RBF-PU method for pricing options under the jump–diffusion model with local volatility

Mollapourasl, Reza; Fereshtian, Ali; Li, Hengguang; Lu, Xihong

doi:10.1016/j.cam.2018.01.002

Cited by 22 publications

(14 citation statements)

References 40 publications

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“…Corollary 1. If the model parameters satisfy (54) then the discretization scheme (31) with the differential matrix D as in (38) applied to a linear test problem is conditionally stable for λ△τ < 1.…”

Section: Stability Analysis Cf [41]mentioning

confidence: 99%

“…with U M the solution to the fully discretized problem (28)- (29) in the European option case and (30)- (31) in the American option case, and with notations…”

Section: Greeksmentioning

confidence: 99%

“…where U k is the numerical solution for the system (30)- (31) and X is the set of interpolation points of [0, 1]. In Figure 5 we provide the plot of the early exercise boundary for an American put option under the Merton model and the Kou model for parameter sets 1 and 2.…”

Section: American Put Option Prices Under the Merton And The Kou Modelmentioning

confidence: 99%

“…To overcome this drawback a local version of the method was proposed in order to have a sparse better-conditioned linear system. Local RBF-methods are applied in, e.g., [32,31] for the pricing of American options with stochastic volatility and jump-diffusion model.…”

Section: Introductionmentioning

confidence: 99%

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An RBF–FD method for pricing American options under jump–diffusion models

Haghi

Mollapourasl

Vanmaele

2018

Computers & Mathematics with Applications

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American option prices under jump-diffusion models are determined as solutions to partial integro-differential equations (PIDE). In this paper a new combination of a time and spatial discretization applied to a linear complementary formulation (LCP) of the free boundary PIDE is proposed. First a coordinate stretching transformation is performed to the asset price so that the computation of the prices can be focused on regions of real interest instead of on the whole solution domain. An implicit-explicit time discretization applied to the reformulated LCP on a uniform temporal grid is followed by a spatial discretization to get a fully discrete system. The radial basis function (RBF) finite difference method is a local method resulting in a sparse linear system in contrast to global RBF-methods which lead to ill-conditioned dense matrix systems. For the corresponding European option we prove consistency, stability and second-order convergence in a discrete L2-norm. We derive mild conditions for the model parameters under which these results hold. Numerical experiments are performed with European and American options, and a comparison with numerical results available in the literature illustrates the accuracy and efficiency of the proposed algorithm.

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Section: Stability Analysis Cf [41]mentioning

confidence: 99%

“…with U M the solution to the fully discretized problem (28)- (29) in the European option case and (30)- (31) in the American option case, and with notations…”

Section: Greeksmentioning

confidence: 99%

Section: American Put Option Prices Under the Merton And The Kou Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An RBF–FD method for pricing American options under jump–diffusion models

Haghi

Mollapourasl

Vanmaele

2018

Computers & Mathematics with Applications

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“…The possibility of using RBFs in a partition of unity scheme was mentioned in [4], further discussed in [49], implemented for interpolation on the sphere, in the plane, and in three-dimensional space in [15,16]. Also, we applied localized partition of unity method for pricing American and European option prices as well as Greeks in [36].…”

Section: Introductionmentioning

confidence: 99%

RBF approximation by partition of unity for valuation of options under exponential Lévy processes

Fereshtian

Mollapourasl

Avram

2019

Journal of Computational Science

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The prices of some European and American-style contracts on assets driven by a class of Markov processes containing, in particular, L\'{e}vy processes of pure jump type with infinite jump activity, are obtained numerically, as solutions of the partial integro-differential equations (PIDEs) they satisfy. This paper overcomes the ill-conditioning inherent in global meshfree methods by using localized RBF approximations known as the RBF partition of unity (RBF-PU) method for (PIDEs) arising in option pricing problems in L\'{e}vy driven assets. Then, Crank-Nicolson, LeapFrog (CNLF) is applied for time discretization. We treat the local term using an implicit step, and the nonlocal term using an explicit step, to avoid the inversion of the nonsparse matrix. For dealing with early exercise feature of American option and solving free boundary problem we use the implicit-explicit method combined with a penalty method. Efficiency and practical performance are demonstrated by numerical experiments for pricing European and American contracts.

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Condensed generalized finite element method

Cui

2020

Numerical Methods Partial

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Generalized or extended finite element methods (GFEM/XFEM) are in general badly conditioned and have numerous additional degrees of freedom (DOF) compared with the FEM due to introduction of enriched functions. In this paper, we develop an approach to establish a subspace of a conventional GFEM/XFEM approximation space using partition of unity (PU) techniques and local least-square procedures. The proposed GFEM is referred to as condensed GFEM (CGFEM), which (i) possesses as many DOFs as the preliminary FEM, (ii) enjoys similar approximation properties with the GFEM/XFEM, and (iii) is well-conditioned in the sense that its conditioning is of the same order as that of the FEM. The fundamental approximation properties of CGFEM are proven mathematically. The CGFEM is applied to a problem of high order polynomial approximations and a Poisson crack problem. The optimal convergence orders of the former are proven rigorously. Numerical experiments and comparisons with the conventional GFEM/XFEM and FEM are presented to verify the theory and effectiveness of CGFEM.

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RBF-PU method for pricing options under the jump–diffusion model with local volatility

Cited by 22 publications

References 40 publications

An RBF–FD method for pricing American options under jump–diffusion models

An RBF–FD method for pricing American options under jump–diffusion models

RBF approximation by partition of unity for valuation of options under exponential Lévy processes

Condensed generalized finite element method

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