The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense

Sawangtong, Panumart; Trachoo, Kamonchat; Sawangtong, Wannika; Wiwattanapataphee, Benchawan

doi:10.3390/math6080129

Cited by 23 publications

(18 citation statements)

References 32 publications

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“…Hence, further investigation of such models model is an active topic of research. [13][14][15] The SVCJ model for the value of asset S t and its variance V t satisfies 1…”

Section: And the References Therein)mentioning

confidence: 99%

“…These mathematical models and their usefulness were later discussed in Eraker et al As a matter of fact, the authors confirmed that the SVIJ and SVCJ models furnish the correct fitting, but SVCJ is more straightforward to approximate because jumps in returns and volatility are simultaneous. Hence, further investigation of such models model is an active topic of research …”

Section: Introductory Notesmentioning

confidence: 99%

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Asset pricing for an affine jump‐diffusion model using an FD method of lines on nonuniform meshes

Soleymani

Akgül

2018

Math Methods in App Sciences

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We present a novel numerical scheme for the valuation of options under a well‐known jump‐diffusion model. European option pricing for such a case satisfies a 1 + 2 partial integro‐differential equation (PIDE) including a double integral term, which is nonlocal. The proposed approach relies on nonuniform meshes with a focus on the discontinuous and degenerate areas of the model and applying quadratically convergent finite difference (FD) discretizations via the method of lines (MOL). A condition for observing the time stability of the fully discretized problem is given. Also, we report results of numerical experiments.

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“…Hence, further investigation of such models model is an active topic of research. [13][14][15] The SVCJ model for the value of asset S t and its variance V t satisfies 1…”

Section: And the References Therein)mentioning

confidence: 99%

Section: Introductory Notesmentioning

confidence: 99%

Asset pricing for an affine jump‐diffusion model using an FD method of lines on nonuniform meshes

Soleymani

Akgül

2018

Math Methods in App Sciences

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“…Chen and Wang [13] presented a second-order Crank-Nicolson alternating direction implicit (ADI) scheme for solving a 2D spatial fractional BS equation. Sawangtong et al [14] proposed the BS equation with two assets based on the Liouville-Caputo fractional derivative. In [15], Zhan et al developed numerical methods for the time fractional BS equation.…”

Section: Introductionmentioning

confidence: 99%

Finite Difference Method for the Multi-Asset Black–Scholes Equations

et al. 2020

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In this paper, we briefly review the finite difference method (FDM) for the Black–Scholes (BS) equations for pricing derivative securities and provide the MATLAB codes in the Appendix for the one-, two-, and three-dimensional numerical implementation. The BS equation is discretized non-uniformly in space and implicitly in time. The two- and three-dimensional equations are solved using the operator splitting method. In the numerical tests, we show characteristic examples for option pricing. The computational results are in good agreement with the closed-form solutions to the BS equations.

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“…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. Yavuz and Ozdemir (2018) initiated a conformable fractional adomian decomposition method (CFADM) and conformable fractional modified HPM to tackle the fractional B-S equation.…”

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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The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense

Cited by 23 publications

References 32 publications

Asset pricing for an affine jump‐diffusion model using an FD method of lines on nonuniform meshes

Asset pricing for an affine jump‐diffusion model using an FD method of lines on nonuniform meshes

Finite Difference Method for the Multi-Asset Black–Scholes Equations

Solving Black–Scholes equations using fractional generalized homotopy analysis method

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