A derivation of the Black-Scholes option pricing model using a central limit theorem argument

TL;DR

This paper presents an elementary derivation of the Black-Scholes option pricing formula using the central limit theorem, making it accessible for undergraduate students unfamiliar with stochastic calculus.

Contribution

It introduces a new, simplified derivation of the Black-Scholes model based on the central limit theorem, avoiding complex stochastic differential equations.

Findings

Derivation is accessible to undergraduates with basic probability knowledge.

Provides an alternative proof of the Black-Scholes formula.

Highlights the connection between the CLT and option pricing.

Abstract

The Black-Scholes model (sometimes known as the Black-Scholes-Merton model) gives a theoretical estimate for the price of European options. The price evolution under this model is described by the Black-Scholes formula, one of the most well-known formulas in mathematical finance. For their discovery, Merton and Scholes have been awarded the 1997 Nobel prize in Economics. The standard method of deriving the Black-Scholes European call option pricing formula involves stochastic differential equations. This approach is out of reach for most students learning the model for the first time. We provide an alternate derivation using the Lindeberg-Feller central limit theorem under suitable assumptions. Our approach is elementary and can be understood by undergraduates taking a standard undergraduate course in probability.