Operator splitting schemes for the two-asset Merton jump-diffusion model

Boen, Lynn; Hout, Karel J. in ’t

doi:10.48550/arxiv.1901.03839

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“…IT splitting is combined with a temporal discretization scheme for semidiscrete PIDEs, so as to yield a temporal discretization of semidiscrete PIDCPs. In [3] a variety of contemporary operator splitting schemes has been studied for the temporal discretization of the semidiscrete two-dimensional Merton PIDE for European option values. Here, the integral part, which corresponds to the dense matrix A pJq , is always conveniently treated in an explicit fashion.…”

Section: It Splittingmentioning

confidence: 99%

“…Boen & in 't Hout [3] subsequently studied seven operator splitting schemes of both the implicitexplicit (IMEX) and the ADI kind in the application to the two-dimensional Merton PIDE, where the two-dimensional integral part is always handled explicitly. It was concluded that, among the schemes considered, the adaptation introduced in [19] of the MCS scheme where the integral part is dealt with in a two-step Adams-Bashforth manner is preferable.…”

Section: Introductionmentioning

confidence: 99%

“…The main aim of the present paper is to introduce and investigate adaptations of IMEX and ADI operator splitting schemes for the two-dimensional Merton PIDCP. We extend all second-order schemes from [3] by using the IT splitting technique and next study their performance in ample numerical experiments. In addition, we consider two penalty type methods that were proposed in [7] and [12].…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 describes the spatial discretization of this PIDCP. Section 4 deals with the temporal discretization of the obtained semidiscrete PIDCP and defines the adaptation of the pertinent IMEX and ADI schemes considered in [3] by means of IT splitting, which will be applied in an iterative manner. Also the two penalty type methods from [7] and [12] are formulated in this section.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Operator splitting schemes for American options under the two-asset Merton jump-diffusion model

Boen

Hout

2020

Applied Numerical Mathematics

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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...

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Section: It Splittingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%