A Nonstandard Approach to Option Pricing

Nonstandard probability theory and stochastic analysis, as developed by Loeb, Anderson, and Keisler, has the attractive feature that it allows one to exploit combinatorial aspects of a well‐understood discrete theory in a continuous setting. We illustrate this with an example taken from financial economics: a nonstandard construction of the well‐known Black‐Scholes option pricing model allows us to view the resulting object at the same time as both (the hyperfinite version of) the binomial Cox‐Ross‐Rubinstein model (that is, a hyperfinite geometric random walk) and the continuous model introduced by Black and Scholes (a geometric Brownian motion). Nonstandard methods provide a means of moving freely back and forth between the discrete and continuous points of view. This enables us to give an elementary derivation of the Black‐Scholes option pricing formula from the corresponding formula for the binomial model. We also devise an intuitive but rigorous method for constructing self‐financing hedge portfolios for various contingent claims, again using the explicit constructions available in the hyperfinite binomial model, to give the portfolio appropriate to the Black‐Scholes model. Thus, nonstandard analysis provides a rigorous basis for the economists' intuitive notion that the Black‐Scholes model contains a built‐in version of the Cox‐Ross‐Rubinstein model.