Nested Multilevel Monte Carlo with Biased and Antithetic Sampling

ArXiv ID: 2308.07835 “View on arXiv”

Authors: Unknown

Abstract

We consider the problem of estimating a nested structure of two expectations taking the form $U_0 = E["\max{“U_1(Y), π(Y)"}”]$, where $U_1(Y) = E[“X\ |\ Y”]$. Terms of this form arise in financial risk estimation and option pricing. When $U_1(Y)$ requires approximation, but exact samples of $X$ and $Y$ are available, an antithetic multilevel Monte Carlo (MLMC) approach has been well-studied in the literature. Under general conditions, the antithetic MLMC estimator obtains a root mean squared error $\varepsilon$ with order $\varepsilon^{"-2"}$ cost. If, additionally, $X$ and $Y$ require approximate sampling, careful balancing of the various aspects of approximation is required to avoid a significant computational burden. Under strong convergence criteria on approximations to $X$ and $Y$, randomised multilevel Monte Carlo techniques can be used to construct unbiased Monte Carlo estimates of $U_1$, which can be paired with an antithetic MLMC estimate of $U_0$ to recover order $\varepsilon^{"-2"}$ computational cost. In this work, we instead consider biased multilevel approximations of $U_1(Y)$, which require less strict assumptions on the approximate samples of $X$. Extensions to the method consider an approximate and antithetic sampling of $Y$. Analysis shows the resulting estimator has order $\varepsilon^{"-2"}$ asymptotic cost under the conditions required by randomised MLMC and order $\varepsilon^{"-2"}|\log\varepsilon|^3$ cost under more general assumptions.

Keywords: Multilevel Monte Carlo (MLMC), Nested Expectations, Risk Estimation, Variance Reduction, Biased Approximations, General Derivatives / Risk Management

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper presents dense theoretical analysis of nested multilevel Monte Carlo methods with bias and antithetic sampling, featuring advanced probability and convergence proofs, but lacks experimental results or implementation details beyond a brief mention of numerical experiments in the summary.

  flowchart TD
    A["Research Goal: Efficient Estimation of Nested Expectations<br>U₀ = E[max{"U₁Y, πY"}"]] --> B["Core Methodology<br>Nested Multilevel Monte Carlo"]
    
    B --> C{"Sampling Approach"}
    
    C --> D["Antithetic MLMC<br>for Outer Expectation U₀"]
    C --> E["Biased MLMC<br>for Inner Expectation U₁Y"]
    
    D --> F["Computational Process<br>Balanced Approximation Strategy"]
    E --> F
    
    F --> G["Key Outcomes & Findings"]
    
    G --> H["Order ε⁻² Cost<br>Under Strong Convergence"]
    G --> I["Order ε⁻²|log ε|³ Cost<br>Under General Assumptions"]