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

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.