Series Academic Report of Jiangsu Provincial Center for Applied Mathematics (China University of Mining and Technology)

Title: Sub-diffusive Black-Scholes Model and Girsanov Transform for Sub-diffusions

Speaker: Professor Zhenqing Chen, University of Washington

Time: July 2, 2025 (Wednesday), 15:30-16:30

Venue: Room A321, School of Mathematics

Abstract:

We propose a novel Black-Scholes model under which the stock price processes are modeled by stochastic differential equations driven by a sub-diffusion. The new framework can capture the less financial activity phenomenon during bear markets while having the classical Black-Scholes model as its special case. The sub-diffusive spot market is arbitrage-free but is generally 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 a risk-neutral probability measure 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 involving a fractional derivative in time, which we coin a time-fractional Black-Scholes PDE. (Based on joint work with Shuaiqi Zhang.)

About the Speaker:

Zhenqing Chen is a tenured professor in the Department of Mathematics at the University of Washington (Seattle). He was elected a Fellow of the Institute of Mathematical Statistics in 2007 and a Fellow of the American Mathematical Society in 2014. His research interests lie in probability theory and stochastic processes, focusing on Markov processes and Dirichlet space theory, potential theory, stochastic differential equations, diffusion processes, stable processes, and probabilistic methods in partial differential equations. He currently serves (or has served) as Editor-in-Chief of the international journalPotential Analysisand an editorial board member of journals includingAnnals of Probability,Annals of Applied Probability,Stochastic Processes and their Applications,Electronic Journal of Probability,Journal of Theoretical Probability, andProceedings of the American Mathematical Society. In 2019, he was awarded the Itô Prize. He has published nearly 200 papers in top journals such asJournal of the European Mathematical Society,Memoirs of the American Mathematical Society,Mathematische Annalen,Advances in Mathematics,Communications in Mathematical Physics,Annals of Probability,Probability Theory and Related Fields,Transactions of the American Mathematical Society, andJournal of Functional Analysis, and has authored one monograph.