Strong approximation of Black--Scholes theory based on simple random walks

TL;DR

This paper presents a strong, pathwise discrete approximation of the Black--Scholes model using simple symmetric random walks, which could aid in teaching financial mathematics without advanced measure theory.

Contribution

It introduces a novel strong approximation method for the Black--Scholes model based on nested simple random walks, extending to key financial quantities.

Findings

Provides a pathwise approximation of stock and option prices

Extends to replicating portfolios and Greeks

Useful for educational purposes with limited mathematical background

Abstract

A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973 (BSM model). A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979 (CRR model). The BSM and the CRR models have been used for example to price European call and put options. Our aim in this work is to give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman--Kac formula as well. It is hoped that such a discrete pathwise approximation can be useful for example when teaching students whose mathematical background is limited, e.g. does not contain measure theory or stochastic analysis.