Activity: Talk or presentation types › Invited talk
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. In fact, both time series data of asset prices and option derived price data indicate that H often takes values close to 0.1 or even smaller, i.e. rougher than Brownian Motion. The non-Markovian nature of the driving fractional Brownian motion in the rough Bergomi model, however, poses a challenge for numerical options pricing. Indeed, 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 ([Neuenkirch and Shalaiko, arXiv:1606.03854]). We prove rate H + 1/2 for the weak convergence of the Euler method and, in the case of quadratic payoff functions, we obtain rate one. Indeed, the problem 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.