Paul Glasserman

Monte Carlo methods for security pricing

The Monte Carlo approach has proved to be a valuable and flexible computational tool in modern finance. This paper discusses some of the recent applications of the Monte Carlo method to security pricing problems, with emphasis on improvements in efficiency. We first review some variance reduction methods that have proved useful in finance. Then we describe the use of deterministic low-discrepancy sequences, also known as quasi-Monte Carlo methods, for the valuation of complex derivative securities. We summarize some recent applications of the Monte Carlo method to the estimation of partial derivatives or risk sensitivities and to the valuation of American options. We conclude by mentioning other applications.

Pricing American-style securities using simulation

We develop a simulation algorithm for estimating the prices of American-style securities, i.e. securities with opportunities for early exercice. Our algorithm provides both point estimates and error bounds for true security price.

A Sotchastic Mesh Method for Pricing High-Dimensional American Options

High-dimensional pricing problems frequently arise with financial options (examples include basket options, outperformance options, interest-rate and foreign currency options) and real options. American versions of these options, i.e., where the owner has the right to exercise early, are particularly challenging to price. We present a new stochastic mesh method for pricing high-dimensional American options when there is a finite, but possibly large, number of exercise dates. The algorithm provides point estimates and confidence intervals and it converges to the correct values as the computational effort increases. Computational evidence is given which indicates the viability of the method.

A Continuity Correction for Discrete Barrier Options

The payoff of a barrier option depends on whether or not a specified asset price, index, or rate reaches a specified level during the life of the option. Most models for pricing barrier options assume continuous monitoring of the barrier; under this assumption, the option can often be priced in closed form. Many (if not most) real contracts with barrier provisions specify discrete monitoring instants; there are essentially no formulas for pricing these options, and even numerical pricing is difficult. We show, however, that discrete barrier options can be priced with remarkable accuracy using continuous barrier formulas by applying a simple continuity correction to the barrier. The correction shifts the barrier away from the underlying by a factor of exp.flae p 1t/, wherefl… 0:5826,ae is the underlying volatility, and1t is the time between monitoring instants. The correction is justified both theoretically and experimentally.

Importance Sampling for Portfolio Credit Risk

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.

Portfolio Value-at-Risk with Heavy-Tailed Risk Factors

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.

Connecting discrete and continuous path-dependent options

Abstract. This paper develops methods for relating the prices of discrete- and continuous-time versions of path-dependent options sensitive to extremal values of the underlying asset, including lookback, barrier, and hindsight options. The relationships take the form of correction terms that can be interpreted as shifting a barrier, a strike, or an extremal price. These correction terms enable us to use closed-form solutions for continuous option prices to approximate their discrete counterparts. We also develop discrete-time discrete-state lattice methods for determining accurate prices of discrete and continuous path-dependent options. In several cases, the lattice methods use correction terms based on the connection between discrete- and continuous-time prices which dramatically improve convergence to the accurate price.

A Comparison of Some Monte Carlo and Quasi Monte Carlo Techniques for Option Pricing

This article compares the performance of ordinary Monte Carlo and quasi Monte Carlo methods in valuing moderate-and high-dimensional options. The dimensionality of the problems arises either from the number of time steps along a single path or from the number of underlying assets. We compare ordinary Monte Carlo with and without antithetic variates against Sobol’, Faure, and Generalized Faure sequences and three constructions of a discretely sampled Brownian path. We test the standard random walk construction with all methods, a Brownian bridge construction proposed by Caflisch and Morokoff with Sobol’ points and an alternative construction based on principal components analysis also with Sobol’ points. We find that the quasi Monte Carlo methods outperform ordinary Monte Carlo; the Brownian bridge construction generally outperforms the standard construction; and the principal components construction generally outperforms the Brownian bridge construction and is more widely applicable. We interpret both the Brownian bridge and principal components constructions in terms of orthogonal expansions of Brownian motion and note an optimality property of the principal components construction.

How Likely is Contagion in Financial Networks

Interconnections among financial institutions create potential channels for contagion and amplification of shocks to the financial system. We estimate the extent to which interconnections increase expected losses, with minimal information about network topology, under a wide range of shock distributions. Expected losses from network effects are small without substantial heterogeneity in bank sizes and a high degree of reliance on interbank funding. They are also small unless shocks are magnified by some mechanism beyond simple spillover effects; these include bankruptcy costs, fire sales, and mark-to-market revaluations of assets. We illustrate the results with data on the European banking system.

Analysis of an importance sampling estimator for tandem queues

We analyze the performance of an importance sampling estimator for a rare-event probability in tandem Jackson networks. The rare event we consider corresponds to the network population reaching K before returning to ø, starting from ø, with K large. The estimator we study is based on interchanging the arrival rate and the smallest service rate and is therefore a generalization of the asymptotically optimal estimator for an M/M/1 queue. We examine its asymptotic performance for large K, showing that in certain parameter regions the estimator has an asymptotic efficiency property, but that in other regions it does not. The setting we consider is perhaps the simplest case of a rare-event simulation problem in which boundaries on the state space play a significant role.