A Nonstandard Approach to Option Pricing

TLDR

Nonstandard probability theory lets one exploit combinatorial aspects of discrete models in a continuous setting, enabling seamless movement between discrete and continuous viewpoints. The paper aims to devise a rigorous method for constructing self‑financing hedge portfolios for contingent claims within the Black‑Scholes framework using hyperfinite binomial model constructions. The authors construct a nonstandard version of the Black‑Scholes model that simultaneously represents a hyperfinite Cox‑Ross‑Rubinstein binomial tree and a geometric Brownian motion, and use this framework to build self‑financing hedge portfolios for contingent claims. They show that the Black‑Scholes pricing formula can be derived elementarily from the binomial model and that nonstandard analysis rigorously underpins the intuition that the Black‑Scholes model embeds a Cox‑Ross‑Rubinstein structure.

Abstract

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.