Monte Carlo simulation is widely used to measure the credit risk in portfolios of loans, corporate bonds, and other instruments subject to possible default. The accurate measurement of credit risk is often a rare-event simulation problem because default probabilities are low for highly rated obligors and because risk management is particularly concerned with rare but significant losses resulting from a large number of defaults. This makes importance sampling (IS) potentially attractive. But the application of IS is complicated by the mechanisms used to model dependence between obligors; and capturing this dependence is essential to a portfolio view of credit risk. This paper provides an IS procedure for the widely used normal copula model of portfolio credit risk. The procedure has two parts: one applies IS conditional on a set of common factors affecting multiple obligors, the other applies IS to the factors themselves. The relative importance of the two parts of the procedure is determined by the strength of the dependence between obligors. We provide both theoretical and numerical support for the method.

This paper develops efficient methods for computing portfolio value-at-risk (VAR) when the underlying risk factors have a heavy-tailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the delta-gamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an effective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome difficulties associated with VAR calculation with heavy-tailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR.

This paper describes, analyzes and evaluates an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation. Obtaining accurate estimates of such loss probabilities is essential to calculating value-at-risk, which is a quantile of the loss distribution. The method employs a quadratic ("delta-gamma") approximation to the change in portfolio value to guide the selection of effective variance reduction techniques; specifically importance sampling and stratified sampling. If the approximation is exact, then the importance sampling is shown to be asymptotically optimal. Numerical results indicate that an appropriate combination of importance sampling and stratified sampling can result in large variance reductions when estimating the probability of large portfolio losses. 1 Introduction An important concept for quantifying and managing portfolio risk is value-at-risk (VAR) [17, 19]. VAR is defined as a quantile of the loss in portfolio value during a holding ...

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This paper develops a variance reduction technique for pricing derivatives in high-dimensional multifactor models, with particular emphasis on term structure models formulated in the HeathJarrow -Morton framework. A premise of this work is that the largest gains in simulation e- ciency come from taking advantage of the structure of both the cashows of a security and the model in which it is priced; for this to be feasible in practice requires that the identication and use of relevant structure be automated. We exploit model and payo structure through a combination of importance sampling and stratied sampling. The importance sampling applies a change of drift to the underlying factors; we select the drift by rst solving an optimization problem. We then identify a particularly eective direction for stratied sampling (which may be thought of as an approximate numerical integration) by solving an eigenvector problem. Examples illustrate that the combination of the methods can pro...

Linear recurrences modulo 2 with long periods have been widely used for contructing (pseudo)random number generators. Here, we use them for quasiMonte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period length and to estimate the integral by the average value of the integrand over all vectors of successive output values produced by the small generator. We examine randomizations of this scheme, discuss criteria for selecting the parameters, and provide examples. This approach can be viewed as a polynomial version of lattice rules. 1 MONTE CARLO VS QUASI-MONTE CARLO 1.1 The Monte Carlo Method The aim of most stochastic simulations is to estimate a mathematical expectation, and this can be put into the framework of estimating the integral of a function f over the t-dimensional unit hypercube [0; 1) t , namely ¯ = Z [0;1) t f(u)du: (1) Randomness in simulations is indeed generated from a se...

. In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple one-dimensional partial dierential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases. Key words: Asian options, Options on a traded account, Brownian motion, xed strike, oating strike. 1 Introduction Asian options are securities with payo which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. A number of approximations that produce closed form expressions have appeared, se...

This paper develops approximations for the distribution of losses from default in a normal copula framework for credit risk. We put particular emphasis on approximating small probabilities of large losses, as these are the main requirement for calculation of value at risk and related risk measures. Our starting point is an approximation to the rate of decay of the tail of the loss distribution for multifactor, heterogeneous portfolios. We use this decay rate in three approximations: a homogeneous single-factor approximation, a saddlepoint heuristic, and a Laplace approximation. The first of these methods is the fastest, but the last two can be surprisingly accurate at very small loss probabilities. The accuracy of the methods depends primarily on the structure of the correlation matrix in the normal copula. Calculation of the decay rate requires solving an optimization problem in as many variables as there are factors driving the correlations, and this is the main computational step underlying these approximations. We also derive a two-moment approximation that fits a homogeneous portfolio by matching the mean and variance of the loss distribution. Whereas

Pricing financial options often requires Monte Carlo methods. One particular case is that of barrier options, whose payoff may be zero depending on whether or not an underlying asset crosses a barrier during the life of the option. This paper develops variance reduction techniques that take advantage of the special structure of barrier options, and are appropriate for general simulation problems with similar structure. We use a change of measure at each step of the simulation to reduce the variance due to the possibility of a barrier crossing at each monitoring date. The paper details the theoretical underpinnings of this method, and evaluates alternative implementations when exact distributions conditional on one-step survival are available and when not available. When these one-step conditional distributions are unavailable, we introduce algorithms that combine change of measure and estimation of conditional probabilities simultaneously. The methods proposed are more generally applicable to terminal reward problems on Markov processes with absorbing states.

This paper proposes and evaluates variance reduction techniques for e#cient estimation of portfolio loss probabilities using Monte Carlo simulation. Precise estimation of loss probabilities is essential to calculating value-at-risk, which is simply a percentile of the loss distribution. The methods we develop build on delta-gamma approximations to changes in portfolio value. The simplest way to use such approximations for variance reduction employs them as control variates; we show, however, that far greater variance reduction is possible if the approximations are used as a basis for importance sampling, stratified sampling, or combinations of the two. This is especially true in estimating very small loss probabilities. 1.1 Introduction Value-at-Risk (VAR) has become an important measure for estimating and managing portfolio risk [Jorion 1997, Wilson 1999]. VAR is defined as a certain quantile of the change in a portfolio's value during a specified holding period. To be more specific...