WSC 2004 Final Abstracts
Sunday 1:00:00 PM 2:30:00 PM
Simulation in Finance
Chair: Jeremy Staum (Northwestern University)
Quasi-Monte Carlo (QMC) methods have been used in a variety of problems in finance over the last few years, where they often provide more accurate estimators than the Monte Carlo (MC) method. These results have led many researchers to try to find reasons for the success of QMC methods in finance. A general explanation is that financial problems often have a structure that interacts in a constructive way with the point set used by the QMC method, thus resulting in estimators with reduced error. This positive interaction can be amplified by various fine-tuning techniques, which we review in the first part of this paper. Leaving aside these techniques, we then choose a few randomized QMC methods and test their "robustness" by comparing their performance against MC on different financial problems. Our results suggest that the chosen methods are efficient in a broad sense for financial simulations.
Efficient Pricing of Barrier Options with the Variance-Gamma Model
Athanassios N. Avramidis (Université de Montréal)
We develop an efficient Monte Carlo algorithm for pricing barrier options with the variance gamma model (Madan, Carr, and Chang 1998). After generalizing the double-gamma bridge sampling algorithm of Avramidis, L'Ecuyer, and Tremblay (2003), we develop conditional bounds on the process paths and exploit these bounds to price barrier options. The algorithm's efficiency stems from sampling the process paths up to a random resolution that is usually much coarser than the original path resolution. We obtain unbiased estimators, including the case of continuous-time monitoring of the barrier crossing. Our numerical examples show large efficiency gain relative to full-dimensional path sampling.
Simulation of Coherent Risk Measures
Vadim Lesnevski, Barry L. Nelson, and Jeremy Staum (Northwestern University)
In financial risk management, a coherent risk measure equals the maximum expected loss under several different probability measures, which are analogous to systems in ranking and selection. Here it is the best system's expected value and not identity that is of interest. We explore the correctness and computational efficiency of simulated confidence intervals for a maximum of several expectations.
Randomized Quasi-Monte Carlo: A Tool for Improving the Efficiency of Simulations in Finance
Christiane Lemieux (University of Calgary)
Sunday 3:00:00 PM 4:30:00 PM
Derivatives Pricing and Risk-sensitive Estimation
Chair: Michael Fu (University of Maryland)
We propose the algorithms for pricing American and European options in incomplete markets. We consider a non-self-financing replicating portfolio and minimize the hedging error consisting of the self-financing error of the portfolio dynamics and the error of the option’s payoff replication. We treat the pricing problem as regression with constraints and reduce it to a quadratic minimization problem. The algorithms of pricing American and European options differ in imposing one additional type of constraints. Prices of options for different initial and strike prices can be found in one optimization run. The algorithms create a table representing the option price as a function of time and the underlying stock price for the whole lifetime of the option. We illustrate the numerical performance of the algorithms with options on futures contracts in natural gas market.
Simulation-Based Pricing of Mortgage-Backed Securities
Jian Chen (Fannie Mae)
Mortgage-Backed-Securities (MBS), as the largest invest-ment class of fixed income securities, have always been hard to price. Because of the following reasons, normal numerical methods like lattice methods, or finite difference method for solving PDEs are hard to apply: 1) the path de-pendence of mortgage pool cash flows. 2) the embedded American call option to prepay. 3) the American put option to default. 4) the fact that mortgage borrower do not/cannot exercise these option optimally. And those reasons make Monte Carlo simulation the best approach to price MBS. A standard MBS pricing framework would consists the fol-lowing parts:1) Interest Rate model. 2) Prepayment model, which consists house turnover model and refinance model. 3) OAS model, which captures risk factors from the market price. Those factors are not accounted for in the previous two models. In order to hedge MBS efficiently and effec-tively, we need to calculate hedging measures quickly and correct. Chen and Fu (2001, 2002, 2003) has developed some efficient hedging algorithm in the past to perform this task.
Risk and Information in the Estimation of Hidden Markov Models
Vahid R. Ramezani, Steven I. Marcus, and Michael Fu (University of Maryland)
In this paper, we consider the relationship between risk-sensitivity and information. Product estimators are introduced as a generalization of Maximum A Posteriori Probability (MAP) estimator for Hidden Markov Models. We study the relationship between the inclusion of higher order moments, the underlying dynamics and the availability of information. Asymptotic periodicity of these estimators and the relationship between risk and information is studied via simulation.
Pricing Derivative Securities in Incomplete Markets
Sergey Sarykalin and Stan Uryasev (University of Florida)
Monday 10:30:00 AM 12:00:00 PM
New Simulation Methodology for Financial Engineering
Chair: Perwez Shahabuddin (Columbia University)
This paper investigates the adequacy of various principal components (p.c.) approaches as data reduction schemes for processing contingent claim valuations on baskets of equities. As a general proposition we are interested in discovering possible features and rules-of-thumb for the applicability of p.c. techniques. In particular, what accuracy does one lose in valuation-hedging schemes as the dimensionality of the p.c. space is reduced? We also have an interest in validating the posted "stylized" facts of implied volatility as they apply to our data sets.
Exact Simulation of Option Greeks under Stochastic Volatility and Jump Diffusion Models
Mark Broadie and Özgür Kaya (Columbia University)
This paper derives Monte Carlo simulation estimators to compute option price derivatives, i.e., the 'Greeks,' under Heston's stochastic volatility model and some variants of it which include jumps in the price and variance processes. We use pathwise and likelihood ratio approaches together with the exact simulation method of Broadie and Kaya (2004) to generate unbiased estimates of option price derivatives in these models. By appropriately conditioning on the path generated by the variance and jump processes, the evolution of the stock price can be represented as a series of lognormal random variables. This makes it possible to extend previously known results from the Black-Scholes setting to the computation of Greeks for more complex models. We give simulation estimators and numerical results for some path-dependent and path-independent options.
A Unified Approach for Finite-Dimensional, Rare-Event Monte Carlo Simulation
Zhi Huang (Lehman Brothers) and Perwez Shahabuddin (Columbia University)
We consider the problem of estimating the small probability that a function of a finite number of random variables exceeds a large threshold. Each input random variable may be light-tailed or heavy-tailed. Such problems arise in financial engineering and other areas of operations research. Specific problems in this class have been considered earlier in the literature, using different methods that depend on the special properties of the particular problem. Using the Laplace principle (in a restricted finite-dimensional setting), this paper presents a unified approach for deriving the log-asymptotics, and developing provably efficient fast simulation techniques using the importance sampling framework of hazard rate twisting.
An Examination of Forward Volatility
Ray Popovic and David Goldsman (Georgia Institute of Technology)
Monday 1:30:00 PM 3:00:00 PM
Tutorial: Portfolio Credit Risk Management
Chair: Paul Glasserman (Columbia University)
Tutorial on Portfolio Credit Risk Management
William J. Morokoff (Moody's KMV)
The distribution of possible future losses for a portfolio of credit risky corporate assets, such as bonds or loans, shows strongly asymmetric behavior and a fat tail as the consequence of the limited upside of credit (the promised coupon payment) and substantial downside if the corporation defaults. Because of correlation, it is not possible to fully diversify away this fat tail. Detailed correlation models require Monte Carlo simulation to determine the loss distribution for a credit portfolio. This tutorial covers the basics of credit risk modeling including an overview of the credit markets, a summary of what data are available for defining and calibrating models, and a discussion of key modeling questions. Finally a detailed discussion of simulation methods used in calculation credit portfolio loss distribution and related credit risk measures is presented.
Monday 3:30:00 PM 5:00:00 PM
Chair: John Charnes (University of Kansas)
Portfolio Credit Risk Analysis Involving CDO Tranches
Menghui Cao and William J. Morokoff (Moody's KMV)
Credit risk analysis for portfolios containing CDO tranches is a challenging task for risk managers. We propose here a basis function approach for CDO tranche valuation and portfolio risk analysis at horizon, based on a multi-step Monte Carlo simulation model. The idea is to approximate the expected value of the tranche at horizon by a linear combination of basis functions, which are chosen to best characterize the current state of the associated CDO. It can be generalized for portfolio risk analysis involving any complex financial instruments.
A Simulation-Based First-to-Default (FtD) Credit Default Swap (CDS) Pricing Approach under Jump-Diffusion
Tarja Joro and Anne R Niu (University of Alberta) and Paul Na (Bayerische Landesbank )
In recent years, credit derivatives market has grown explo-sively and credit derivatives have become popular tools for hedging credit risk of financial institutions. Among the more sophisticated credit derivatives are the ones where the contingent payoffs depend on the dependence relation-ship among several firms in a basket such as First-to-Default Credit Default Swap. In this paper, we present a simulation-based First-to-Default Credit Derivative Swap pricing approach under jump-diffusion and compares it with the popular default-time approach via Copula.
Approximating Free Exercise Boundaries for American-Style Options Using Simulation and Optimization
Barry R. Cobb and John M. Charnes (The University of Kansas)
Monte Carlo simulation can be readily applied to asset pricing problems with multiple state variables and possible path dependencies because convergence of Monte Carlo methods is independent of the number of state variables. This paper applies Monte Carlo simulation to the problem of determining free exercise boundaries for pricing American-style options. We use a simulation-optimization method to identify approximately optimal exercise thresholds that are defined by a minimal number of parameters. We demonstrate that asset prices calculated using this method are comparable to those found using other numerical asset pricing methods.
Tuesday 8:30:00 AM 10:00:00 AM
Tutorial: QMC Methods in Finance and Risk Analysis
Chair: William Morokoff (Moody's KMV)
Quasi-Monte Carlo Methods in Finance
Pierre L'Ecuyer (Université de Montréal)
We review the basic principles of Quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, and the main classes of constructions underlying their implementations: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate integrals over the s-dimensional unit hypercube, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We give examples and provide computational results that illustrate the efficiency improvement achieved.
Tuesday 10:30:00 AM 12:00:00 PM
Applications of Simulation in Pricing and Hedging Derivatives
Chair: William Morokoff (Moody's KMV)
Monte Carlo Methods for American Options
Russel E. Caflisch and Suneal Chaudhary (University of California at Los Angeles)
We review the basic properties of American options and the difficulties of applying Monte Carlo valuation to American options. Recent progress on the Least Squares Monte Carlo (LSM) method is described, including the use of quasi-random sequences in LSM. A particle approach to evaluation of American options is formulated. Conclusions and prospects for future research are discussed.
Calibrating Credit Portfolio Loss Distributions
Menghui Cao and William J. Morokoff (Moody's KMV)
Determination of credit portfolio loss distributions is essential for the valuation and risk management of multi-name credit derivatives such as CDOs. The default time model has recently become a market standard approach for capturing the default correlation, which is one of the main drivers for the portfolio loss. However, the default time model yields very different default dependency compared with a continuous-time credit migration model. To build a connection between them, we calibrate the correlation parameter of a single-factor Gaussian copula model to portfolio loss distribution determined from a multi-step credit migration simulation. The deal correlation is produced as a measure of the portfolio average correlation effect that links the two models. Procedures for obtaining the portfolio loss distributions in both models are described in the paper and numerical results are presented.
An Importance Sampling Method for Portfolios of Credit Risky Assets
William J. Morokoff (Moody's KMV)
The distribution of possible future losses for a portfolio of credit risky corporate assets, such as bonds or loans, shows strongly asymmetric behavior and a fat tail as the consequence of the limited upside of credit (the promised coupon payment) and substantial downside if the corporation defaults. Because of correlation, it is not possible to fully diversify away this fat tail. Detailed correlation models require Monte Carlo simulation to determine the loss distribution for a credit portfolio. This paper describes an importance sampling method that provides substantial speed up for computing economic capital, the rare event quantile of the loss distribution that must be held in reserve by a lending institution for solvency. The method, based solely on correlation information, provides accuracy in the tail while maintaining suitable performance for statistics related to the center of the distribution. It is also suitable for long/short portfolios.
Tuesday 1:30:00 PM 3:00:00 PM
Tutorial: Asset Liability Management
Chair: Michael Fu (University of Maryland)
We shall examine the principles behind contemporary approaches to insurance risk management. Furthermore, we shall consider various methodologies, some successful, that have been or are currently employed to implement those principles. We shall illustrate these with several specific studies that show the identification, using simulation, of close-to-optimal investment strategies.
Fads and Fallacies in Asset Liability Management for Life Insurance
Graham Lord (Princeton University)
Tuesday 3:30:00 PM 5:00:00 PM
Tutorial: CrystalBall Software
Chair: Lawrence Goldman (Decisioneering, Inc.)
Crystal Ball ® and Design for Six Sigma
Lawrence I. Goldman and Crystal Campbell (Decisioneering, Inc.)
In today’s competitive market, businesses are adopting new practices like Design for Six Sigma (DFSS), a customer driven, structured methodology for faster-to-market, higher quality, and less costly new products and services. Monte Carlo simulation and stochastic optimization can help DFSS practitioners understand the variation inherent in a new technology, process, or product, and can be used to create and optimize potential designs. The benefits of understanding and controlling the sources of variability include reduced development costs, minimal defects, and sales driven through improved customer satisfaction. This tutorial uses Crystal Ball Professional Edition, a suite of easy-to-use Microsoft® Excel-based software, to demon-strate how stochastic simulation and optimization can be used in all five phases of DFSS to develop the design for a new compressor.