# A Conditional Monte Carlo Method for Simulating Conditional Expectations

Project: Research

## Description

In the context of financial engineering and risk management, many quantities of interest are represented by conditional expectations. For instance, in the problem of pricing American options, option price at a given time is denoted by a conditional expectation under the risk-neutral measure; in credit portfolio risk management, capital allocation of the credit risk is characterized by conditional expectations. Moreover, it has been shown that under very general conditions, the sensitivities of Value-at-Risk, probability functions, and financial option prices can be represented by conditional expectations.In many practical problems these conditional expectations are not readily computable, and hence one has to resort to Monte Carlo simulation to estimate them. However, estimating conditional expectations is usually a difficult task in simulation. The major difficulty lies on the fact that conditional expectation is usually specified on a probability zero event. In other words, simulating an observation fulfilling the condition is a probability zero event, which makes the estimation extremely difficult. Kernel method is widely used to estimate conditional expectations. Essentially it uses the observations in a neighborhood of the probability zero event and smoothes them to estimate the conditional expectations. It is easy to implement but has several drawbacks including that it has slow convergence rate and that its performance relies on selection of an appropriate neighborhood. As a result, more efficient estimators are desirable for estimating conditional expectations, for both academia and practitioners.This project aims to develop such efficient estimators for estimating conditional expectations using simulation. The essential idea is to transform a conditional expectation into a ratio of two ordinary expectations by using the techniques of conditional Monte Carlo. Based on this transformation, we devise efficient estimators. Asymptotic behaviors of the proposed estimators will be analyzed. We expect that theoretically the proposed method has a faster convergence rate than the kernel method, and does not involve any selection of neighborhood.Furthermore, we will also apply the proposed method to a number of importance applications, including the problem of pricing American options, the problem of simulating price sensitivities of exotic options, and the problem of measuring capital allocation for portfolio credit risk.

## Detail(s)

Project number 9041601 GRF Finished 1/01/11 → 30/09/13