Sub-diffusive Black-Scholes model and Girsanov transform for sub-diffusions
Shuaiqi Zhang, Zhen-Qing Chen
TL;DR
This work extends the Black-Scholes framework by modeling stock prices with general sub-diffusions, capturing reduced market activity in bear markets while recovering the classical model as a special case. It develops a Girsanov transform tailored to sub-diffusions, proving the existence and uniqueness of a sub-diffusion-based risk-neutral measure under which discounted prices are martingales. The authors derive explicit European option pricing formulas within this incomplete, arbitrage-free setting and show that option pricing is governed by a time-fractional Black-Scholes PDE, naturally incorporating the inverse-subordinator dynamics. The approach provides a principled link between sub-diffusive asset dynamics and fractional-time PDEs, with potential implications for pricing and hedging in markets with stochastic activity levels.
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.