Multilevel Path Branching for Digital Options

TL;DR

This work introduces a branching path estimator for digital options modeled by SDEs, integrating repeated path splitting with MLMC to create correlated particle paths that share Brownian increments. By averaging over $2^{\hat{\ell}}$ particles and carefully choosing branching times, the estimator achieves favorable variance and cost characteristics, with optimal branching scaling $\tau_{\ell'}\propto 2^{-\eta\ell'}$ and $\eta=2/(p+1)$, leading to MLMC complexities that can match or surpass those for Lipschitz payoffs. The analysis covers both standard Euler–Maruyama and Milstein discretizations, and extends to antithetic Milstein variants that avoid Lévy-area calculations while maintaining variance reductions. Theoretical results are complemented by numerical experiments (GBM, Clark–Cameron SDE) showing reduced variance, bounded kurtosis, and improved computational efficiency over conventional MLMC. The paper also develops density-concentration bounds for elliptic SDEs, enabling broader applicability to density estimation and Greeks, and outlines future work in extending the results to exponentials, higher moments, and SPDE settings.

Abstract

We propose a new Monte Carlo-based estimator for digital options with assets modelled by a stochastic differential equation (SDE). The new estimator is based on repeated path splitting and relies on the correlation of approximate paths of the underlying SDE that share parts of a Brownian path. Combining this new estimator with Multilevel Monte Carlo (MLMC) leads to an estimator with a computational complexity that is similar to the complexity of a MLMC estimator when applied to options with Lipschitz payoffs. This preprint includes detailed calculations and proofs (in grey colour) which are not peer-reviewed and not included in the published article.

Figures (8)

• Figure 1: An illustration of the branching estimator $\Delta \mathcal{P}_{4}$ defined in \ref{['def:est-main']} for $\tau_{\ell'}=2^{-\ell'-1}$ and $h_{\ell}=2^{-\ell}$. \ref{['fig:branching-est-tree']} shows the logical structure ending up in the eight correlated samples of $\Delta P_{4}$. \ref{['fig:branching-est-tree-bm']} shows the eight underlying, correlated Brownian paths.
• Figure 2: Outline of the analysis presented in the current work. Rectangles are assumptions while ellipses are lemmas and theorems. An arrow indicates implication under sufficient but not necessary conditions.
• Figure 3: Numerical verification for \ref{['eq:est-assumpt-cross']} with $p{=}1/2$ and $h_{\ell} {=} 2^{-14}$ for the GBM example in \ref{['eq:gbm']} when using Euler-Maruyama.
• Figure 4: The GBM example in \ref{['eq:gbm']} for $d{=}1$ when using Euler-Maruyama (solid) and Milstein (dashed) in the traditional, $\Delta P_{\ell}$, and branching, $\Delta \mathcal{P}_{\ell}$, estimators. \ref{['fig:Vl-gbm-1d']} shows numerical verification of the variance convergence of $\Delta \mathcal{P}_{\ell}$. \ref{['fig:Wl-gbm-1d']} The work estimate per sample, based on the number of generated samples from the standard normal distribution. The work estimates when using the Milstein scheme are identical. \ref{['fig:kurt-gbm-1d']} The kurtosis of $\Delta \mathcal{P}_{\ell}$. \ref{['fig:total-work-gbm-1d']} The total work estimate of MLMC for different tolerances. This figure illustrates the improved computational complexity of MLMC when using the new branching estimator.
• Figure 5: Numerical verification for \ref{['eq:est-assumpt-cross']} with $h_{\ell} = 2^{-14}$ for the Clark-Cameron example in \ref{['eq:CC']} when using Euler-Maruyama (solid) and antithetic Milstein (dashed). For \ref{['fig:tau-conv-CC-el1d']}, $S = \br{x \in \rset^{2} : \minp{x_{1},x_{2}} \geq 1}$ while for \ref{['fig:tau-conv-CC-1d']} we choose $S = \br{x \in \rset^{2} : x_{2} \geq 1}$.

Theorems & Definitions (26)

• Definition 2.1: Branching Brownian Motion
• Definition 2.2: Branching estimator
• Theorem 2.4: Work and variance
• proof
• Remark 2.5: Optimal eta
• Remark 2.6: Number of branches
• Corollary 2.7
• proof
• Remark 2.8
• proof