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Nested Multilevel Monte Carlo with Preintegration for Efficient Risk Estimation

Yu Xu, Xiaoqun Wang

Abstract

Nested Monte Carlo is widely used for risk estimation, but its efficiency is limited by the discontinuity of the indicator function and high computational cost. This paper proposes a nested Multilevel Monte Carlo (MLMC) method combined with preintegration for efficient risk estimation. We first use preintegration to integrate out one outer random variable, which effectively handles the discontinuity of the indicator function, then we construct the MLMC estimator with preintegration to reduce the computational cost. Our theoretical analysis proves that the strong convergence rate of the MLMC combined with preintegration reaches -1, compared with -1/2 for the standard MLMC. Consequently, we obtain a nearly optimal computational complexity. Besides, our method can also handle the high-kurtosis phenomenon caused by indicator functions. Numerical experiments verify that the smoothed MLMC with preintegration outperforms the standard MLMC and the optimal computational cost can be attained. Combining our method with quasi-Monte Carlo further improves its performance in high dimensions. Keywords: Nested simulation, Multilevel Monte Carlo, Risk estimation, Preintegration

Nested Multilevel Monte Carlo with Preintegration for Efficient Risk Estimation

Abstract

Nested Monte Carlo is widely used for risk estimation, but its efficiency is limited by the discontinuity of the indicator function and high computational cost. This paper proposes a nested Multilevel Monte Carlo (MLMC) method combined with preintegration for efficient risk estimation. We first use preintegration to integrate out one outer random variable, which effectively handles the discontinuity of the indicator function, then we construct the MLMC estimator with preintegration to reduce the computational cost. Our theoretical analysis proves that the strong convergence rate of the MLMC combined with preintegration reaches -1, compared with -1/2 for the standard MLMC. Consequently, we obtain a nearly optimal computational complexity. Besides, our method can also handle the high-kurtosis phenomenon caused by indicator functions. Numerical experiments verify that the smoothed MLMC with preintegration outperforms the standard MLMC and the optimal computational cost can be attained. Combining our method with quasi-Monte Carlo further improves its performance in high dimensions. Keywords: Nested simulation, Multilevel Monte Carlo, Risk estimation, Preintegration

Paper Structure

This paper contains 14 sections, 7 theorems, 79 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nested simulation and preintegration
  3. Problem setting and nested simulation
  4. Preintegration
  5. MLMC combined with preintegration
  6. Error analysis
  7. Strong convergence results for MLMC with preintegration
  8. Complexity analysis
  9. Robustness analysis
  10. Numerical experiments
  11. Multiple assets
  12. Conclusion
  13. Appendix
  14. Proof of Lemma \ref{['lemma: boundary_condition_error growth']}

Key Result

Theorem 1

Let $f(\cdot)$ be the probability density function of $\varphi(\bm \omega)$. Assume the following: Then the bias of the nested estimator $\hat{\theta}_{n,m}$ asymptotically satisfies where and $\sigma^2(\boldsymbol{y})$ denotes the conditional variance of $X$ (conditioned on $\bm\omega = \boldsymbol{y}$). $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Expectation for dimensions $d=4,8,16$
  • Figure 2: Variance for dimensions $d=4,8,16$
  • Figure 3: Kurtosis for dimensions $d=4,8,16$
  • Figure 4: Cost for dimensions $d=4,8,16$
  • Figure 5: Variance for dimensions $d=32, 50$
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • ...and 7 more