An accurate solution for the generalized Black-Scholes equations governing option pricing

Ashish Awasthi; Riyasudheen TK; Ashish Awasthi; Riyasudheen TK

doi:10.3934/math.2020147

AIMS Mathematics

2020, Volume 5, Issue 3: 2226-2243. doi: 10.3934/math.2020147

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Research article

An accurate solution for the generalized Black-Scholes equations governing option pricing

Ashish Awasthi ^,,
Riyasudheen TK

Department of Mathematics, National Institute of Technology Calicut, Calicut - 673601, India

Received: 29 November 2019 Accepted: 17 February 2020 Published: 28 February 2020
MSC : 62P05, 65N40, 65N12, 65N15

Today industries related to finance are essentially implementing advanced mathematical tools. In 1973, Fisher Black and Myron Scholes developed an eminent stochastic model which later coined as Black-Scholes differential equations for option pricing. This paper illustrates a convenient time integration scheme based on the generalized trapezoidal formulas (GTF $[\alpha = \frac{1}{3}]$) introduced by Chawla et al. in 1996. GTF is applied for the temporal discretization along with the classical finite difference schemes in space direction. The proposed scheme yields the (uniform) stability employing the uniform bound of the inverse operator, as well as second-order spatial accuracy and third-order temporal accuracy under reasonable conditions. Finally, the numerical illustrations and comparison with existing schemes demonstrate the stability and accuracy of the method.
- Black-Scholes equation,
- option pricing,
- european options,
- generalized trapezoidal formulas,
- uniform boundedness
Citation: Ashish Awasthi, Riyasudheen TK. An accurate solution for the generalized Black-Scholes equations governing option pricing[J]. AIMS Mathematics, 2020, 5(3): 2226-2243. doi: 10.3934/math.2020147

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Abstract

Today industries related to finance are essentially implementing advanced mathematical tools. In 1973, Fisher Black and Myron Scholes developed an eminent stochastic model which later coined as Black-Scholes differential equations for option pricing. This paper illustrates a convenient time integration scheme based on the generalized trapezoidal formulas (GTF $[\alpha = \frac{1}{3}]$) introduced by Chawla et al. in 1996. GTF is applied for the temporal discretization along with the classical finite difference schemes in space direction. The proposed scheme yields the (uniform) stability employing the uniform bound of the inverse operator, as well as second-order spatial accuracy and third-order temporal accuracy under reasonable conditions. Finally, the numerical illustrations and comparison with existing schemes demonstrate the stability and accuracy of the method.

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