Application of Operator Splitting Methods in Finance

Hout, Karel in ’t; Toivanen, Jari

doi:10.1007/978-3-319-41589-5_16

Cited by 13 publications

(10 citation statements)

References 61 publications

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“…The combination of IT splitting with IMEX schemes was introduced in [23] and next investigated in e.g. [18,26,28] for one-and two-dimensional PIDCPs. The combination of this technique with ADI schemes was introduced in [11] and subsequently studied in [10,20] for two-dimensional PDCPs (without integral term).…”

Section: It Splittingmentioning

confidence: 99%

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%

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

Boen

Hout

2020

Applied Numerical Mathematics

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“…Salmi & Toivanen [36] recommended the IMEX CNAB scheme, where the PDE part is handled in a Crank-Nicolson manner and the integral part is treated in a two-step Adams-Bashforth fashion. This second-order two-step IMEX scheme has next been applied to two-dimensional option valuation PIDEs in for example [23,37].…”

Section: Introductionmentioning

confidence: 99%

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

Boen

Hout

2021

Journal of Computational and Applied Mathematics

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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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“…Also PDE methods have been extended to multivariate problems in finance. For instance using operator splitting methods as in [28], principal component analysis and expansions as in [42], and wavelet compression techniques proposed in [36,25,26].…”

Section: Introductionmentioning

confidence: 99%

Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing

Glau¹,

Kreßner²,

Statti³

2019

Preprint

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Treating high dimensionality is one of the main challenges in the development of computational methods for solving problems arising in finance, where tasks such as pricing, calibration, and risk assessment need to be performed accurately and in real-time. Among the growing literature addressing this problem, Gass et al. [14] propose a complexity reduction technique for parametric option pricing based on Chebyshev interpolation. As the number of parameters increases, however, this method is affected by the curse of dimensionality. In this article, we extend this approach to treat high-dimensional problems: Additionally exploiting low-rank structures allows us to consider parameter spaces of high dimensions. The core of our method is to express the tensorized interpolation in tensor train (TT) format and to develop an efficient way, based on tensor completion, to approximate the interpolation coefficients. We apply the new method to two model problems: American option pricing in the Heston model and European basket option pricing in the multi-dimensional Black-Scholes model. In these examples we treat parameter spaces of dimensions up to 25. The numerical results confirm the low-rank structure of these problems and the effectiveness of our method compared to advanced techniques.

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Application of Operator Splitting Methods in Finance

Cited by 13 publications

References 61 publications

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

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

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

Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing

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