Alternative Approach for the Derivation of Black-Scholes Partial Differential Equation in the Theory of Options Pricing Using Risk Neutral Binomial Process

Abstract

This paper presents a risk neutral binomial process as an alternative approach for the derivation of analytic pricing equation called “Black-Scholes Partial differential Equation” in the theory of option pricing. Binomial option pricing is a powerful technique that can be used to solve many complex option-pricing problems. In contrast to the Black-Scholes model and other option pricing models that require solutions to stochastic differential equations, the binomial model is mathematically simple. Binomial model is based on the assumption of no arbitrage. The assumption of no arbitrage implies that all risk-free investments earn the risk-free rate of return and no investment opportunity exists that requires zero amounts of investment but yield positive returns. We derive Black-Scholes partial differential equation using risk neutral binomial process. We also discuss the convergence of binomial model to the analytic pricing formula, the Black-Scholes model for pricing options. Binomial model has the Black-Scholes analytic formula as the limiting case as the number of steps tends to infinity. This model is much more capable of handling options with early exercise because it considers the cash flow at each time period rather than just the cash flows at expiration.