In quantitative finance, modeling the volatility structure of underlying assets is a key component in the pricing of options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, H < 1/2, over reasonable timescales. Both time series data of asset prices and option derived price data indicate that H often takes values close to 0.1 or smaller, i.e. rougher than Brownian Motion. This change greatly improves the fit to time series data of underlying asset prices as well as to option prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in the rough Bergomi model poses severe challenges for theoretical and numerical analyses as well as for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi model, the strong rate of convergence is only H. We prove rate H + 1/2 for the weak convergence of the Euler method and, surprisingly, in the case of quadratic payoff functions we obtain rate one. Indeed, the problem of weak convergence for rough Bergomi is very subtle; we provide examples demonstrating that the rate of convergence for payoff functions well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof relies on Taylor expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments.
In quantitative finance, modeling the volatility structure of underlying assets is a key component in the pricing of options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, H < 1/2, over reasonable timescales. Both time series data of asset prices and option derived price data indicate that H often takes values close to 0.1 or smaller, i.e. rougher than Brownian Motion. This change greatly improves the fit to time series data of underlying asset prices as well as to option prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in the rough Bergomi model poses severe challenges for theoretical and numerical analyses as well as for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi model, its strong rate of convergence is only H. For a simplified rough Bergomi model, we prove rate H + 1/2 for the weak convergence of the Euler method and, surprisingly, in the case of quadratic payoff functions we obtain rate one. Indeed, the problem of weak convergence for rough Bergomi is very subtle; we provide examples demonstrating that the rate of convergence for payoff functions well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof relies on Taylor expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments.
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For a simplified rough Bergomi model, we