Importance Sampling in the Heath-Jarrow-Morton Framework PAUL GLASSERMAN, PHILIP HEIDELBERGER, AND PERWEZ SHAHABUDDIN PAUL This article develops a variance-reduction tech- measuring the risks in derivatives portfo- GLASSERMAN is a nique for pricing derivatives by simulation in high- lios. The more realistic — and thus more professor at the Grad- dimensional multifactor models. A premise of this complex — the pricing model used, the uate School of Busi- ness at Columbia work is that the greatest gains in simulation effi- more likely that Monte Carlo will be the University in New ciency come from taking advantage of the structure only viable numerical method for working York. of both the cash flows of a security and the model with the model. The applicability of simu- in which it is priced. For this to be feasible in prac- lation is relatively insensitive to model PHILIP tice requires automating the identification and use details and — in sharp contrast with deter- HEIDELBERGER is a research staff member of relevant structure. ministic numerical methods — to model at the IBM Research dimension. The relevant notion of dimen- Division in Yorktown We exploit model and payoff structure through a sion can vary from one setting to another, Heights, New York. combination of importance sampling and stratified but in general increasing the number of sampling. The importance sampling applies a underlying assets, the number of factors, or PERWEZ change of drift to the underlying factors; we select the number of time steps all increase SHAHABUDDIN is an associate professor the drift by first solving an optimization problem. dimension. Multifactor models of the evo- in the IEOR depart- We then identify a particularly effective direction lution of the yield curve are a particularly ment at Columbia for stratified sampling (which may be thought of as important class of high-dimensional prob- University. an approximate numerical integration) by solving lems often requiring simulation. an eigenvector problem. The main limitation of Monte Carlo simulation is that it is rather slow; put a dif- Examples illustrate that the combination of the ferent way, the results obtained from Monte methods can produce enormous variance reduction Carlo in a short amount of computing time even in high-dimensional multifactor models. The can be very imprecise. Because numerical method introduces some computational overhead in results obtained through Monte Carlo are solving the optimization and eigenvector problems; statistical estimates, their precision is best to address this, we propose and evaluate approxi- measured through their standard error, mate solution procedures, which enhance the appli- which is ordinarily the ratio of a standard cability of the method. deviation per observation to the square root of the number of observations. It follows that there are two ways of onte Carlo simulation has increasing precision: reducing the numera- become an essential tool for tor, or increasing the denominator. Given a pricing and hedging complex fixed time allocated for computing a price, Mderivative securities and for increasing the number of observations 32 IMPORTANCE SAMPLING IN THE HEATH-JARROW-MORTON FRAMEWORK FALL 1999 entails using a faster machine or finding programming eigenvector problem. The calculation of these quanti- speed-ups; such opportunities are fairly quickly ties is a “preprocessing” step of the method (executed exhausted. This leaves the numerator of the standard before any paths are simulated) that systematically iden- error as the main opportunity for improvements. Vari- tifies structure to be exploited by the variance-reduc- ance-reduction techniques attempt to improve the tion methods. precision of Monte Carlo estimates by reducing the In GHS we give a theoretical analysis of this standard deviation (and thus variance) per observation. method based, in part, on scaling the randomness in The literature on Monte Carlo simulation offers the underlying model by a parameter e and investigat- a broad range of methods for attempting to reduce vari- ing asymptotics as eÆ0. Asymptotic optimality prop- ance. The effectiveness of these methods varies widely erties of the method are established in GHS. We test- across applications. In practice, the most commonly ed the method on three types of examples: Asian used methods are the simplest ones, particularly anti- options, a stochastic volatility model, and the Cox- thetic variates and control variates. These can be very Ingersoll-Ross short rate model. Our numerical results effective in some cases, and provide almost no benefit indicate substantial potential for variance reduction in others. Some of the most powerful methods — using the method. importance sampling is a good example — get much The purpose of this article is to develop and less use, in part because they are more difficult to work investigate the use of the method in a much more ambi- with, but also because if used improperly they can give tious class of models — multifactor models of the entire disastrous results. This situation can leave users of term structure of interest rates of the Heath-Jarrow- Monte Carlo in a quandary, not knowing what meth- Morton [1992] type. Using, for example, quarterly rates ods to use in what settings. with a twenty-year horizon makes the state vector for A premise of our work is that the greatest gains such a model eighty-dimensional. Simulating this vector from the use of variance-reduction techniques rely on for m steps in a d-factor model corresponds to sampling exploiting the special structure of a problem or model. in an md-dimensional space, easily making the dimen- The identification of special structure should be auto- sion very large. mated to the extent possible so that each application The preprocessing phase of our method selects a does not require a separate investigation. This perspec- drift for the model — tailored to the factor structure tive is particularly relevant to importance sampling, and the payoff of the instrument to be priced — that which seeks to improve precision by focusing simula- drives the evolution of the forward curve to the most tion effort on the most important regions of the space important region. “Importance” is measured by the from which samples are drawn. Which regions are most product of the discounted payoff from a path and the important depends critically on the underlying model probability density of the path. To further reduce vari- and also on the form of the payoff of the particular ance, we stratify linear combinations of the input ran- security to be priced. The use of importance sampling dom variables, the choice of linear combination also thus requires adaptation to each payoff and each model; tailored to the factor structure and the payoff. The for this to be feasible in practice requires that the adap- complexity of the HJM setting necessitates some sim- tation be automated. plification in the implementation of the preprocessing In Glasserman, Heidelberger, and Shahabuddin calculations, so in addition to investigating the use of [1999] (henceforth GHS), we propose and analyze a the basic method from GHS, we propose and evaluate variance-reduction technique that combines impor- some approximations. tance sampling (based on a change of drift in the under- We test the methods in pricing caps and swap- lying stochastic processes) with stratified sampling. As tions (important for fast calibration), yield spread general methods for variance reduction, these are both options, and flex caps. Our numerical results support the reasonably standard; the innovation in GHS [1999] lies viability of the method in the HJM setting. in the approach used to select the change of drift and the directions along which to stratify. I. OVERVIEW OF THE METHOD In the most general version of the method, the new drift is computed by solving an optimization The general simulation method in GHS [1999], problem, and the stratification direction solves an based on combining importance sampling and strati- FALL 1999 THE JOURNAL OF DERIVATIVES 33 fied sampling, applies to the estimation of an expres- Expectations of functions of X^ may be viewed as sion of the form E[G(Z)] where Z has the standard n- integrals with respect to this density. dimensional normal distribution, and G takes non- More specifically, suppose (1) and (2) give the negative values. G is the discounted payoff of a dynamics of all relevant state variables, including the derivative security, and Z the vector of stochastic value of whatever asset is chosen as numeraire, under the inputs to the simulation. martingale measure associated with the chosen numeraire. (Later, we specialize to the case where Xt The Setting records the forward curve at time t, the numeraire is the money market account, and the martingale measure is Although it is not required for the method, it is the usual risk-neutral measure.) useful to frame the general setting as one of simulating By including enough information in the state a vector diffusion process of the form vector Xt, we can ordinarily make the discounted payoff of a derivative security a deterministic func- ^ dXt = m(Xt, t)dt + s(Xt, t)dWt (1) tion of the path of Xt (or its approximation Xina simulation). The price of the derivative security is the expectation of this discounted payoff. But the where Xt and m(Xt, t) are k-vectors; s(Xt, t) is a k ¥ d path of X^ is itself some deterministic function of the matrix; and Wt is a d-dimensional standard Brownian motion. Processes of this form are commonly simulat- n-dimensional random vector Z, so by letting G ed through a discrete-time approximation (an Euler denote the composition of these two functions we scheme) of the form may denote by G(Z) the discounted payoff of the derivative security associated with input Z. The price of the derivative is XXXitXitZˆˆˆ( ˆ ,) ˆ( ˆ ,) iii++11=+msDD + i i(2) ^ ^ af==ÚEGZ[ ( )]n Gz ()n () zdz (3) where m and s are discrete approximations to the con- R tinuous coefficients; Dt is the simulation time step; and Pricing by Monte Carlo may be viewed as a way of esti- mating such an integral.