Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities

TL;DR

This work extends multilevel Monte Carlo to incorporate numerical smoothing, targeting the computation of probabilities and densities for SDEs with discontinuous or non-Lipschitz functionals. By coupling a root-finding based smoothing step with a 1D pre-integration and a telescoping MLMC estimator, the authors establish error decompositions and prove that the level-difference variance decays at rate $\mathcal{O}(\Delta t_{\ell})$ under Euler–Maruyama, enabling significant complexity reductions. Theoretical results show near-optimal MLMC complexities with smoothing: $\mathcal{O}(\text{TOL}^{-2-2/s} (\log \text{TOL})^{2})$ for probabilities and $\mathcal{O}(\text{TOL}^{-2} (\log \text{TOL})^{2})$ for densities with Euler, and canonical $\mathcal{O}(\text{TOL}^{-2})$ for probabilities (and, in many cases, densities) with Milstein. Robustness improvements are demonstrated via bounded kurtosis ($\kappa_{\ell}=\mathcal{O}(1)$) at deep levels, countering the high-kurtosis issue of standard MLMC. Numerical experiments on GBM and Heston models for digital options and densities corroborate the theoretical gains, showing dramatic reductions in variance, kurtosis, and overall computational cost.

Abstract

The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in (Quantitative Finance, 23(2), 209-227, 2023), in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence, and consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler--Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions.

Figures (9)

• Figure 4.1: Probability/Digital option under GBM: Convergence plots for MLMC combined with the Euler--Maruyama scheme. The bottom left plot corresponds to a scaled expected cost per level, i.e., ${\mathcal{O}}\left(2^{\gamma \ell}\right)$ (without tracking the constant), with $\gamma$ being the work growth rate.
• Figure 4.2: Probability/Digital option under GBM: Convergence plots for MLMC combined with the Milstein scheme.
• Figure 4.3: Digital option under the GBM model: Comparison of the numerical complexity (expected work (in seconds), $\mathrm{E}\left[W\right]$, vs tolerance, $\text{TOL}$, in a log--log scale) of standard MLMC, and MLMC with numerical smoothing, combined with the Euler or Milstein schemes. When both Euler and Milstein schemes, MLMC combined with numerical smoothing outperforms standard MLMC, and achieves a better numerical complexity rate. The canonical MLMC complexity (i.e., ${\mathcal{O}}\left(\text{TOL}^{-2}\right)$) is obtained when using the Milstein scheme with our approach.
• Figure 4.4: Probability/Digital option under Heston: Convergence plots for MLMC combined with the Euler--Maruyama scheme and FT.
• Figure 4.5: Digital option/probability under the Heston model: Comparison of the numerical complexity (expected work (in seconds), $\mathrm{E}\left[W\right]$, vs tolerance, $\text{TOL}$, in a log--log scale) of the standard MLMC and MLMC with numerical smoothing. MLMC combined with numerical smoothing outperforms standard MLMC and achieves a better numerical complexity rate.

Theorems & Definitions (23)

• Remark 3.6: On the relaxation of Assumption \ref{['ass: uniform boundedness of the drift and diffusion coefficients']}
• Theorem 3.7: Variance estimates for probabilities computation
• Theorem 3.8: Variance estimates for densities estimation
• proof : Proof of Theorem \ref{['corrol: Lipschitz of the integrands']}
• proof : Proof of Theorem \ref{['corrol: Lipschitz of the integrands_density']}
• Corollary 3.9: Complexity of MLMC with numerical smoothing
• proof
• Remark 3.10: About high-order schemes
• Corollary 3.11: Bounded Kurtosis for MLMC with numerical smoothing
• proof