We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the

We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. We adapt flux limiting techniques originally developed in the field of computational fluid dynamics in order to rapidly obtain accurate solutions. We show that flux limiting methods are total variation diminishing (and hence free of spurious oscillations) for non-conservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit schemes. We also modify the van Leer flux limiter so that the second-order total variation diminishing property is preserved for non-uniform grid spacing. 1 Introduction Asian options are securities with payoffs...

comments of the editor and an anonymous referee, who have helped tremendously in improving the content and exposition of the paper. Thanks also to Marco Avellaneda, Steven Evans, Eric

. A partial differential equation method based on using auxiliary variables is described for pricing discretely monitored Asian options with or without barrier features. The barrier provisions can be applied to either the underlying asset or to the average. They may also be of either instantaneous or delayed effect (i.e. Parisian style). Numerical examples demonstrate that this method can be used for pricing floating strike, fixed strike, American, or European options. In addition, examples are provided which indicate that an upstream biased quadratic interpolation is superior to linear interpolation for handling the jump conditions at observation dates. Moreover, it is shown that defining the auxiliary variable as the average rather than the running sum is more rapidly convergent for American-Asian options. Keywords: Asian options, Barrier options, Parisian options, PDE option pricing Running Title: Discrete Asian Barrier Options Acknowledgment: This work was supported by the Nation...

version 1.7 A semi-Lagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semi-Lagrangian timestepping. Crank-Nicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. With low volatility values, it is observed that the non-smoothness at the strike in the payoff affects the convergence rate; sub-quadratic convergence rate is observed.

This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorov-type operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, are also announced.

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American-Asian options are average-price options that allow early exercise. In this paper, we first derive structural properties of the optimal exercise policy for these call options in a general setting. In particular, we show that the optimal policy is a threshold policy: the option should be exercised as soon as the average asset price reaches a characterized threshold, which can be written as a function of asset price at that time. After further characterizing the exercise boundary, we parameterize it, and then derive gradient estimators with respect to the parameters of the model. Implementing these estimators in an iterative gradient-based stochastic approximation algorithm, we approximate the optimal exercise boundary and consequently obtain an estimate for the price of the American-Asian option. Numerical experiments carried out indicate that the algorithm performs extremely well.

One method for valuing path-dependent options is the augmented state space approach described in Hull and White (1993) and Barraquand and Pudet (1996), among others. In certain cases, interpolation is required because the number of possible values of the additional state variable grows exponentially. We provide a detailed analysis of the convergence of these algorithms. We show that it is possible for the algorithm to be non-convergent, or to converge to an incorrect answer, if the interpolation scheme is selected inappropriately. We concentrate on Asian options, due to their popularity and because of some errors in the previous literature.

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