A Dynamic Programming Procedure for Pricing American-Style Asian Options

Abstract

Pricing European-style Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no easily computable 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. A procedure for pricing American-style Asian options of the Bermudan flavor, based on dynamic programming combined with finite-element piecewise-polynomial approximation of the value function, is developed here. A convergence proof is provided. Numerical experiments illustrate the consistency and efficiency of the PROCEDURE. Theoretical properties of the value function and of the optimal exercise strategy are also established.Option Pricing, Asian Options, Path-Dependent Options, American Options, Bermudan Options, Dynamic Programming, Piecewise Polynomials

Pricing of early-exercise Asian options under Lévy processes based on Fourier cosine expansions

“…The CPU times are not comparable since they refer to different computers. However, the CPU time recorded in [1] for American option is less than the CPU time for the European counterpart with the same number of grid points which is counterintuitive.…”

“…In this subsection, we compare the early-exercise Asian option prices under the BS model by our ASCOS method with the results in [1]. Results with M = 13 are compared for different parameter sets.…”

American option pricing with randomized quasi-Monte Carlo simulations

“…We made additional experiments with an Asian American call option example, taken from Ben Ameur, Breton, and L'Ecuyer (2002), and an American call over the maximum of 5 independent assets, taken from Broadie and Glasserman (1997) and Longstaff and Schwartz (2001). For the Asian American option, the VRFs were over 100 for RQMC with BB or PCA, whereas those factors where around 7 for the maximum call option.…”

“…A popular approach is to select a finite set of real-valued basis functions defined over the state space, write the value function as a linear combination of those basis functions, and estimate the coefficients by least squares from the computed values at a set of evaluation points, at each step of the DP algorithm. If the dimension of the state space is not too large, the evaluation points can be carefully selected to form a grid that covers the relevant part of the state space, as done in Haurie and L'Ecuyer (1986), L'Ecuyer (1989), and Ben Ameur, Breton, and L'Ecuyer (2002), for example.…”

Free boundary and optimal stopping problems for American Asian options

“…While there are several papers on the valuation of Asian options with early exercise (for instance, Barraquand and Pudet [2], Barles [1], Hansen and Jorgensen [14], Meyer [27], Wu, You and Kwok [33], Fu and Wu [13], Jiang and Dai [20], Ben-Ameur, Breton and L'Ecuyer [4], Marcozzi [26], Dai and Kwok [8], Huang and Thulasiram [17]), most of these are devoted to numerical issues (the development of numerical techniques for pricing and determining the exercise boundary) by some means assuming as established the existence and regularity of the solution to the free boundary or optimal stopping problem. To a certain extent, using the weak notion of viscosity solution, it is possible to obtain general existence results.…”