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 [Formula: see text]. 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 [Formula: see text] for non Lipschitz payoffs. Finally we show some numerical simulations supporting our theoretical results.

We consider the pricing of derivatives when the evolution of the underlying is given by a continuous time finite state Markov chain. We present a semianalytic approach that consists in (i) simulating the number of transitions of the underlying up to a given time horizon, (ii) computing via an explicit analytic formula the derivative price for each simulated number of transitions, and (iii) approximating the actual price by the empirical average over the values computed in (ii). This corresponds to a Monte Carlo approach with variance reduction by conditioning and, with respect to a plain Monte Carlo, it thus leads to a smaller variance in addition to more precise values. The method is applied, in particular, to path dependent derivatives, and numerical results are presented and discussed.