Variational Formulation of American Option Prices in the Heston Model

Abstract

We give an analytical characterization of the price function of an American option in Heston-type models. Our approach is based on variational inequalities and extends recent results of Feehan (2011, 2016) and . We study the existence and uniqueness of a weak solution of the associated degenerate parabolic obstacle problem. Then, we use suitable estimates on the joint distribution of the log-price process and the volatility process in order to characterize the analytical weak solution as the solution to the optimal stopping problem. We also rely on semi-group techniques and on the affine property of the model.It should be emphasized that a clear analytic characterization of the price function allows not only to formally justify the theoretical convergence of some classical pricing algorithms but also to investigate the regularity properties of the price function (see [13] for the case of the Black and Scholes models).Concerning the existing literature, E. Ekstrom and J. Tysk in [7] give a rigorous and complete analysis of these issues in the case of European options, proving that, under some regularity assumptions on the payoff functions, the price function is the unique classical solution of the associated PDE with a certain boundary behaviour for vanishing values of the volatility. However, the payoff functions they consider do not include the case of standard put and call options.Recently, P. Daskalopoulos and P. Feehan studied the existence, the uniqueness, and some regularity properties of the solution of this kind of degenerate PDE and obstacle problems in the elliptic case, introducing suitable weighted Sobolev spaces which clarify the behaviour of the solution near the degenerate boundary (see [5,6]). In another paper ([8]) P. Feehan and C. Pop addressed the issue of the probabilistic representation of the solution, but we do not know if their assumptions on the solution of the parabolic obstacle problem are satisfied in the case of standard American options. Note that Feehan and Pop did prove regularity results in the elliptic case, see [9]. They also announce results for the parabolic case in [8].The aim of this paper is to give a precise analytical characterization of the American option price function for a large class of payoffs which includes the standard put and call options. In particular, we give a variational formulation of the American pricing problem using the weighted Sobolev spaces and the bilinear form introduced in [5]. The paper is organized as follows. In Section 2, we introduce our notations and we state our main results. Then, in Section 3, we study the existence and uniqueness of the solution of the associated variational inequality, extending the results obtained in [5] in the elliptic case. The proof relies, as in [5], on the classical penalization technique introduced by Bensoussan and Lions [4] with some technical devices due to the degenerate nature of the problem. We also establish a Comparison Theorem. Finally, in section 4, we prove that the solution of the variational ineq...

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