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 + 2?
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 86 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,o) according to the following rule V(S,t) = eax + aoU(x, o) where S = ex , 2? , ? 2 ? 2r ? ? = ? 1 ? 2 ? 1? , 2 ?? ? t =T ? ? 2r ? ? = ? ? 2 + 1? . ?? ? 1 4 2 Then U(x, o) satisfies the basic diffusion equation ? U ? 2U = 2 . ?? ?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. revious ‘partial derivative exercises’ f om r 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. 87 ? ln( S / S ‘ ) + (r ? ? )(T ? t ) 2 ? ? ? 2 G( S , t ) = ? exp ? ? ? 2 2? (T ? t ) ? S ‘ 2? (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 ) > ? ( 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 a(S,S’) is that its integral is equal to one: +? ? ? ? ( S, S ‘ )dS ‘ = 1. Another very important property of the delta-function is +? ?? ? f (S ‘ )? ( 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 olution 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 88 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)? (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’