A penalty method for American options with jump diffusion processes

Abstract

The fair price for an American option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem. We develop an implicit discretization method for pricing such American options. The jump diffusion correlation integral term is computed using an iterative method coupled with an FFT while the American constraint is imposed by using a penalty method. We derive sufficient conditions for global convergence of the discrete penalized equations at each timestep. Finally, we present numerical tests which illustrate such convergence. Subject Classification (1991): 65M12, 65M60, 91B28 Mathematics

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

“…1) The function v n can be written as the value function of an optimal stopping problem: 11) in which σ n is the n-th jump time of the Poisson process N t .…”

“…We compare our prices to the ones obtained by [11,12]. [11] used a penalty method to approximate the American option price, while we analyze the variational inequalities directly (see (3.5) and (3.10)). Moreover, our approximating sequence is monotone (see Proposition 3.1).…”

“…We will iterate (3.10) to approximate the solution of (3.5), which can be seen as a global fixed point iteration algorithm. This global fixed point algorithm is different from the local fixed point algorithm in [11], where d'Halluin et al implemented the Crank Nicolson time stepping of a non-linear integro-partial differential equation coming from an alternative representation (due to the penalty method) of the American option price function. Also see [12] for the case of European options.…”

“…Each of these iterations provide strictly decreasing free boundary curves with the same regularity and jump properties as the free boundary curve for the American option price function, see Remarks 2.1 and 4.2. The approximating sequence in [11] does not carry the same meaning, it is a technical step to carry out the Crank Nicolson time stepping of their non-linear integro-PDE.…”

An iterative method for pricing American options under jump-diffusion models

“…Several methods have been proposed to approximate the linear complementarity problems resulting from American option pricing. These include a penalty method presented by d'Halluin, Forsyth and Labahn in [8] and an operator splitting method presented by Ikonen and Toivanen in [13,15]. One alternative approach to PIDEs is to employ a riskneutral valuation formula and evaluate it quickly using FFT (Fast Fourier Transform) [17].…”