European Option Pricing of Fractional Black-Scholes Model Using Sumudu Transform and its Derivatives

Khan, Waseem Asghar; Ansari, Faryal Aijaz

doi:10.31559/glm2016.1.3.1

Cited by 7 publications

(8 citation statements)

References 17 publications

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“…Chen et al investigated predictorcorrector approaches for the solution of American option in 2015 [34]. Khan and Ansari [35] were the first to use the Sumudu transform to solve the fractional model of European options. In the same year, Zhang provided an unconditionally stable implicit numerical scheme for the model by changing the Riemann-Liouville derivatives to the Caputo derivative [32].…”

Section: Introductionmentioning

confidence: 99%

Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

Bajalan

2021

Complexity

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The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.

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Section: Introductionmentioning

confidence: 99%

Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

Bajalan

2021

Complexity

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“…investigated Predictor-corrector approaches for the solution of American option in 2015 (Chen (2015b)). Khan and Ansari (Khan & Ansari (2016)) were the first to use the Sumudu transform to solve the fractional model of European options. In the same year, Zhang provided an unconditionally stable implicit numerical scheme for the model by changing the Riemann-Liouville derivatives to Caputo derivative (Zhang et al, 2016b).…”

Section: Introductionmentioning

confidence: 99%

Novel ANN method for solving ordinary and fractional Black-Scholes equation

Bajalan,

Bajalan

2021

Preprint

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The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of Ordinary Differential Equations (ODE). Then each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine tuning for speeding up the process and domain mapping to confront infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of Black-Scholes model are reported.

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“…Ravi Kanth and Aruna (2016) employed the fractional differential transform method and modified the fractional differential transform method to solve the time-fractional B-S European option pricing equation. Khan and Ansari (2016) solved the fractional B-S European option pricing equation using the Sumudu transform and its derivatives. Granada et al (2017) applied the HPM model to solve the B-S equation.…”

Section: Introductionmentioning

confidence: 99%

“…Khan and Ansari ( 2016 ) solved the fractional B–S European option pricing equation using the Sumudu transform and its derivatives. Granada 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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European Option Pricing of Fractional Black-Scholes Model Using Sumudu Transform and its Derivatives

Cited by 7 publications

References 17 publications

Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

Novel ANN method for solving ordinary and fractional Black-Scholes equation

Solving Black–Scholes equations using fractional generalized homotopy analysis method

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