This paper deals with the efficient numerical solution of the two-dimensional partial integrodifferential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Merton jump-diffusion model. We consider the adaptation of various operator splitting schemes of both the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind that have recently been studied for partial integro-differential equations (PIDEs) in [3]. Each of these schemes conveniently treats the nonlocal integral part in an explicit manner. Their adaptation to PIDCPs is achieved through a combination with the Ikonen-Toivanen splitting technique [14] as well as with the penalty method [32]. The convergence behaviour and relative performance of the acquired eight operator splitting methods is investigated in extensive numerical experiments for American put-on-the-min and put-on-theaverage options. Zvan, Forsyth & Vetzal [32] proposed the penalty method for solving the Heston PDCP. The properties of this method were rigorously analyzed in Forsyth & Vetzal [9] for the one-dimensional Black-Scholes PDCP. The penalty method was generalized by d'Halluin et al. [7,8] to one-dimensional PIDCPs for American option values, in particular under the Merton and Kou models, and subsequently applied by Clift & Forsyth [6] to two-dimensional PIDCPs. Here a fixed-point iteration is used to efficiently handle the integral part.Ikonen & Toivanen [14] introduced an alternative approach for solving the one-dimensional Black-Scholes PDCP. Here the PDCP is reformulated by means of an auxiliary variable that facilitates, in each step of a given temporal discretization scheme, an effective splitting between the PDE part and the early exercise constraint. This IT splitting technique has next been employed in [15] for the Heston PDCP and in [30] for the Kou PIDCP. For treating the integral part in the latter case, an iterative method has been applied that is similar to a fixed-point iteration.Haentjens et al. [10,11] considered the Heston PDCP and combined alternating direction implicit (ADI) schemes for directional splitting of PDEs with the IT splitting technique for the early exercise constraint, defining the so-called ADI-IT methods. These methods were next studied in [20] for the one-and two-dimensional Black-Scholes PDCPs, where also a useful interpretation of this combined splitting approach was provided.Complementary to this, the adaptation of ADI schemes to two-dimensional partial integro-differential equations (PIDEs) has recently been investigated by in 't Hout & Toivanen [19]. Here three novel adaptations of the well-established modified Craig-Sneyd (MCS) scheme [21] were analyzed and applied for the valuation of European options under the Bates model, where the mixed derivative term and the integral part are conveniently treated in an explicit fashion.Boen & in 't Hout [3] subsequently studied seven operator splitting schemes of both the implicitexplicit (IMEX) and the ADI kind in the application to th...
This paper deals with the numerical solution of the two-dimensional time-dependent Merton partial integro-differential equation (PIDE) for the values of rainbow options under the two-asset Merton jump-diffusion model. Key features of this well-known equation are a twodimensional nonlocal integral part and a mixed spatial derivative term. For its efficient and stable numerical solution, we study seven recent and novel operator splitting schemes of the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind. Here the integral part is always conveniently treated in an explicit fashion. The convergence behaviour and the relative performance of the seven schemes are investigated in ample numerical experiments for both European put-on-the-min and put-on-the-average options.
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