A New Version of Black–Scholes Equation Presented by Time-Fractional Derivative

Farhadi, A.; Salehi, Mohsen; Erjaee, G.H.

doi:10.1007/s40995-017-0244-7

Cited by 17 publications

(13 citation statements)

References 15 publications

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“…For the time-fractional Black-Scholes model, finite-difference approximations were also considered by Koleva and Vulkov [20]. The effect of trend memory in financial option pricing was described by Farhadi et al [14] using the time-fractional Black-Scholes equation. Chen et al [3] recently considered an operator splitting method for the evaluation of American options under the same model.…”

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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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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“…Recently, a good amount of progress has been made to model the variety of financial contracts through fractional evolution equations. For instance, Wyss in [61], Jumarie in [22], and A. Farhadi in [2] studied fractional versions of the Black-Scholes equation modeling a variety of financial derivatives for risk hedging. Wyss in [61] introduced time-fractional Black-Scholes equation to price European style option.…”

Section: Introductionmentioning

confidence: 99%

“…Jumarie in [22] introduced two versions of fractional Black-Scholes equation and numerically computer their solutions. A. Farhadi in [2] incorporated tend memory in the pricing of financial derivatives and derived new time-fractional order Black-Scholes equation, and RVIM algorithm was used to compute the solution. Some of the recent work on treatments of fractional Black-Scholes equation includes, [13] by Prathumwan, which focuses on 2D time fractional-order Black-Scholes equation for a European put option; [41] by Mianfu, Lili, Renxuan, Dongfang developed a novel scheme for time-fractional based on the Chebyshev-Galerkin spectral method.…”

Section: Introductionmentioning

confidence: 99%

On numerical pricing of put-call parities for Asian options driven by new time-fractional Black-Scholes evolution equation

Hussain

Khan

2021

Filomat

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The objective of this paper is twofold. Firstly, to derive time-fractional evolution equation modeling the No-Arbitrage premium of Asian option (with arithmetic and geometric averages ) contingent upon an underlying asset that satisfies the fractional stochastic differential equation, in a setting when the strike price is fixed and floating. Secondly, we have computed the four versions of the put-call parities for Asian options, by solving the time-fractional Black-Scholes evolution modeling the difference of the premiums of put and call Asian options, through Fractional Reduced Differential Transform (FRDT) algorithm. We have also established the convergence and the error estimates for the FRDT Algorithm for the two independent variables.

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“…Ouafoudi and Gao (2018) used both HPM and modified HPM and Sumudu transform to yield solutions for the B-S equation in the form of convergence power series with a regularly calculated element. Farhadi and Erjaee (2018) introduced a timefractional derivative for solving the B-S equation. Sawangtong et al (2018) investigated an analytical solution for undertaking the B-S equation with two assets using the Laplace transform HPM approach in the Liouville-Caputo fractional derivative sense.…”

Section: Introductionmentioning

confidence: 99%

“…Ouafoudi and Gao ( 2018 ) used both HPM and modified HPM and Sumudu transform to yield solutions for the B–S equation in the form of convergence power series with a regularly calculated element. Farhadi and Erjaee ( 2018 ) introduced a time-fractional derivative for solving the B–S equation. Sawangtong et al.…”

Section: Introductionmentioning

confidence: 99%

Solving Black–Scholes equations using fractional generalized homotopy analysis method

Saratha¹,

Krishnan

Bagyalakshmi

et al. 2020

Comp. Appl. Math.

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This paper aims to solve the Black-Scholes (B-S) model for the European options pricing problem using a hybrid method called fractional generalized homotopy analysis method (FGHAM). The convergence region of the B-S model solutions are clearly identified using h-curve and the closed form series solutions are produced using FGHAM. To verify the convergence of the proposed series solutions, sequence of errors are obtained by estimating the deviation between the exact solution and the series solution, which is increased in number of terms in the series. The convergence of sequence of errors is verified using the convergence criteria and the results are graphically illustrated. Moreover, the FGHAM approach has overcome the difficulties of applying multiple integration and differentiation procedures while obtaining the solution using well-established methods such as homotopy analysis method and homotopy perturbation method. The computational efficiency of the proposed method is analyzed using a comparative study. The advantage of the proposed method is shown with a numerical example using the comparative study between FGHAM and Monte Carlo simulation. Using the numerical example, analytical expression for the implied volatility is derived and the non-local behavior is studied for the various values of the fractional parameter. The results of FGHAM are statistically validated with the exact solution and the other existing computational methods.

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A New Version of Black–Scholes Equation Presented by Time-Fractional Derivative

Cited by 17 publications

References 15 publications

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

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

On numerical pricing of put-call parities for Asian options driven by new time-fractional Black-Scholes evolution equation

Solving Black–Scholes equations using fractional generalized homotopy analysis method

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