Spectral Methods

We present a new spectral element method for solving partial integro-differential equations for pricing European options under the Black-Scholes and Merton jump diffusion models. Our main contributions are: (i) using an optimal set of orthogonal polynomial bases to yield banded linear systems and to achieve spectral accuracy; (ii) using Laguerre functions for the approximations on the semi-infinite domain, to avoid the domain truncation; and (iii) deriving a rigorous proof of stability for the time discretizations of European put options under both the Black-Scholes model and the Merton jump diffusion model. The new method is flexible for handling different boundary conditions and non-smooth initial conditions for various contingent claims. Numerical examples are presented to demonstrate the efficiency and accuracy of the new method.

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“…We can then conclude (3.31) by applying the discrete Gronwall's inequality (see, for instance, Lemma B.10 in [32]) to the above.…”

Section: Lemma 34 (L 2 -Bound) Rv(s) 2 ≤ C(δ) V(s)mentioning

confidence: 92%

A New Spectral Element Method for Pricing European Options Under the Black–Scholes and Merton Jump Diffusion Models

Feng

Shen

2011

J Sci Comput

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“…The weighted residual method forms the basis for the development of these approximation techniques and will help to distinguish them [57].…”

Section: Galerkin Numerical Methodsmentioning

confidence: 99%

“…Spectral methods, which have gained much popularity due to their spectral accuracy and versatility [56,57], can be formulated for both local (element-based) and global approximations. Consider the task of discretizing the dynamics of problem (2.5)-(2.8), 52) where t → (x, u) ∈ R Nx × R Nu , with initial conditions, 53) and endpoint conditions,…”

Section: Numerical Solutions To Differential Equationsmentioning

confidence: 99%

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Galerkin Optimal Control

Boucher

Kang

Gong

2016

J Optim Theory Appl

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“…One of the most important applications of polynomial differentiation in other bases is in spectral methods like collocation method [14]. Differentiation matrices for Chebyshev and Jacobi polynomials were computed [13]. The differentiation of Jacobi polynomials through Bernstein basis was studied [12].…”

Section: Introductionmentioning

confidence: 99%