Importance Sampling for Heston Option Pricing in Rare-Event Regimes

Summary

This paper develops importance-sampling schemes to price European call options under the Heston stochastic-volatility model in two settings where ordinary Monte Carlo can have high variance: very short maturities and deeply out-of-the-money strikes. Using large-deviation analysis of the log-price cumulant generating function, it constructs a state-dependent change of measure intended to make rare pricing events more likely in simulation.

For short maturities, the authors prove logarithmic efficiency by showing that their proposed drift minimizes the asymptotic decay rate of the estimator’s second moment. For deep out-of-the-money options, they introduce a scaling in which variance mean reversion slows as log-moneyness grows, then use Riccati analysis to establish the method’s asymptotic properties. Numerical experiments report variance reductions of several orders of magnitude relative to standard estimators in both regimes. These conclusions concern the stated asymptotic settings; the excerpt does not provide implementation details, specific option market comparisons, or evidence about performance outside them.

Key ideas

• The proposed importance-sampling methods target high-variance rare events in short-maturity and deep out-of-the-money Heston option pricing.
• A state-dependent change of measure is derived using large-deviation behavior of the log-price distribution.
• In the short-maturity regime, the paper proves logarithmic efficiency for its proposed drift.
• For deep out-of-the-money options, the method uses a slow mean-reversion scaling for variance and a specialized Riccati analysis.
• Numerical experiments report substantial variance reduction against standard estimators in both regimes.

Tags

# Efficient Importance Sampling under Heston Model: Short Maturity and Deep Out-of-the-Money Options


# Efficient Importance Sampling under Heston Model: Short Maturity and Deep Out-of-the-Money Options









This paper investigates asymptotically optimal importance sampling (IS) schemes for pricing European call options under the Heston stochastic volatility model. We focus on two distinct rare-event regimes where standard Monte Carlo methods suffer from significant variance deterioration: the limit as maturity approaches zero and the limit as the strike price tends to infinity. Leveraging the large deviation principle (LDP), we design a state-dependent change of measure derived from the asymptotic behavior of the log-price cumulant generating functions. In the short-maturity regime, we rigorously prove that our proposed IS drift, inspired by the variational characterization of the rate function, achieves logarithmic efficiency (asymptotic optimality) by minimizing the decay rate of the second moment of the estimator. In the deep OTM regime, we introduce a novel slow mean-reversion scaling for the variance process, where the mean-reversion speed scales as the inverse square of the small-noise parameter (defined as the reciprocal of the log-moneyness). We establish that under this specific scaling, the variance process contributes non-trivially to the large deviation rate function, requiring a specialized Riccati analysis to verify optimality. Numerical experiments demonstrate that the proposed method yields substantial variance reduction--characterized by factors exceeding several orders of magnitude--compared to standard estimators in both asymptotic regimes.

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