A general valuation framework for rough stochastic local volatility models and applications

Rough volatility models are a new class of stochastic volatility models that have been shown to provide a consistently good fit to implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with the corresponding Hurst parameter less than a half. Albeit the empirical success, the valuation of derivative securities under rough volatility models is challenging. The reason is that it is neither a semi-martingale nor a Markov process. This paper proposes a novel valuation framework for rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semi-martingale approximation for the RSLV model and establish its existence, uniqueness, Markovian representation and convergence. Then we propose a fast continuous-time Markov chain (CTMC) approximation algorithm to the PSLV model and establish its convergence. Numerical experiments demonstrate the convergence of our approximation method to the true prices, and also the remarkable accuracy and efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.