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

Yun-Feng Tu, Chuan-Hsiang Han

TL;DR

This work develops an asymptotically optimal importance sampling strategy for pricing European calls under the Heston model by exploiting large deviation principles. It introduces a state-dependent drift affine in the square root of the variance, achieving logarithmic efficiency in the short-maturity regime and, via a slow mean-reversion scaling, nontrivial large-deviation behavior in the deep out-of-the-money regime. The core contributions include a Riccati-based analysis that connects cumulant generating functions to optimal drifts, a rigorous treatment of two distinct asymptotic limits, and substantial numerical variance reductions demonstrated through extensive experiments. The approach preserves the affine structure of the Heston dynamics, enabling tractable computations while delivering dramatic variance reductions in regimes where standard Monte Carlo fails. The results offer a practical path to efficient evaluation of rare-event probabilities in stochastic-volatility models and suggest extensions to path-dependent or rough volatility settings.

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.

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

TL;DR

This work develops an asymptotically optimal importance sampling strategy for pricing European calls under the Heston model by exploiting large deviation principles. It introduces a state-dependent drift affine in the square root of the variance, achieving logarithmic efficiency in the short-maturity regime and, via a slow mean-reversion scaling, nontrivial large-deviation behavior in the deep out-of-the-money regime. The core contributions include a Riccati-based analysis that connects cumulant generating functions to optimal drifts, a rigorous treatment of two distinct asymptotic limits, and substantial numerical variance reductions demonstrated through extensive experiments. The approach preserves the affine structure of the Heston dynamics, enabling tractable computations while delivering dramatic variance reductions in regimes where standard Monte Carlo fails. The results offer a practical path to efficient evaluation of rare-event probabilities in stochastic-volatility models and suggest extensions to path-dependent or rough volatility settings.

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.

Paper Structure

This paper contains 57 sections, 11 theorems, 119 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Importance Sampling and Large Deviations
  3. Existing Approaches and Limitations
  4. Main Contributions
  5. Problem Formulation and Preliminaries
  6. The Heston Stochastic Volatility Model
  7. Large Deviation Principle
  8. Importance Sampling Framework
  9. Asymptotic Optimality.
  10. Change of Measure and Drift Design.
  11. Construction of the Proposed Measure
  12. Heuristic for Choosing $\bar{h}$.
  13. Short-Maturity Asymptotics ($T \to 0$)
  14. First Moment Analysis
  15. Second Moment Analysis
  16. ...and 42 more sections

Key Result

Theorem 2.2

If the limiting scaled cumulant generating function (SCGF) exists and is essentially smooth. Let $\mathcal{D}_\Gamma \coloneqq \{ \lambda \in \mathbb{R}^d : \Gamma(\lambda) < \infty \}$ denote the effective domain of $\Gamma$. Then, $\{Z^\varepsilon\}$ satisfies an LDP with a good rate function $\Lambda(x)$ given by the Fenchel-Legendre transform of $\Gamma

Figures (3)

  • Figure 1: VRR vs. Moneyness for $T=1/252$ (1 Day). The VRR grows exponentially with moneyness, exceeding 2500 for deep OTM strikes.
  • Figure 2: VRR vs. Moneyness for $T=21/252$ (1 Month). Significant variance reduction is maintained, peaking around 450.
  • Figure 3: VRR vs. Moneyness for $T=1$ (1 Year). The VRR saturates below 250, indicating the diminishing effectiveness of the specific drift as the time horizon allows for more complex path dynamics.

Theorems & Definitions (15)

  • Definition 2.1: Large Deviation Principle
  • Theorem 2.2: Gärtner-Ellis Theorem
  • Lemma 3.1: Forde and Jacquier forde2011small
  • Proposition 3.2: First Moment Decay Rate
  • Proposition 3.3: Limit of Integrated Variance Moment
  • Proposition 3.4: Auxiliary SCGF $\Gamma_{II}$
  • Theorem 3.5: Short-Maturity Asymptotic Optimality
  • proof
  • Proposition 4.1: Limiting SCGF
  • Proposition 4.2: First Moment Decay Rate
  • ...and 5 more