A number of Monte Carlo simulation-based approaches have been proposed within the past decade to address the problem of pricing American-style derivatives. The purpose of this paper is to empirically test some of these algorithms on a common set of problems in order to be able to assess the strengths and weaknesses of each approach as a function of the problem characteristics. In addition, we introduce another simulation-based approach that parameterizes the early exercise curve and casts the valuation problem as an optimization problem of maximizing the expected payoff (under the martingale measure) with respect to the associated parameters, the optimization problem carried out using a simultaneous perturbation stochastic approximation (SPSA) algorithm.

Abstract We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.

A general framework is developed to analyze the optimal stopping (exercise) regions of American path dependent options with either Asian feature or lookback feature. We examine the monotonicity properties of the option values and stopping regions with respect to the interest rate, dividend yield and time. From the ordering properties of the values of American lookback options and American Asian options, we deduce the corresponding nesting relations between the exercise regions of these American options. We illustrate how some properties of the exercise regions of the American Asian options can be inferred from those of the American lookback options.

. Pricing Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no analytical solution is available. Pricing their American-style counterparts, which provide early exercise opportunities, poses the additional di#culty of solving a dynamic optimization problem to determine the optimal exercise strategy. We develop a numerical method for pricing American-style Asian options based on dynamic programming combined with finite-element piecewise-polynomial approximation of the value function. Numerical experiments show convergence, consistency, and e#ciency. Some theoretical properties of the value function and of the optimal exercise strategy are also established. (Option pricing, Asian Options, Path-dependent options, American Options, Dynamic Programming, Piecewise Polynomials) 1 1 Introduction A financial derivative is a contract which provides its holder a future payment that depends on...

A number ofM onte Carlo simulation-based approaches have been proposed within the past decade to address the problem of pricing American-style derivatives. The purpose of this paper is to empirically test some of these algorithms on a common set of problems in order to be able to assess the strengths and weaknesses of each approach as a function of the problem characteristics. In addition, we introduce another simulation-based approach that parameterizes the early exercise curve and casts the valuation problem as an optimization problem of maximizing the expected payo# (under the martingale measure) with respect to the associated parameters, the optimization problem carried out using a simultaneous perturbation stochastic approximation (SPSA) algorithm. Keywords: American options, M nte Carlo simulation, options pricing, stochastic approximation, early exercise. # This work was su1 orted in part by the NSFu27 Grant DMI-9713720,and by theSemicondu49 Research Corporationsutio Grant 97-F...

Managerial flexibility has value. The ability of their managers to make smart decisions in the face of volatile market and technological conditions is essential for firms in any competitive industry. This advanced tutorial describes the use of Monte Carlo simulation and stochastic optimization for the valuation of real options that arise from the abilities of managers to influence the cash flows of the projects under their control. Option pricing theory supplements discounted cash flow methods of valuation by considering managerial flexibility. Managers ’ options to take actions that affect real investment projects are comparable to options on the sale or purchase of financial assets. Just as a financial option derives much of its value from the potential price movements of the underlying financial asset, a real option derives much of its value from the potential fluctuations of the cash flows generating the value of the investment project. 1

Abstract. Pricing Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no analytical solution is available. Pricing their American-style counterparts, which provide early exercise opportunities, poses the additional difficulty of solving a dynamic optimization problem to determine the optimal exercise strategy. We develop a numerical method for pricing American-style Asian options based on dynamic programming combined with finite-element piecewise-polynomial approximation of the value function. Numerical experiments show convergence, consistency, and efficiency. Some theoretical properties of the value function and of the optimal exercise strategy are also established.