Rate of convergence of Monte Carlo simulations for the Hobson-Rogers model

The Hobson–Rogers model is used to price derivative securities under the no-arbitrage condition in a stochastic volatility setting, preserving the completeness of the market. Here we are studying the rate of convergence of the Euler/Monte Carlo approximations, when pricing European, Asian and digital type options. The aim of the present work is to express the approximation error in terms of the time step size, denoted by h, used for the Euler scheme. We recover an already known result, obtained by other authors using PDE approximations, for European options. Namely we show that for a Lipschitz coefficient of the driving equations for the asset price and Lipschitz payoffs, we obtain an error of the order of square root of h. Moreover, using Malliavin Calculus techniques, we show that with a regular coefficient we may attain an error of the order of h for regular payoffs and of the order of for non Lipschitz payoffs. Finally we show some numerical simulations supporting our theoretical results.