Risk Estimation via Regression

TLDR

The paper proposes a regression‑based nested Monte Carlo simulation method to estimate financial risk. The method employs an outer simulation to generate risk factors, an inner simulation to price securities and compute portfolio losses, and then applies regression across realizations to estimate the loss function. The regression approach attains an MSE convergence rate of k⁻¹ (up to a bias term) versus k⁻²⁄³ for standard nested simulation, and numerical experiments confirm the theoretical advantage and outperform other methods.

Abstract

We introduce a regression-based nested Monte Carlo simulation method for the estimation of financial risk. An outer simulation level is used to generate financial risk factors and an inner simulation level is used to price securities and compute portfolio losses given risk factor outcomes. The mean squared error (MSE) of standard nested simulation converges at the rate k −2/3 , where k measures computational effort. The proposed regression method combines information from different risk factor realizations to provide a better estimate of the portfolio loss function. The MSE of the regression method converges at the rate k −1 until reaching an asymptotic bias level which depends on the magnitude of the regression error. Numerical results consistent with our theoretical analysis are provided and numerical comparisons with other methods are also given.