The classical models for asset processes in math finance are SDEs driven by Brownian motion of the following type $X_t=x+\int_0^tb(s,X_s)ds+\int_0^t\sigma(s,X_s)dB_s$. Then $u(t,X_t)=\mathbb E[{g(X_T)}|\mathcal F_{t}^X]$ is a deterministic function of $X_t$ and $u(t,x)$ solves a parabolic PDE. In this talk, I will talk about two types of path-dependent option pricing problems. In the first scenario, the option function depends on the whole path of the Markov process $X$, the option pricing problem is indeed to compute $\mathbb E^{\mathbb P}[g(X_{[0,T]})]$; In the second scenario, we consider the option pricing problem for rough volatility models, where the volatility of the asset process follows a Volterra SDE. The option function depends on the terminal value of a non-Markovian asset process. In both cases, the function $u(t,\cdot)$ solves the so-called Path-Dependent PDEs. Due to the path-dependent feature, the standard numerical algorithms are not efficient for both cases.

We introduce the "Signature" idea from the Rough Path theory to our numerical algorithms to improve the efficiency. Our first algorithm is based on "deep signature" and deep learning methods for BSDEs; Our second algorithm is based on cubature formula for "Volterra signature", which is motived from the "cubature formula" for the signature of Brownian motion.

The talk is based on two joint works with Man Luo, Zhaoyu Zhang, and Jianfeng Zhang. Speaker(s): Qi Feng (UM)