WSC 2008 Final Abstracts

Monte Carlo Simulation of Diffusions
Peter W Glynn (Stanford University)

Abstract:
This tutorial is intended to provide an overview of the key algorithms that are used to simulate sample paths of diffusion processes, as well as to offer an understanding of their fundamental approximation properties.

Simulating Point Processes by Intensity Projection
Kay Giesecke, Hossein Kakavand, and Mohammad Mousavi (Stanford University)

Abstract:
Point processes with stochastic intensities are ubiquitous in many application areas, including finance, insurance, reliability and queuing. They can be simulated from standard Poisson arrivals by time-scaling with the cumulative intensity, whose path is typically generated with a discretization method. However, discretization introduces bias into the simulation results. This paper proposes a method for the exact simulation of point processes with stochastic intensities. The method leads to unbiased estimators. It is illustrated for a point process whose intensity follows an affine jump-diffusion process.

Beta Approximations for Bridge Sampling
Paul Glasserman and Kyoung-Kuk Kim (Columbia University, Graduate School of Business)

Abstract:
We consider the problem of simulating X conditional on the value of X+Y, when X and Y are independent positive random variables. We propose approximate methods for sampling (X|X+Y) by approximating the fraction (X/z|X+Y=z) with a beta random variable. We discuss applications to Levy processes and infinitely divisible distributions, and we report numerical tests for Poisson processes, tempered stable processes, and the Heston stochastic volatility model.

Connecting the Top-Down to the Bottom-Up: Pricing CDO under a Conditional Survival (CS) Model
Steven S. G. Kou and Xianhua Peng (Columbia University)

Abstract:
In this paper, we use exact simulation to price CDO under a new dynamic model, the Conditional Survival (CS) model, which provided excellent calibration to both iTraxx tranches and underlying single name CDS spreads on March 14, 2008, the day before the collapse of Bear Sterns, when the market was highly volatile. The distinct features of the CS model include: (1) it is able to generate clustering of defaults occurring dynamically in time and strong cross-sectional correlation, i.e., the simultaneous defaults of many names, both of which have been evidenced in the current subprime mortgage crisis; (2) it incorporates idiosyncratic default risk of single names but does not specify concrete models for them; (3) it provides automatic calibration to underlying single name CDS; (4) it allows fast CDO tranche pricing and calculation of sensitivity of CDO tranches to underlying single name CDS.

Reducing the Variance of Likelihood Ratio Greeks in Monte Carlo
Luca Capriotti (Global Modelling and Analytics Group, Credit Suisse Investment Banking Division)

Abstract:
We investigate the use of Antithetic Variables, Control Variates and Importance Sampling to reduce the statistical errors of option sensitivities calculated with the Likelihood Ratio Method in Monte Carlo. We show how Antithetic Variables solve the well-known problem of the divergence of the variance of Delta for short maturities and small volatilities. With numerical examples within a Gaussian Copula framework, we show how simple Control Variates and Importance Sampling strategies provide computational savings up to several orders of magnitude.

Revisit of Stochastic Mesh Method for Pricing American Options
Guangwu Liu and Jeff Hong (Hong Kong University of Science and Technology)

Abstract:
We revisit the stochastic mesh method for pricing American options, from a conditioning viewpoint, rather than the importance sampling viewpoint of Broadie and Glasserman (1997). Starting from this new viewpoint, we derive the weights proposed by Broadie and Glasserman (1997) and show that their weights at each exercise date use only the information of the next exercise date (therefore, we call them forward-looking weights). We also derive new weights that exploit not only the information of the next exercise date but also the information of the last exercise date (therefore, we call them binocular weights). We show how to apply the binocular weights to the Black-Scholes model, more general diffusion models, and the variance-gamma model. We demonstrate the performance of the binocular weights and compare to the performance of the forward-looking weights through numerical experiments.

Valuation of Variable Annuity Contracts with Cliquet Options in Asia Markets
Ming-hua Hsieh (National Chengchi University)

Abstract:
Variable annuities are very appealing to the investor. For example, in United States, sales volume on variable annuities grew to a record 184 billion in calendar year 2006. However, due to their complicated payoff structure, their valuation and risk management are challenges to the insurers. In this paper, we study a variable annuity contract with cliquet options in Asia markets. The contact has quanto feature. We propose an efficient Monte Carlo method to value the contract. Numerical examples suggest our approach is quite efficient.

Efficient Tail Estimation for Sums of Correlated Lognormals
Jose Blanchet (Columbia University), Sandeep Juneja (Tata Institute of Fundamental Research) and Leonardo Rojas-Nandayapa (Univeristy of Aarhus)

Abstract:
Our focus is on efficient estimation of tail probabilities of sums of correlated lognormals. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose three different procedures that can be rigorously shown to be asymptotically optimal as the tail probability of interest decreases to zero. The first algorithm is based on importance sampling and is as easy to implement as crude Monte Carlo. The second algorithm is based on an elegant conditional Monte Carlo strategy which involves polar coordinates and the third one is an importance sampling algorithm that can be shown to be strongly efficient.

A Rate Result for Simulation Optimization with Conditional Value-at-Risk Constraints
Soumyadip Ghosh (IBM TJ Watson Research Center)

Abstract:
We study a stochastic optimization problem that has its roots in financial portfolio design. The problem has a specified deterministic objective function and constraints on the conditional value-at-risk of the portfolio. Approximate optimal solutions to this problem are usually obtained by solving a sample-average approximation. We derive bounds on the gap in the objective value between the true optimal and an approximate solution so obtained. We show that under certain regularity conditions the approximate optimal value converges to the true optimal at the canonical rate O(n^-1/2), where n represents the sample size. The constants in the expression are explicitly defined.

Optimizing Portfolio Tail Measures: Asymptotics and Efficient Simulation Optimization
Sandeep Juneja (Tata Institute of Fundamental Research)

Abstract:
We consider a portfolio allocation problem where the objective function is a tail event such as probability of large portfolio losses. The dependence between assets is captured through multi-factor linear model. We address this optimization problem using two broad approaches. We show that a suitably scaled asymptotic of the probability of large losses can be developed that is a simple convex function of the allocated resources. Thus, asymptotically, portfolio allocation problem is approximated by a convex programming problem whose solution is easily computed and provides significant managerial insight. We then solve the original problem using sample average simulation optimization. Since rare events are involved, naive simulation may perform poorly. To remedy this, we introduce change-of-variable based importance sampling technique and develop a single change of measure that asymptotically optimally estimates tail probabilities across the entire space of feasible allocations.

Response Surface Methodology for Simulating Hedging and Trading Strategies
Evren Baysal, Barry L. Nelson, and Jeremy Staum (Northwestern University)

Abstract:
Suppose that one wishes to evaluate the distribution of profit and loss (P&L) resulting from a dynamic trading strategy. A straightforward method is to simulate thousands of paths (i.e., time series) of relevant financial variables and to track the resulting P&L at every time at which the trading strategy rebalances its portfolio. In many cases, this requires numerical computation of portfolio weights at every rebalancing time on every path, for example, by a nested simulation performed conditional on market conditions at that time on that path. Such a two-level simulation could involve many millions of simulations to compute portfolio weights, and thus be too computationally expensive to attain high accuracy. We show that response surface methodology enables a more efficient simulation procedure: in particular, it is possible to do far fewer simulations by using kriging to model portfolio weights as a function of underlying financial variables.

Supply Chain Risks Analysis by Using Jump-Diffusion Model
Xianzhe Chen and Jun Zhang (North Dakota State University)

Abstract:
This paper investigates the effects of demand risk on the performance of supply chain in continuous time setting. The inventory level has been modeled as a jump-diffusion process and the two-number inventory policy has been implemented in the supply chain system. The simulated annealing algorithm has been used to search the optimal critical values of the two-number policy. The jump magnitude has been considered in two cases: constant and Laplace distribution which has favorable property, i.e. leptokurtic. Numerical studies have been conducted for various scenarios to provide insights of effects of demand disruption on the performance of the supply chain.

A Particle Filtering Framework for Randomized Optimization Algorithms
Enlu Zhou, Michael C. Fu, and Steven I. Marcus (University of Maryland, College Park)

Abstract:
We propose a framework for optimization problems based on particle filtering (also called Sequential Monte Carlo method). This framework unifies and provides new insight into randomized optimization algorithms. The framework also sheds light on developing new optimization algorithms through the freedom in the framework and the various improving techniques for particle filtering.