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Group Classification (1+2)-dimensional Linear Equation of Asian Options Pricing

Stanislav V. Spichak, Valeriy I. Stogniy, Inna M. Kopas

TL;DR

The paper conducts a comprehensive group classification of a broad class of (1+2)-dimensional linear PDEs arising in Asian options pricing. It employs classical Lie group methods to determine the kernel algebra, derives the equivalence group, and reduces the problem to a set of canonical forms for the arbitrary function f(x). The authors show that equations with eight-dimensional invariance can be transformed to the linear Kolmogorov equation and obtain invariant exact solutions via symmetry reductions, producing canonical models with explicitly characterized symmetry algebras. This framework clarifies the symmetry structure of Asian options PDEs and provides analytic tools for constructing solutions in financially relevant settings.

Abstract

We consider a class of (1+2)-dimensional linear partial differential of Asian options pricing. Special cases have been used to models of financial mathematics. We carry out group classification of a class equations. In particular, the maximum dimension Lie invariance algebra within the above class is eight-dimensional. It is shown that an equation with such an algebra can be transformed into the linear Kolmogorov equation with the help of the point transformations of variables. Using the operators of invariance algebra symmetry reduction is carried out and invariant exact solutions are constructed for some equations.

Group Classification (1+2)-dimensional Linear Equation of Asian Options Pricing

TL;DR

The paper conducts a comprehensive group classification of a broad class of (1+2)-dimensional linear PDEs arising in Asian options pricing. It employs classical Lie group methods to determine the kernel algebra, derives the equivalence group, and reduces the problem to a set of canonical forms for the arbitrary function f(x). The authors show that equations with eight-dimensional invariance can be transformed to the linear Kolmogorov equation and obtain invariant exact solutions via symmetry reductions, producing canonical models with explicitly characterized symmetry algebras. This framework clarifies the symmetry structure of Asian options PDEs and provides analytic tools for constructing solutions in financially relevant settings.

Abstract

We consider a class of (1+2)-dimensional linear partial differential of Asian options pricing. Special cases have been used to models of financial mathematics. We carry out group classification of a class equations. In particular, the maximum dimension Lie invariance algebra within the above class is eight-dimensional. It is shown that an equation with such an algebra can be transformed into the linear Kolmogorov equation with the help of the point transformations of variables. Using the operators of invariance algebra symmetry reduction is carried out and invariant exact solutions are constructed for some equations.

Paper Structure

This paper contains 3 sections, 4 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Algebra $A^{ker}$ and Group of Equivalence Transformations
  3. Classification of Lie symmetries

Key Result

Theorem 1

For arbitrary function $f(x)$ the Lie symmetry algebra of equation Sp-St-1 is where $\beta(t,x)$ is an arbitrary solutions of equation $\beta_t=x^2\beta_{xx}$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4