WSC 2006 Abstracts

A New Efficient Simulation Strategy for Pricing Path-Dependent Options
Gang Zhao, Yakun Zhou, and Pirooz Vakili (Boston University)

Abstract:
The purpose of this paper is twofold. First, it serves to describe a new strategy, called Structured Database Monte Carlo (SDMC), for efficient Monte Carlo simulation. Its second aim is to show how this approach can be used for efficient pricing of path-dependent options via simulation. We use efficient simulation of a sample of path-dependent options to illustrate the application of SDMC. Extensions to other path-dependent options are straightforward.

Applying Model Reference Adaptive Search to American-Style Option Pricing
Huiju Zhang and Michael Fu (University of Maryland)

Abstract:
This paper considers the application of stochastic optimization methods to American-style option pricing. We apply a randomized optimization algorithm called Model Reference Adaptive Search (MRAS) to pricing American-style options by parameterizing the early exercise boundary. Numerical results are provided for pricing American-style call and put options written on underlying assets following geometric Brownian motion and Merton jump-diffusion processes. The results from the MRAS algorithm are also compared with the Cross-Entropy (CE) method.

American Options From Mars
Samuel M. T. Ehrlichman and Shane G. Henderson (Cornell University)

Abstract:
We develop a class of control variates for the American option pricing problem that are constructed through the use of MARS -- multivariate adaptive regression splines. The splines approximate the option's value function at each time step, and the value function approximations are then used to construct a martingale that serves as the control variate. Significant variance reduction is possible even in high dimensions. The primary restriction is that we must be able to compute certain one-step conditional expectations.

Using Copulas in Risk Analysis
Dalton F Andrade, Pedro Barbetta, Paulo José Freitas, and Ney A. M. Zunino (Federal University of Santa Catarina - UFSC) and Carlos Magno Jacinto (Petrobras)

Abstract:
Practically every well installation process nowadays relies on some sort of risk assessment study, given the high costs involved. Those studies focus mostly on estimating the total time required by the well drilling and completion operations, as a way to predict the final costs. Among the different techniques employed, the Monte Carlo Simulation currently stands out as the preferred method. One relevant aspect which is frequently left out from simulation models is the dependence relationship among the processes under consideration. That omission can have a serious impact on the results of risk assessment and, consequently, on the conclusions drawn from them. In general, practitioners do not incorporate the dependence information because that is not always an easy task. This paper intends to show how Copula functions may be used as a tool to build correlation-aware Monte Carlo simulation models.

An Adaptive Procedure for Estimating Coherent Risk Measures Based on Generalized Scenarios
Vadim Lesnevski, Barry L. Nelson, and Jeremy Staum (Northwestern University)

Abstract:
Coherent risk measures based on generalized scenarios can be viewed as estimating the maximum expected value from among a collection of simulated "systems." We present a procedure for generating a fixed-width confidence interval for this coherent risk measure. The procedure improves upon previous methods by being reliably efficient for simulation of generalized scenarios and portfolios with heterogeneous characteristics.

Efficient Importance Sampling for Reduced Form Models in Credit Risk
Achal Bassamboo (Kellogg School of Management) and Sachin Jain (Amaranth Group Inc.)

Abstract:
In this paper we study the problem of estimating probability of large losses in the framework of doubly stochastic credit risk models. We derive a logarithmic asymptote for the probability of interest in a specific asymptotic regime and propose an asymptotically optimal importance sampling algorithm for efficiently estimating the same. The numerical results in the last section corroborate our theoretical findings.

Efficient Simulation for Risk Measurement in Portfolio of CDOs
Michael Gordy (The Federal Reserve Board) and Sandeep Juneja (Tata Institute of Fundamental Research)

Abstract:
We consider a portfolio containing CDO tranches and ordinary bonds. Our interest is in large loss probabilities and risk measures such as value-at-risk. When loss is measured on mark-to-market basis, estimation via simulation requires a nested procedure: In the outer step one draws realizations of all risk factors up to the horizon, and in the inner step one re-prices each instrument in the portfolio at the horizon conditional on the drawn risk factors. Practitioners perceive the computational burden of such nested schemes to be unacceptable, and adopt variety of somewhat ad hoc measures to avoid the inner simulation. In this paper, we question whether such short cuts are necessary. We show that relatively small number of trials in the inner step can yield accurate estimates, and analyze how fixed computational budget may be allocated to inner and outer steps to minimize the mean square error of the resultant estimator.

Efficient Simulation for Large Deviation Probabilities of Sums of Heavy-tailed Random Variables
Jose Blanchet and Jingchen Liu (Harvard University)

Abstract:
We describe an efficient state-dependent importance sampling algorithm for estimating large deviation probabilities for sums of iid rv's with finite variance and regularly varying tails. Our algorithm can be shown to be strongly efficient basically throughout the whole large deviations region as the number of increments increases. Our techniques combine results of the theory of large deviations for sums of regularly varying distributions and the basic ideas can be applied to other rare-event simulation problems involving both light and heavy-tailed features.

The BEST Algorithm for Solving Stochastic Mixed Integer Programs
Susan Sanchez and Kevin Wood (Naval Postgraduate School)

Abstract:
We present a new algorithm for solving two-stage stochastic mixed-integer programs (SMIPs) having discrete first-stage variables, and continuous or discrete second-stage variables. For a minimizing SMIP, the BEST algorithm (1) computes an upper Bound on the optimal objective value (typically a probabilistic bound), and identifies a deterministic lower-bounding function, (2) uses the bounds to Enumerate a set of first-stage solutions that contains an optimal solution with pre-specified confidence, (3) for each first-stage solution, Simulates second-stage operations by repeatedly sampling random parameters and solving the resulting model instances, and (4) applies statistical Tests (e.g., "screening procedures") to the simulated outcomes to identify a near-optimal first-stage solution with pre-specified confidence. We demonstrate the algorithm's performance on a stochastic facility-location problem.

Quasi-Monte Carlo Strategies for Stochastic Optimization
Shane Drew (Northwestern University) and Tito Homem-de-Mello (Nortwestern University)

Abstract:
In this paper we discuss the issue of solving stochastic optimization problems using sampling methods. Numerical results have shown that using variance reduction techniques from statistics can result in significant improvements over Monte Carlo sampling in terms of the number of samples needed for convergence of the optimal objective value and optimal solution to a stochastic optimization problem. Among these techniques are stratified sampling and Quasi-Monte Carlo sampling. However, for problems in high dimension, it may be computationally inefficient to calculate Quasi-Monte Carlo point sets in the full dimension. Rather, we wish to identify which dimensions are most important to the convergence and implement a Quasi-Monte Carlo sampling scheme with padding, where the important dimensions are sampled via Quasi-Monte Carlo sampling and the remaining dimensions with Monte Carlo sampling. We then incorporate this sampling scheme into an external sampling algorithm (ES-QMCP) to solve stochastic optimization problems.

Jackknife Estimators for Reducing Bias in Asset Allocation
Amit Partani and David P Morton (The University of Texas at Austin) and Ivilina Popova (Seattle University)

Abstract:
We use jackknife-based estimators to reduce bias when estimating the optimal value of a stochastic program. Our discussion focuses on an asset allocation model with a power utility function. As we will describe, estimating the optimal value of such a problem plays a key role in establishing the quality of a candidate solution, and reducing bias improves our ability to do so efficiently. We develop a jackknife estimator that is adaptive in that it does not assume the order of the bias is known a priori.