Application of the Laplace Homotopy Perturbation Method to the Black–Scholes Model Based on a European Put Option with Two Assets

Prathumwan, Din; Trachoo, Kamonchat

doi:10.3390/math7040310

Cited by 5 publications

(3 citation statements)

References 40 publications

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“…On the other hand, Ref. [12] determined the solution to the Black-Scholes equation using the homotopy perturbation combination method and the Laplace transform. Then, Ref.…”

Section: Introductionmentioning

confidence: 99%

A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method

Sugandha,

Rusyaman,

Sukono

et al. 2023

Mathematics

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The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation’s assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems.

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“…On the other hand, Ref. [12] determined the solution to the Black-Scholes equation using the homotopy perturbation combination method and the Laplace transform. Then, Ref.…”

Section: Introductionmentioning

confidence: 99%

A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method

Sugandha,

Rusyaman,

Sukono

et al. 2023

Mathematics

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“…There are many researchers who investigated analytical and approximate solutions of the Black-Scholes equation. Various effective methods have been used to solve the Black-Scholes equation, for example, the finite difference method [37], finite element method [15,38], homotopy perturbation method [39,40], the Mellin transform method [41], Adomian decomposition method [42], the variational iteration method [43,44], radial basis function partition of unity method (RBF-PUM) [45,46], and adaptive moving mesh method [47].…”

Section: Introductionmentioning

confidence: 99%

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

Prathumwan

Trachoo

2020

Adv Differ Equ

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The purpose of this paper was to investigate the dynamics of the option pricing in the market through the two-dimensional time fractional-order Black-Scholes equation for a European put option. The Liouville-Caputo derivative was used to improve the ordinary Black-Scholes equation. The analytic solution is a powerful tool for describing the behavior of the option price in the European style market. In this study, analytic solution is carried out by the Laplace homotopy perturbation method. Moreover, the obtained solution showed that the Laplace homotopy perturbation method was an efficient method for finding an analytic solution of two-dimensional fractional-order differential equation.

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“…Jena and Chakraverty ( 2019 ) introduced a residual power series (RPS) method for finding the analytical solution for the fractional B–S equation with an initial condition in the European option pricing problem. Prathumwan and Trachoo ( 2019 ) applied the Laplace HPM to yield an approximate solution for the B–S partial differential equation in the European put option with two assets. Uddin and Taufiq ( 2019 ) solved the time-fractional B–S equation through a transformation method with the radial basis kernel.…”

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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Application of the Laplace Homotopy Perturbation Method to the Black–Scholes Model Based on a European Put Option with Two Assets

Cited by 5 publications

References 40 publications

A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method

A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method

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

Solving Black–Scholes equations using fractional generalized homotopy analysis method

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