A Nonlinear Option Pricing Model Through the Adomian Decomposition Method

González-Gaxiola, O.; Chávez, Juan Ruíz de; Santiago, José

doi:10.1007/s40819-015-0070-6

Cited by 22 publications

(15 citation statements)

References 30 publications

(45 reference statements)

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“…This is possible because the Lie derivative of GL + (1) (22) can also be represented byẼ r 1 from (73). Hence, we can write (24) in terms of the affine cotangent bundle group Lie derivatives as,…”

Section: Backward Compatibility Of Propagation On the Cotangent Bundlmentioning

confidence: 99%

“…The application of mean and covariance propagation to A f f + (1) becomes the main subject of the paper. There exist other methods to approximately solve PDEs in mathematical finance, such as finite difference methods [21,22], finite element methods [23] and the Adomian decomposition method [24,25]. In this paper, we also make a comparison between the mean and covariance propagation technique and a standard finite difference scheme used to solve the governing equations.…”

Section: Introductionmentioning

confidence: 99%

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Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Jayaraman

Campolo

Chirikjian

2020

Entropy

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The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

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Section: Backward Compatibility Of Propagation On the Cotangent Bundlmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Jayaraman

Campolo

Chirikjian

2020

Entropy

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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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“…In literature, a lot of solution methods have been considered by a good number of researchers. These include: Adomian decomposition method (ADM), variational iteration method (VIM), Modified ADM (MADM), homotopy perturbation method (HPM), Differential transformation method (DTM), projected DTM (PDTM) [18][19][20][21][22][23][24][25]. He's polynomials method was initiated in [26,27] by Ghorbani et al, where the nonlinear terms were expressed as series of polynomials calculated with the aid of HPM.…”

Section: Introductionmentioning

confidence: 99%

Closed-form Solutions of the Time-fractional Standard Black-Scholes Model for Option Pricing using He-separation of Variable Approach

Edeki

Akinlabi

Egara

et al. 2020

Wseas Transactions on Environment and Development

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The Black-Scholes option pricing model in classical form remains a benchmark model in Financial Engineering and Mathematics concerning option valuation. Though, it has received a series of modifications as regards its initial constancy assumptions. Most of the resulting modifications are nonlinear or time-fractional, whose exact or analytical solutions are difficult to obtain. This paper, therefore, presents exact (closed-form) solutions to the time-fractional classical Black-Scholes option pricing model by means of the He-Separation of Variable Transformation Method (HSVTM). The HSVTM combines the features of the He's polynomials, the Homo-separation variable, the modified DTM, which increases the efficiency and effectiveness of the proposed method. The proposed method is direct and straight forward. Hence, it is recommended for obtaining solutions to financial models resulting from either Ito or Stratonovich Stochastic Differential Equations (SDEs).

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A Nonlinear Option Pricing Model Through the Adomian Decomposition Method

Cited by 22 publications

References 30 publications

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

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

Closed-form Solutions of the Time-fractional Standard Black-Scholes Model for Option Pricing using He-separation of Variable Approach

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