Sub-diffusive Black-Scholes model and Girsanov transform for sub-diffusions

TL;DR

This paper introduces a sub-diffusive Black-Scholes model capturing market inactivity during bear markets, utilizing Girsanov transforms and fractional PDEs to price European options.

Contribution

It develops a novel sub-diffusive Black-Scholes framework with explicit pricing formulas and a fractional PDE approach, extending classical models to account for market sub-diffusion.

Findings

The model is arbitrage-free but incomplete.

Explicit European call option pricing formula derived.

Time-fractional Black-Scholes PDE established.

Abstract

We propose a novel Black-Scholes model under which the stock price processes are modeled by stochastic differential equations driven by sub-diffusions. The new framework can capture the less financial activity phenomenon during the bear markets while having the classical Black- Scholes model as its special case. The sub-diffusive spot market is arbitrage-free but is in general incomplete. We investigate the pricing for European-style contingent claims under this new model. For this, we study the Girsanov transform for sub-diffusions and use it to find risk-neutral probability measures for the new Black-Scholes model. Finally, we derive the explicit formula for the price of European call options and show that it can be determined by a partial differential equation (PDE) involving a fractional derivative in time, which we coin a time-fractional Black-Scholes PDE.