Monte Carlo pricing under fast mean-reverting stochastic volatility: the multi-scale limit epsilon to zero
Abstract
We compute E[(ST−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→0 the price converges, at rate O(eps), to the Black--Scholes price with the averaged volatility σ̅, and the implied-volatility smile flattens to σ̅. Finally, we test a martingale control variate built on the Black--Scholes delta with volatility σ̅. If the martingale is driven by the true volatility σ(Yt), 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 σ̅, the variance is essentially not reduced. Last, we calibrate the model with ρ≠0 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.
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BibTeX
@misc{mertz2026monte,
title = {{Monte Carlo pricing under fast mean-reverting stochastic volatility: the multi-scale limit epsilon to zero}},
author = {Laurent Mertz and Olivier Pironneau},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05765731v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Monte Carlo pricing under fast mean-reverting stochastic volatility: the multi-scale limit epsilon to zero
AU - Laurent Mertz
AU - Olivier Pironneau
PY - 2026
DA - 2026/09/26
PB - HAL
UR - https://hal.science/hal-05765731v1
AB - 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→0 the price converges, at rate O(eps), to the Black--Scholes price with the averaged volatility σ̅, and the implied-volatility smile flattens to σ̅. Finally, we test a martingale control variate built on the Black--Scholes delta with volatility σ̅. If the martingale is driven by the true volatility σ(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 σ̅, the variance is essentially not reduced. Last, we calibrate the model with ρ≠0 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.
ER -