Black-Scholes without stochastics or PDEs

TL;DR

This paper presents a derivation of the Black-Scholes model and its extension to exchange options using a binomial tree approach, avoiding stochastic calculus and PDEs, making the concepts more accessible.

Contribution

It introduces a novel derivation method for Black-Scholes and exchange options based solely on binomial trees, without stochastic calculus or PDEs.

Findings

Black-Scholes model derived from binomial tree limit

Extension to exchange options demonstrated

Method accessible without advanced stochastic calculus

Abstract

We show how to derive the Black-Scholes model and its generalisation to the `exchange-option' (to exchange one asset for another) via the continuum limit of the Binomial tree. No knowledge of stochastic calculus or partial differential equations is assumed, as we do not use them.