The Order Barrier for Strong Approximation of Rough Volatility Models
Provides a theoretical lower bound (order barrier) for strong approximation of rough volatility, relevant for financial mathematics and stochastic simulation.
The paper establishes that for strong approximation of rough volatility models with Hurst parameter H<1/2, the optimal convergence rate using equidistant discretization is n^{-H}, achieved by Euler and Euler-trapezoidal schemes.
We study the strong approximation of a rough volatility model, in which the log-volatility is given by a fractional Ornstein-Uhlenbeck process with Hurst parameter $H<1/2$. Our methods are based on an equidistant discretization of the volatility process and of the driving Brownian motions, respectively. For the root mean-square error at a single point the optimal rate of convergence that can be achieved by such methods is $n^{-H}$, where $n$ denotes the number of subintervals of the discretization. This rate is in particular obtained by the Euler method and an Euler-trapezoidal type scheme.