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Importance sampling and active subspace method in quasi-Monte Carlo

Jiaxin Yu, Xiaoqun Wang

TL;DR

This work investigates the combination of importance sampling and the active subspace method under the quasi-Monte Carlo framework and proposes a three-step approach, referred to as the IS-AS-preintegration method, which sequentially applies importance sampling, active subspace, and preintegration.

Abstract

The quasi-Monte Carlo method is widely used in computational finance, whose efficiency strongly depends on the smoothness and effective dimension of the integrand. In this work, we investigate the combination of importance sampling and the active subspace method under the quasi-Monte Carlo framework and propose a three-step approach, referred to as the IS-AS-preintegration method, which sequentially applies importance sampling, active subspace, and preintegration. The proposed method is applied to the option pricing and sensitivity analysis problems in finance, and its performance is evaluated through extensive numerical experiments. The results demonstrate that the proposed method is highly competitive compared with existing popular methods. In particular, for out-of-the-money and deep out-of-the-money options, the proposed approach overcomes the limitations of the preintegration via active subspace method and achieves superior variance reduction, while maintaining comparable performance for other moneyness cases.

Importance sampling and active subspace method in quasi-Monte Carlo

TL;DR

This work investigates the combination of importance sampling and the active subspace method under the quasi-Monte Carlo framework and proposes a three-step approach, referred to as the IS-AS-preintegration method, which sequentially applies importance sampling, active subspace, and preintegration.

Abstract

The quasi-Monte Carlo method is widely used in computational finance, whose efficiency strongly depends on the smoothness and effective dimension of the integrand. In this work, we investigate the combination of importance sampling and the active subspace method under the quasi-Monte Carlo framework and propose a three-step approach, referred to as the IS-AS-preintegration method, which sequentially applies importance sampling, active subspace, and preintegration. The proposed method is applied to the option pricing and sensitivity analysis problems in finance, and its performance is evaluated through extensive numerical experiments. The results demonstrate that the proposed method is highly competitive compared with existing popular methods. In particular, for out-of-the-money and deep out-of-the-money options, the proposed approach overcomes the limitations of the preintegration via active subspace method and achieves superior variance reduction, while maintaining comparable performance for other moneyness cases.
Paper Structure (31 sections, 8 theorems, 89 equations, 6 figures, 3 tables, 4 algorithms)

This paper contains 31 sections, 8 theorems, 89 equations, 6 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Preintegration
  4. Basic Principle
  5. Variance Reduced by Preintegration
  6. Variable Separability and Smoothing Effect
  7. Active Subspace Method
  8. Active Subspace
  9. Dimension Reduction Effect
  10. Active Subspace and Preintegration
  11. Importance Sampling
  12. Basic Principle
  13. Optimal Drift Importance Sampling
  14. Laplace Importance Sampling
  15. Importance Sampling and Preintegration-GPCA
  16. ...and 16 more sections

Key Result

Theorem 3.1

Let $g_1(\bm{x})$ be differentiable almost everywhere, and let $U$ be an orthogonal matrix. By using the coordinate transformation $\bm{y}=U^T\bm{x}$, and defining $g_2(\bm{y}) := g_1(U\bm{y})$, we have that $g_1(\bm{x})$ and $g_2(\bm{y})$ have the same active subspaces.

Figures (6)

  • Figure 1: Convergence behavior in the deep in-the-money case ($K=50$).
  • Figure 2: Convergence behavior in the in-the-money case ($K=80$).
  • Figure 3: Convergence behavior in the at-the-money case ($K=100$).
  • Figure 4: Convergence behavior in the out-of-the-money case ($K=120$).
  • Figure 5: Convergence behavior in the deep out-of-the-money case ($K=150$).
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 3.1
  • Proof 1
  • Corollary 3.2
  • Proof 2
  • Lemma 3.3
  • Proof 3
  • Corollary 3.4
  • Theorem 3.5
  • Proof 4
  • Theorem 3.6
  • ...and 3 more