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In quantitative finance, modeling the volatility structure of underlyingassets is vital to pricing options. Rough stochastic volatility models, such asthe rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6),887-904, 2016], seek to fit observed market data based on the observation thatthe log-realized variance behaves like a fractional Brownian motion with smallHurst parameter, $H < 1/2$, over reasonable timescales. Both time series ofasset prices and option-derived price data indicate that $H$ often takes valuesclose to $0.1$ or less, i.e., rougher than Brownian motion. This changeimproves the fit to both option prices and time series of underlying assetprices while maintaining parsimoniousness. However, the non-Markovian nature ofthe driving fractional Brownian motion in rough volatility models poses severechallenges for theoretical and numerical analyses and for computationalpractice. While the explicit Euler method is known to converge to the solutionof the rough Bergomi and similar models, its strong rate of convergence is only$H$. We prove rate $H + 1/2$ for the weak convergence of the Euler method forthe rough Stein-Stein model, which treats the volatility as a linear functionof the driving fractional Brownian motion, and, surprisingly, we prove rate onefor the case of quadratic payoff functions. Our proof uses Talay-Tubaroexpansions and an affine Markovian representation of the underlying and isfurther supported by numerical experiments. These convergence results provide afirst step toward deriving weak rates for the rough Bergomi model, which treatsthe volatility as a nonlinear function of the driving fractional Brownianmotion.

Weak error rates for option pricing under linear rough volatility