Group Analysis of Differential Equations and Integrable Systems − 2018

Stanislav Spichak (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine)
Valeriy Stogniy and Inna Kopas (National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine)

Symmetries and fundamental solutions of (2+1)-dimensional linear equation of pricing of Asian option

There was investigated the equation describing the pricing of Asian option in continuous time $t\in[0;T]$: $$ \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S} +S\frac{\partial V}{\partial A}-rV=0, $$ where $T$ is the term of the contract, $V=V(t,S,A)$ is the function of the option value, $S$ is the current stock price, $A$ is the average value of all available prices $S$ of the underlying assets by the time $t,$ $r$ and $\sigma$ are the constants describing the risk-free interest rate and stock volatility respectively. We found the maximal algebra of invariance for this equation. Then, by using the Aksenov-Berest approach, the invariance subalgebra for fundamental solutions is constructed. Basing on the operator of this subalgebra we can reduce the equation of pricing of Asian option to the one that has known fundamental solution. Thus, an invariant fundamental solution for the initial equation can be constructed. Using the results obtained we apply the ones to resolving problems with call or put boundary conditions.