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arXiv · 2511.10371

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

Shuaiqi Zhang·Zhen-Qing Chen
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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.

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Shuaiqi Zhang, Zhen-Qing Chen. 2025-11-13. Sub-diffusive Black-Scholes model and Girsanov transform for sub-diffusions. https://arxiv.org/abs/2511.10371

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