[Paper Review] Accurate Computation of the Distribution of Sums of Dependent Log-Normals with Applications to the Black-Scholes Model

Zdravko I. Botev, Robert Salomone+1 more|arXiv (Cornell University)|May 9, 2017

Probability and Risk Models4 citations

TL;DR

This paper introduces a novel Monte Carlo methodology for accurately estimating the cumulative distribution function (CDF), probability density function (PDF), and complementary CDF (right tail) of sums of dependent log-normal random variables—critical in financial risk modeling and option pricing. The method achieves superior accuracy and efficiency, especially in the tails, by leveraging conditional Monte Carlo and quasi-Monte Carlo techniques, with theoretical guarantees of second-order efficiency and vanishing relative error in the left tail.

ABSTRACT

We present a new Monte Carlo methodology for the accurate estimation of the distribution of the sum of dependent log-normal random variables. The methodology delivers statistically unbiased estimators for three distributional quantities of significant interest in finance and risk management: the left tail, or cumulative distribution function, the probability density function, and the right tail, or complementary distribution function of the sum of dependent log-normal factors. In all of these three cases our methodology delivers fast and highly accurate estimators in settings for which existing methodology delivers estimators with large variance that tend to underestimate the true quantity of interest. We provide insight into the computational challenges using theory and numerical experiments, and explain their much wider implications for Monte Carlo statistical estimators of rare-event probabilities. In particular, we find that theoretically strongly-efficient estimators should be used with great caution in practice, because they may yield inaccurate results in the pre-limit. Further, this inaccuracy may not be detectable from the output of the Monte Carlo simulation, because the simulation output may severely underestimate the true variance of the estimator.

Motivation & Objective

• Address the lack of reliable, accurate methods for computing the distribution of sums of dependent log-normal random variables in high-dimensional or tail regimes.
• Overcome the limitations of existing Monte Carlo estimators that suffer from high variance and underestimation in the presence of positive correlation among log-normal factors.
• Develop a unified framework that efficiently estimates the CDF, PDF, and right tail (complementary CDF) of the sum of dependent log-normals across the entire distributional range, including non-asymptotic regions.
• Ensure robustness and accuracy in practical financial applications such as pricing Asian options and computing value-at-risk under the Black-Scholes model with correlated assets.
• Provide a theoretically grounded, second-order efficient estimator whose standard error can be reliably estimated from simulation, enabling trustworthy uncertainty quantification.

Proposed method

• Proposes a conditional Monte Carlo approach that decomposes the sum of dependent log-normals by conditioning on the maximum component, enabling more accurate tail estimation.
• Uses a change-of-measure technique combined with importance sampling to construct an asymptotically efficient estimator for the left tail of the CDF.
• Introduces a smooth, infinitely differentiable estimator for the PDF that enables accelerated convergence in quasi-Monte Carlo (QMC) settings beyond the standard $\mathcal{O}(\sqrt{n})$ rate.
• Derives a refined tail asymptotic for the log-normal distribution (Lemma 2), which underpins the theoretical justification for second-order efficiency.
• Employs a bivariate Gaussian tail probability lemma (Lemma 4) to establish that joint tail probabilities decay faster than individual tail probabilities, enabling rigorous error control.
• Validates the estimator's performance through numerical experiments and theoretical analysis, showing it outperforms existing methods by orders of magnitude in correlated settings.

Experimental results

Research questions

• RQ1 Why do existing Monte Carlo estimators for the right tail of the sum of dependent log-normals fail in the presence of positive correlation, and what causes their high variance and underestimation?
• RQ2 Can a single Monte Carlo estimator be constructed that accurately estimates the CDF, PDF, and right tail of the sum of dependent log-normals across all regions of the distribution?
• RQ3 What conditions ensure that a Monte Carlo estimator for rare-event probabilities is not only asymptotically efficient but also second-order efficient, allowing reliable standard error estimation?
• RQ4 How does the smoothness of the PDF estimator impact convergence rates in quasi-Monte Carlo simulations, and why is this property unique to the proposed method?
• RQ5 To what extent can theoretical refinements of tail asymptotics improve the accuracy and robustness of rare-event simulation in dependent log-normal models?

Key findings

• The proposed CDF estimator is not only asymptotically efficient but also strongly efficient with vanishing relative error in the left tail, ensuring increasing accuracy as the tail becomes more extreme.
• The PDF estimator is infinitely smooth in model parameters, enabling convergence rates in quasi-Monte Carlo simulations that exceed the canonical $\mathcal{O}(\sqrt{n})$ rate, a unique advantage over competing methods.
• In settings with positively correlated log-normal factors—common in financial risk models—the proposed right-tail estimator is more accurate than existing methods by several orders of magnitude, while existing methods exhibit exploding variance.
• Theoretical analysis proves the estimator is second-order efficient, meaning its standard error can be reliably estimated from simulation output, a property absent in prior methods.
• Numerical experiments confirm that many existing estimators for the right tail fail catastrophically under positive correlation, even when they perform well under independence.
• The refined tail asymptotics (Lemma 2) and the bivariate Gaussian tail lemma (Lemma 4) provide a rigorous foundation for proving the estimator’s second-order efficiency and robustness in high-dimensional, dependent settings.