Application of Lie symmetries to Solving Partial Differential Equations associated with the Mathematics of Finance

In financial markets one is sometimes confronted with a complicated system of partial differential equations arising from some physical important problem, and the discovery of the explicit solution of the problem can result with very useful information. That is, the explicit solutions of the financial market models can be used as benchmarks for testing numerical methods of physical experiments. This fact is evidenced by the work of economists Black and Scholes, the Black-Scholes model, whereby they deduced the financial models from solving a linear parabolic partial diiierential equation that were then used in the iinance literature as the main vehicle for pricing contingent claims such as call and put options, together with all other financial derivatives. Due to their work a rich arsenal of methods of theory of partial differential equations were suddenly available for mathematicians working in the area of mathematical finance. Adopting their approach of deducing prices of contingent claim via solving the associated PDE models, we apply the algorithmic quantitative theory of Lie, the Lie symmetry analysis, to derive and solve the models associated with interest rate derivatives whose price dynamics comprise of partial differential equations in their set up. The interest rate derivative model that we consider is of great importance because it deviates from the usual models that are depended on the usual Vasicek model which has a disadvantage of producing negative interest rates. Our interest rate derivative PDE model is depended on the functional interest rate model that satisfies all properties of an interest rate model and produces positive interest rates upon certain restriction put on the co-domain. We obtain their Lie point symmetries and transformations that we then use to deduce their exact group-invariant solutions. In particular, we analyse a zero-coupon bond pricing PDE model and obtain its various reductions that we then use to solve and produce the pricing models for the aforementioned contingent claim. A systematic reductions on optimal Lie algebra is further performed to obtain optimal invariant solutions of the model as well. The resulting analytical expressions in both cases can then be used to add to the minute number of pricing models for the interest rate derivatives instruments in the literature; also play a vital role as benchmarks to verify real world data that is analysed numerically by numerical methods in financial markets.