Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance

Open Physics

Khalique Chaudry Masood¹  Fatima Aeeman¹  Aziz Taha² 

[1] International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa;School of Computer, Statistical and Mathematical Sciences, North-West University, Potchefstroom Campus, Private Bag X2520, Potchefstroom, South Africa;

关键词: bond-pricing equation;    invariant method;    fundamental solutions;    cauchy problem for the bond-pricing model;    02.20.sv;    02.30.jr;    04.20.jb;   

DOI  :  10.1515/phys-2018-0096

来源: DOAJ

【 摘 要 】

The invariant approach is employed to solve the Cauchy problem for the bond-pricing partial differential equation (PDE) of mathematical finance. We first briefly review the invariant criteria for a scalar second-order parabolic PDE in two independent variables and then utilize it to reduce the bond-pricing equation to different Lie canonical forms. We show that the invariant approach aids in transforming the bond-pricing equation to the second Lie canonical form and that with a proper parametric selection, the bond-pricing PDE can be converted to the first Lie canonical form which is the classical heat equation. Different cases are deduced for which the original equation reduces to the first and second Lie canonical forms. For each of the cases, we work out the transformations which map the bond-pricing equation into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the bond-pricing model via these transformations by utilizing the fundamental solutions of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the Cauchy initial value problems for the bond-pricing model with proper choice of terminal conditions are obtained.

【 授权许可】

Unknown