Rough volatility, path-dependent PDEs and weak rates of convergence

Ofelia Bonesini, Antoine Jacquier and Alexandre Pannier

Papers from arXiv.org

Abstract: In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of convergence for discretised stochastic integrals of smooth functions of a Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in (0,1/2)$. These integrals approximate log-stock prices in rough volatility models. We obtain the optimal weak error rates of order~$1$ if the test function is quadratic and of order~$(3H+1/2)\wedge 1$ if the test function is five times differentiable; in particular these conditions are independent of the value of~$H$.

Date: 2023-04, Revised 2025-01
New Economics Papers: this item is included in nep-rmg
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