Randomized Quasi-Monte Carlo and Importance Sampling for Super-Fast Growing Functions with Applications to Finance

TL;DR

The paper tackles efficient high-dimensional integration for discontinuous payoff functions in finance, where standard RQMC struggles due to critical exponential growth. It develops a generalized RQMC-IS framework with growth Ce^{A|x|^2+B|x|} and two tail conditions on the IS density to achieve near-optimal convergence, including the challenging A=1/2 case. It proves Gaussian proposals used in ODIS satisfy the light-tailed condition and that preintegration preserves growth, enabling application to preintegrated payoffs. Numerical experiments on Black-Scholes Asian options, basket options, and Heston models confirm substantial variance reduction and convergence rates predicted by theory.

Abstract

Many problems can be formulated as high-dimensional integrals of discontinuous functions that exhibit significant boundary growth, challenging the error analysis and applications of randomized quasi-Monte Carlo (RQMC) methods. This paper studies RQMC methods for super-fast growing functions satisfying generalized exponential growth conditions, with a special focus on financial derivative pricing. The main contribution of this paper is threefold. First, by combining RQMC with importance sampling (IS), we derive a new error bound for a class of integrands, whose values and derivatives are bounded by the critical growth function $e^{A|\boldsymbol{x}|^2}$ with $A = 1/2$. This result extends the existing results in the literature, which are limited to the case $A < 1/2$. We demonstrate that by imposing a light-tailed condition on the proposal distribution of IS, RQMC can achieve an error rate of $O(n^{-1 + ε})$ with a sample size n and an arbitrarily small $ε>0$. Second, we verify that the Gaussian proposals used in Optimal Drift Importance Sampling (ODIS) satisfy the required light-tailed condition, providing a rigorous theoretical guarantees for RQMC-ODIS in critical growth scenarios. Third, for discontinuous integrands from finance, we prove that the integrands after preintegration satisfy the exponential growth condition. This ensures that the preintegrated functions can be seamlessly incorporated into our RQMC-IS framework. Numerical experiments on financial derivative pricing validate our theory, showing that the RQMC-IS with preintegration is effective in handling problems with discontinuous payoffs, successfully achieving the expected convergence rates.

Figures (6)

• Figure 1: RMSEs for the single asset Asian option with $d=80$. The RMSEs are computed based on $100$ repetitions. The figure has two reference lines proportional to labeled powers of $n$.
• Figure 2: RMSEs for the basket Asian option with $2d=30$. The RMSEs are computed based on $100$ repetitions. The figure has two reference lines proportional to labeled powers of $n$.
• Figure 3: RMSEs for the basket Asian option with $2d=60$. The RMSEs are computed based on $100$ repetitions. The figure has two reference lines proportional to labeled powers of $n$.
• Figure 4: RMSEs for the Heston model with 2d=10. The RMSEs are computed based on 100 repetitions.The figure has two reference lines proportional to labeled powers of n.
• Figure 5: RMSEs for the Heston model with $2d=30$. The RMSEs are computed based on $100$ repetitions.The figure has two reference lines proportional to labeled powers of $n$.

Theorems & Definitions (20)

• Definition 1
• Proposition 2
• Proposition 3
• Remark 1
• Theorem 4
• Lemma 5
• Lemma 6
• proof : Proof of Theorem \ref{['thm:MainTheorem']}
• Theorem 7
• proof