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Mathematics Colloquium

Qi Feng
University of Michigan

Title: Signature and Deep Learning Methods for Path-Dependent Problems in Mathematical Finance
Date: Wednesday, January 18, 2023
Place and Time: fsu.zoom.us/s/97976878227, 3:05-3:55 pm


Path dependence is the dependence of financial assets on the path of previous outcomes rather than simply on current states. For a path-dependent process, "history matters"-it has an enduring influence. This talk will discuss two types of path-dependent problems arising from option pricing and optimal stopping in the financial market. In the first scenario, the asset price process has stochastic volatility modeled through the stochastic Volterra integral equation; In the second scenario, the option payoff depends on the whole paths of the asset price modeled by a semimartingale process. In both cases, the solutions to the option pricing problems are the solutions for two different pathdependent PDEs(PPDEs). We focus on designing efficient numerical algorithms to solve path-dependent option pricing problems (i.e., PPDEs). To overcome the path dependency, we introduce the "Signature" (i.e., iterated integrals of the paths) idea from the Rough Paths theory to our algorithms. The first algorithm introduces the "Volterra signature" to construct the cubature formula for stochastic volatility model. The second algorithm combines "deep signature" and deep learning methods for forward-backward stochastic differential equations(FBSDEs); In the end, we will discuss the connection and other applications of signature and deep learning in mathematical finance. This talk combines recent joint works with Erhan Bayraktar, Man Luo, Zhaoyu Zhang, and Jianfeng Zhang.