −t ) N ( d 2 ), ρ put = − E (T − t )e −r (T −t ) N (− d 2 ).

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Rho (Call) 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 0

0.5

1

1.5

2

Below you can see a table of the “Greek” signs for calls and puts

Long call Short call Long put Short put

Delta Positive Negative Negative Positive

Gamma Positive Negative Positive Negative

Theta Negative Positive Negative Positive

From the BS equation it follows that when the theta is positive, the gamma is negative and vice versa. Otherwise a certain profit would systematically accompany a delta neutral position with a positive gamma and a positive theta. Financial institutions that sell options tend to have a negative gamma and a positive theta. To Re-hedge Or Not To Re-hedge? It would be wrong to give the impression that option traders are continually rebalancing their portfolios to maintain delta-neutrality, gamma-neutrality, veganeutrality, and so on. In real world, transactions costs make frequent rebalancing very expensive. Rather than trying to eliminate all risks, option traders, therefore, usually concentrate on assessing risks and deciding whether they are acceptable. They tend to use delta, gamma, and vega measures to quantify the different aspects of the risk inherent in their option portfolios. If the downside risk is acceptable, no adjustment is made to the portfolio; if it is unacceptable, they take an appropriate position in either the underlying security or another derivative. In addition to monitoring risks such as delta, gamma, and vega, option traders often also carry out a scenario analysis. This involves calculating the gain/loss on their portfolio over a specified period under a variety of different scenarios. Typically, it is one day, one week, or one month. The scenarios can be either chosen by management or generated by a model.

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Try To Answer The Following questions 1) What are the boundary conditions for vanilla options? 2) How does Green’s function enter the formula for European option values? What is its main property? Can you show that the option values given by integrals satisfy the correct boundary conditions? 3) What are the Greeks? Can you explain how a portfolio of options can be made gamma-neutral? Can a portfolio of options be simultaneously gamma and vega-neutral?

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