A Finite Difference Scheme for Option Pricing in Jump-diffusion and Exponential Levy Models

Cont, Rama; Voltchkova, Ekaterina

doi:10.2139/ssrn.458200

Cited by 138 publications

(266 citation statements)

References 24 publications

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“…For each n ≥ 1, the pricing function v n is the unique solution of the classical free-boundary problem (instead of a free boundary problem with an integro-diffential equation) 9) in which t → s n (t) is the free-boundary (the optimal exercise boundary) which needs to be determined (see Lemma 3.5 of [6]). Now starting from v 0 , we can calculate {v n } n≥0 sequentially.…”

Section: A Sequence Of Optimal Stopping Problems For Geometric Brownimentioning

confidence: 99%

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Bayraktar

Xing

2009

Math Meth Oper Res

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Abstract. We approximate the price of the American put for jump diffusions by a sequence of functions, which are computed iteratively. This sequence converges to the price function uniformly and exponentially fast. Each element of the approximating sequence solves an optimal stopping problem for geometric Brownian motion, and can be numerically computed using the classical finite difference methods. We prove the convergence of this numerical scheme and present examples to illustrate its performance.

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Section: A Sequence Of Optimal Stopping Problems For Geometric Brownimentioning

confidence: 99%

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Bayraktar

Xing

2009

Math Meth Oper Res

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“…Recently, Lévy process models have become popular in the financial literature [1,2,3,4,5,6,7]. Option pricing, under exponential Lévy process with finite activity [5,8,9,10,11,12] and infinite activity [13,14,15,16,17] has been extensively studied. In these papers, various numerical methods were proposed for solving the option pricing Partial IntegroDifferential Equation (PIDE).…”

Section: Introductionmentioning

confidence: 99%

Robust numerical valuation of European and American options under the CGMY process

Wang¹,

Wan²,

Forsyth³

2007

JCF

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We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semi-Lagrangian scheme. The resulting Partial Integro-Differential Equation (PIDE) is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.

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“…Viscosity solutions of the PIDE are considered in [11] for the pure jump model, i.e., σ (S, T ) ≡ 0. In the case of the European option under the Merton jump diffusion, U i 's in (2.4) are identically, independently and normally distributed.…”

Section: Markets Driven By Lévy Processesmentioning

confidence: 99%

“…While it is well-known that the solution of (2.10) goes to zero at the right infinity (cf, [11]), the rate of decay can be very slow in the Merton jump diffusion model. Consequently, when S is large, domain truncation may introduce significant errors.…”

Section: With the Substitution V(s τ ) = V (S T − τ )mentioning

confidence: 99%

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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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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A Finite Difference Scheme for Option Pricing in Jump-diffusion and Exponential Levy Models

Cited by 138 publications

References 24 publications

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Robust numerical valuation of European and American options under the CGMY process

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

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