Discrete symmetry analysis of partial differential equations for bond pricing
We show how to compute the discrete symmetries for a given Black-Scholes (B-S) partial differential equation (PDE) with the aid of the full automorphism group of the Lie algebra associated to the standard B-S PDE. The paper determines the discrete symmetries using two methods. The first is by G. Silberberg which determines the full automorphism group by constructing the symmetry generators' centralizer and Lie algebra's radical. The other is by P. Hydon which is based on the observation that the adjoint action of any point symmetry of a partial differential equation is an automorphism of the PDE's Lie point symmetry algebra [27]. Automorphisms are essential for constructing discrete symmetries of a given partial differential equation. How does one _t in this mathematical concept in the application of finance? The concept of arbitrage which in certain circumstances allows us to establish the precise relationship between prices and thence how to determine prices, underlies the theory of financial derivatives pricing and hedging [40]. We use arbitrage together with the Black-Scholes model for asset price movements when trading derivative securities. 1Arbitrage is used to creating a portfolio and the discrete symmetries show how to create a portfolio. Gazizov and Ibragimov [10], computed the Lie point symmetries of the Black-Scholes PDE and found an infinite dimensional Lie algebra of infinitesimal symmetries generated by the operators. Discrete symmetries are more effective on PDEs since they are not held back by boundary conditions and are used in1. equivalent bifurcation theory; 2. construction of invariant solutions; 3. simplification of numerical schemes. 4. used in put-call parity relationship (see application in finance); 5. used in put-call symmetry relationship (see application in finance)
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Department of Mathematics and Applied Mathematics