| Abstract: |
| 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
. Automorphisms are essential for constructing discrete symmetries of a given partial
differential equation. How does one fit 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. We use arbitrage together with the Black-
Scholes model for asset price movements when trading derivative securities. Arbitrage is
used to creating a portfolio and the discrete symmetries show how to create a portfolio.
Gazizov and Ibragimov, computed the Lie point symmetries of the Black-Scholes PDE
and found an infinite dimensional Lie algebra of infinitesimal symmetries generated by the
operators. |
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