Nested Simulation in Portfolio Risk Measurement

Abstract

Risk measurement for derivative portfolios almost invariably calls for nested simulation. In the outer step one draws realizations of all risk factors up to the horizon, and in the inner step one re-prices each instrument in the portfolio at the horizon conditional on the drawn risk factors. Practitioners may perceive the computational burden of such nested schemes to be unacceptable, and adopt a variety of second-best pricing techniques to avoid the inner simulation. In this paper, we question whether such short cuts are necessary. We show that a relatively small number of trials in the inner step can yield accurate estimates, and analyze how a fixed computational budget may be allocated to the inner and the outer step to minimize the mean square error of the resultant estimator. Finally, we introduce a jackknife procedure for bias reduction and a dynamic allocation scheme for improved efficiency.

Adaptive importance sampling for extreme quantile estimation with stochastic black box computer models

“…They design the IS density with the exponential tilting by changing the density parameter in an exponential distribution family. Gordy and Juneja (2010) dealt with the risk measurement problem that inevitably requires nested simulation due to the uncertainty between risk evaluation point and the horizon. They used two risk measures: value-at-risk and the probability of large loss.…”

“…Similarly, Broadie, Du, and Moallemi (2011) consider the probability of large loss as a risk measure and propose a sequential approach for allocating more simulation budget to the inner simulation of the outer scenarios located close to the boundary of the tail probability, that is, close to y for the estimator of P(Y > y ), using the optimization problem that maximizes the probability of a sign change. Gordy and Juneja (2010) and Broadie et al (2011), however, do not consider the IS scheme. Recently Hong et al (2017) use the kernel smoothing to estimate the conditional expectation of the portfolio loss given the risk factor, but they do not use the kernel estimator in Monte Carlo simulation.…”

Regression and Kriging Metamodels with Their Experimental Designs in Simulation: Review

“…Applications of jackkni…ng in simulation are numerous; see Gordy and Juneja (2010) and Kleijnen (2015). Distribution-free bootstrapping or nonparametric bootstrapping is another general statistical method that does not assume normality.…”

Efficient Monte Carlo Counterparty Credit Risk Pricing and Measurement

“…Depending on the complexity of the payoff function of the derivative contracts, the valuation step could take straightforward Black-Scholes-type analytical calculations, or it could demand approximations that depending on the desired level of accuracy might be computationally intensive. These approximations could also involve Monte Carlo simulation: Nested Monte Carlo refers to the use of a second layer of Monte Carlo simulation in the valuation step of the above procedure, (see [16]), and regression-based Monte Carlo (see [4]) uses ideas from regression-based Monte Carlo American option pricing, (see Chapter 8 of [13]).…”