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

ArXiv ID: 2511.19826 “View on arXiv”

Authors: Yun-Feng Tu, Chuan-Hsiang Han

Abstract

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.

Keywords: Importance Sampling, Heston Model, Large Deviation Principle (LDP), Monte Carlo, Stochastic Volatility, Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring advanced concepts like large deviation principles, Riccati analysis, and asymptotic proofs, but its empirical implementation is limited to theoretical numerical experiments without actual backtest-ready data or code, placing it in the ‘Lab Rats’ quadrant.

  flowchart TD
    A["Research Goal<br>Efficient Importance Sampling for Heston Model<br>Short Maturity & Deep OTM Regimes"] --> B["Methodology: Large Deviation Principle LDP<br>Analyze asymptotic behavior of log-price C.G.F."]
    
    B --> C{"Two Rare-Event Regimes"}
    
    C --> D["Regime 1: Short Maturity<br>State-dependent change of measure<br>Minimize 2nd moment decay rate"]
    C --> E["Regime 2: Deep OTM<br>Slow mean-reversion scaling<br>Specialized Riccati analysis"]
    
    D --> F["Computational Process<br>Monte Carlo Pricing<br>Variance Reduction Analysis"]
    E --> F
    
    F --> G["Outcomes<br>Logarithmic Efficiency achieved<br>Proven asymptotic optimality<br>Numerical variance reduction >10^3"]