On the solution of two-dimensional fractional Black–Scholes equation for European put option

Prathumwan, Din; Trachoo, Kamonchat

doi:10.1186/s13662-020-02554-8

Cited by 20 publications

(10 citation statements)

References 46 publications

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“…Further, the relation (17) with the initial condition in (22), yields an algebraic system of (N κ + 1) equations…”

Section: Initial Statementioning

confidence: 99%

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An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

Singh¹,

Kumar²

2022

Preprint

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In this paper a time-fractional Black-Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and 2 − µ is time, where µ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option.

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“…Further, the relation (17) with the initial condition in (22), yields an algebraic system of (N κ + 1) equations…”

Section: Initial Statementioning

confidence: 99%

“…Liang et al [15] derived a bi-fractional Black-Merton-Scholes model. With growth of applications of fractional models in financial field, researchers have shown interest in solving them analytically [16][17][18][19][20] and numerically [21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

Singh¹,

Kumar²

2022

Preprint

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“…The use of the fractional BS model for the high volatility of the stock market is one such generalization. There are two types of fractional derivatives as space-fractional [4,5] and time-fractional derivatives [6,7]. Regarding the time-fractional model, researchers have focused on the analytical [8][9][10] and numerical [11][12][13] methods.…”

Section: Introductionmentioning

confidence: 99%

Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

Kim

Lee

et al. 2021

Journal of Function Spaces

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In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-order α in the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

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“…More recently, a two-dimensional fractional partial differential equation (FPDE) has been established, based on the two-dimensional FMLS model for option pricing [10]. Recently, several researchers [6,17,23,25,[27][28][29] have investigated the problem of option pricing under the Black-Scholes fractional framework or the generalized Black-Scholes equation. Such low-order convergence in both space and time implies that we need a lot of computational nodes in both dimensions to reach reasonable accuracies.…”

Section: Introductionmentioning

confidence: 99%

Spectrally Accurate Option Pricing Under the Time-Fractional Black–scholes Model

Tour¹,

Thakoor²,

Tangman³

2021

ANZIAM J.

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We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular $L1$ finite difference approximation, which converges with order $\mathcal {O}((\Delta \tau )^{2-\alpha })$ for functions which are twice continuously differentiable. However, when using the $L1$ scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

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On the solution of two-dimensional fractional Black–Scholes equation for European put option

Cited by 20 publications

References 46 publications

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

Spectrally Accurate Option Pricing Under the Time-Fractional Black–scholes Model

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