Stochastic PDEs and Quantitative Finance: The Black-Scholes-Merton Model of Options Pricing and Riskless Trading

TL;DR

This paper explores the Black-Scholes-Merton stochastic differential equation model for options pricing, providing historical context, derivation, implementation, and proposing improved trading strategies for high-volatility markets using continuous trading and Student's t-distribution.

Contribution

It introduces an enhanced trading approach with 24-hour trading and incorporates Student's t-distribution into the Black-Scholes model for better handling of market volatility.

Findings

Continuous 24-hour trading improves market responsiveness.

Using Student's t-distribution better captures fat tails in market data.

The improved model reduces arbitrage opportunities in volatile markets.

Abstract

Differential equations can be used to construct predictive models of a diverse set of real-world phenomena like heat transfer, predator-prey interactions, and missile tracking. In our work, we explore one particular application of stochastic differential equations, the Black-Scholes-Merton model, which can be used to predict the prices of financial derivatives and maintain a riskless, hedged position in the stock market. This paper is intended to provide the reader with a history, derivation, and implementation of the canonical model as well as an improved trading strategy that better handles arbitrage opportunities in high-volatility markets. Our attempted improvements may be broken into two components: an implementation of 24-hour, worldwide trading designed to create a continuous trading scenario and the use of the Student's t-distribution (with two degrees of freedom) in evaluating the Black-Scholes equations.