The Black-Scholes Differential Equation

Abstract

Having used arbitrage considerations to derive various properties of derivatives, in particular of option prices (upper and lower bounds, parities, etc.), we now demonstrate how such arbitrage arguments, with the help of results from stochastic analysis, namely Ito’s formula 3.19, can be used to derive the famous Black-Scholes equation. Along with the Assumptions 1, 2, 3, 4 and 5 from Section 5, the additional assumption that continuous trading is possible is essential to establishing the equation, i.e. in the following we assume that Assumption 6 from Section 5 holds. The Black-Scholes equation is a partial differential equation which must be satisfied by every price function of path-independent European derivatives on a single underlying1. Consequently, one method of pricing derivatives consists in solving this differential equation satisfying the boundary conditions corresponding to the situation being investigated. In fact, even quite a number of path-dependent options obey this differential equation. A prominent example is the barrier option. In general however, the price of path-dependent options cannot be represented as a solution to the Black-Scholes equation. It is possible to surmount these difficulties by imbedding the state space in a higher dimensional space defining one or several additional variables in an appropriate manner to represent the different paths. This method is demonstrated explicitly by Wilmott for Asian options with arithmetic means [166]. As we will see below, the valuation of American options can also be accomplished via the Black-Scholes equation (with free boundary conditions).