11 - Option Pricing and Stochastic Processes

Abstract

Stochastic processes and the theory of stochastic differential equations have played a fundamental role in the theory of option pricing. An option gives the holder or owner of it the right to buy or sell something at his discretion before a certain prescribed date at a price specified in advance. Black and Scholes were the first who derived the equilibrium price of an option which gives the owner the right to buy a stock before a certain date at a fixed price. They used a stochastic process to model the price of the stock. The theory of stochastic differential equations had been developed to obtain a direct solution of the partial differential equation in terms of the expected value of a certain stochastic variable, without any reference to physics. The so-called Kolmogorov-backward equation gives the direct solution. The formula of Black and Scholes is not only widely accepted by academic theorists but also used as a black-box by market participants to calculate prices for options they want to sell or buy. But the Black–Scholes formula cannot be used to explain price behavior for all kinds of options. Stochastic differential equations are not only used to price options but are also applied in a more general way to describe optimal consumption and investment decisions in a continuous-time setting.