2023/04/06 by Ofelia Bonesini, Bonesini, Ofelia, Antoine Jacquier +3
Economics, Econometrics and Finance · Social Sciences · #Stochastic processes and financial applications #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management
paper · doi:10.48550/arxiv.2304.03042
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 Itô 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 ∈ (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))\wedge1 if the test function is five times differentiable; in particular these conditions are independent of the value of H.