WSC 2007 Final Abstracts

Monte Carlo Simulation in Financial Engineering
Nan Chen (Chinese University of Hong Kong) and L. Jeff Hong (Hong Kong University of Science and Technology)

Abstract:
This paper reviews the use of Monte Carlo simulation in the field of financial engineering. It focuses on several interesting topics and introduces their recent development, including path generation, pricing American-style derivatives, evaluating Greeks and estimating value-at-risk. The paper is not intended to be comprehensive survey of the research literature.

Sensitivity Estimates From Characteristic Functions
Paul Glasserman and Zongjian Liu (Columbia University)

Abstract:
The likelihood ratio method (LRM) estimates parameter sensitivities by multiplying the output of a simulation by a random weight. This weight is a differentiated log density, also called a score function. We investigate the problem of computing LRM estimators when the relevant densities are unknown but their characteristic functions are available, a situation that arises in financial applications. We analyze the various sources of error introduced when the score function is computed through numerical transform inversion.

Kernel Estimation for Quantile Sensitivities
Guangwu Liu and L. Jeff Hong (The Hong Kong University of Science and Technology)

Abstract:
Quantiles, also known as value-at-risk in financial applications, are important measures of random performance. Quantile sensitivities provide information on how changes in the input parameters affect the output quantiles. In this paper, we study the estimation of quantile sensitivities using simulation. We propose a new estimator by employing kernel method and show its consistency and asymptotic normality for i.i.d. data. Numerical results show that our estimator works well for the test problems.

A Confidence Interval for Tail Conditional Expectation Via Two-level Simulation
Hai Lan, Barry L. Nelson, and Jeremy Staum (Northwestern University)

Abstract:
We develop and evaluate a two-level simulation procedure that produces a confidence interval for tail conditional expectation, otherwise known as conditional tail expectation. This risk measure is closely related to conditional value-at-risk, expected shortfall, and worst conditional expectation. The outer level of simulation generates risk factors and the inner level estimates each expected loss conditional on the risk factor. Our procedure uses the statistical theory of empirical likelihood to construct a confidence interval, and it uses tools from the ranking-and-selection literature to make the simulation efficient.

Efficient Monte Carlo Methods for Convex Risk Measures in Portfolio Credit Risk Models
Joern Dunkel (Universitat Augsburg) and Stefan Weber (Cornell University)

Abstract:
We discuss efficient Monte Carlo (MC) methods for the estimation of convex risk measures within the portfolio credit risk model CreditMetrics. Our focus lies on the Utility-based Shortfall Risk (SR) measures, as these avoid several deficiencies of the current industry standard Value-at-Risk (VaR). It is demonstrated that the importance sampling method exponential twisting provides computationally efficient SR estimators. Numerical simulations of test portfolios illustrate the good performance of the proposed algorithms.

Estimating Tranche Spreads by Loss Process Simulation
Kay Giesecke and Baeho Kim (Stanford University)

Abstract:
A credit derivative is a path dependent contingent claim on the aggregate loss in a portfolio of credit sensitive securities. We estimate the value of a credit derivative by Monte Carlo simulation of the affine point process that models the loss. We consider two algorithms that exploit the direct specification of the loss process in terms of an intensity. One algorithm is based on the simulation of intensity paths. Here discretization introduces bias into the results. The other algorithm facilitates exact simulation of default times and generates an unbiased estimator of the derivative price. We implement the algorithms to value index and tranche swaps, and we calibrate the loss process to quotes on the CDX North America High Yield index.

Approximations and Control Variates for Pricing Portfolio Credit Derivatives
Zhiyong Chen (Bear, Stearns & Co. Inc.) and Paul Glasserman (Columbia Business School)

Abstract:
Portfolio credit derivatives that depend on default correlation are increasingly widespread in the credit market. Valuing such products often entails Monte Carlo simulation. However, for large portfolios, plain Monte Carlo simulation can be slow. In this paper, we develop approximation methods for pricing collateralized debt obligation (CDO) tranches in the widely used factor copula approach. We also discuss using the approximations as control variates to improve the precision of Monte Carlo estimates. These approximation methods and control variate techniques could be applied to pricing other portfolio credit derivatives as well.

Efficient Estimation of Option Price and Price Sensitivities via Structured Database Monte Carlo (SDMC)
Gang Zhao, Tarik Borogovac, and Pirooz Vakili (Boston University)

Abstract:
We describe how to develop generic efficient simulation algorithms for estimating price and price sensitivities (the Greeks) of financial options using the Structured Database Monte Carlo (SDMC) approach. These algorithms are based on stratification, control variate and a combination of the two in an SDMC setting. Experimental results and some discussion of the effectiveness of the approach are provided. The algorithms also serve as illustrations of the basic approach of developing variance reduction algorithms in an SDMC setting that are not necessarily limited to stratification and control variate techniques.

American Option Pricing Under Stochastic Volatility: A Simulation-based Approach
Arunachalam Chockalingam (Purdue University) and Kumar Muthuraman (University of Texas)

Abstract:
We consider the problem of pricing American options when the volatility of the underlying asset price is stochastic. No specific stochastic volatility model is assumed for the stochastic process. We propose a simulation-based approach to pricing such options. Iteratively, the method determines the optimal exercise boundary and the associated price function for a general stochastic volatility model. Given an initial guess of the optimal exercise boundary, the Retrospective Approximation (RA) technique is used to calculate the associated value function. Using this function, the exercise boundary is improved and the process repeated till convergence. This method is a simulation based variant of the exercise-policy improvement scheme developed in Chockalingam and Muthuraman (2007). An illustration of the method is provided when using the Heston (1993) model to represent the dynamics of the volatility, together with comparisons against existing methods to validate our numerical results.

Monte Carlo Methods for Valuation of Ratchet Equity Indexed Annuities
Ming-hua Hsieh and Yu-fen Chiu (National Chengchi University)

Abstract:
Equity Indexed Annuities (EIAs) are popular insurance contracts. EIAs provide the insured with a guaranteed accumulation rate on their premium at maturity. In addition, the insured may receive extra benefit if the return of the linked index is high enough. There are a few variations of EIAs. We consider two types of EIAs: compound ratchet and simple ratchet. Under the geometric Brownian motion assumption for the equity index, plain compound ratchet options is known to have closed form solutions, but plain simple ratchet option is not. In this paper, we derive a closed form solution for plain simple ratchet option. For more exotic options, Monte Carlo methods are usually used for their valuation. To improve their efficiency, we propose two control variates based on the analytical solutions for the price of plain ratchet options. The effectiveness of the proposed control variates is examined via numerical examples of a typical contract.

Non-Gaussian Asset Allocation in the Federal Thrift Savings Plan
Scott T. Nestler (University of Maryland)

Abstract:
Historical data suggest that returns of stocks and indices are not distributed independent and identically Normal, as is commonly assumed. Instead, returns of financial assets are often skewed and have higher kurtosis. In this study, we investigate how the optimal investment choices in the federal government's Thrift Savings Plan (TSP) change when a non-Gaussian factor model for returns, generated with independent components analysis (ICA) and follow-ing the Variance Gamma (VG) process, is used in place of the usual Normally-distributed returns model. Using back-testing and simulation, we hope to show how this method could benefit the more than 3 million TSP participants in achieving their retirement savings objectives.

Path-wise Estimators and Cross-path Regressions: An Application to Evaluating Portfolio Strategies
Martin B. Haugh (None) and Ashish Jain (Columbia Business School)

Abstract:
Recently developed dual techniques allow us to evaluate a given sub-optimal dynamic portfolio policy by using the policy to construct an upper bound on the optimal value function. Since it is easy to construct a lower bound by simulating the given policy, we may use the distance from the lower bound to the upper bound to assess the quality of the policy. One of the difficulties that arises when computing the upper bound, however, is that we need to know the sub-optimal policy's value function and its partial derivatives with respect to all state variables. In this paper we show how path-wise Monte-Carlo estimators together with the cross-path regression approach can be used to estimate the sub-optimal value function and its derivatives, thereby enabling us to compute the more theoretically satisfying upper bound on the optimal value function.

An Empirical Comparison Between Nonlinear Programming Optimization and Simulated Annealing (SA) Algorithm Under a Higher Moments Bayesian Portfolio Selection Framework
Jingjing Lu and Merrill Liechty (Drexel University)

Abstract:
The optimal portfolio selection problem has long been of interest to both academics and practitioners. A higher moments Bayesian portfolio optimization model can overcome the shortcomings of the traditional Markowitz approach and take into consideration the skewness of asset returns and parameter uncertainty. This paper presents a comparison between the simulated annealing and the nonlinear programming methods of optimization for the Bayesian portfolio selection problem in which the objective function includes the portfolio mean, variance and skewness. We make the comparison for a utility function that is easily optimized using both methods. In particular we maximize a cubic utility function, and our results show that to achieve the same level of accuracy, the CPU time for the nonlinear programming optimization will be shorter than for the simulated annealing algorithm. Though it is slower, the simulated annealing algorithm is still a viable option for this utility function.