Monte Carlo pricing under fast mean-reverting stochastic volatility: the multi-scale limit ${\epsilon\to 0}$

Abstract

We compute $\E[(S_T-K)^+]$ by Monte Carlo for a scalar stochastic-volatility model with a fast mean-reverting factor of time scale $\eps$, for $\eps$ ranging from $1$ down to $10^{-3}$. A conditional (mixing) estimator gives finite variance, whereas the direct estimator has infinite variance for this model. The volatility factor is simulated with its exact Ornstein--Uhlenbeck transition. As $\eps\to0$ the price converges, at rate $O(\eps)$, to the Black--Scholes price with the averaged volatility $\bar\sigma$, and the implied-volatility smile flattens to $\bar\sigma$. Finally, we test a martingale control variate built on the Black--Scholes delta with volatility $\bar\sigma$. If the martingale is driven by the true volatility $\sigma(Y_t)$, the variance is reduced by a factor that grows like $1/\eps$, about $160$ at $\eps=10^{-3}$. If it is driven by the constant $\bar\sigma$, the variance is essentially not reduced. Last, we calibrate the model with $\rho\neq0$ to S\&P~500 implied volatilities by full simulation of $S$. At the same number of paths the control variate reduces the variance by a factor $1.7$--$16$ (median $5.6$) and gives more accurate calibrated parameters; at equal CPU time it pays off only if the delta is rebalanced on a coarser grid than the time step. For a basket of four indices (state dimension $8$) the gain is larger (median $8$) and the relative cost smaller, so that the control variate is about $4$--$5$ times more efficient than plain Monte Carlo at equal CPU time.