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...
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