Invariant approaches for the analytic solution of the stochastic Black-Derman toy model

Abstract

We work on the analytical solution of the stochastic differential equations (SDE) via invariant approaches. In particularly, we focus on the stochastic Black-Derman Toy (BDT) interest rate model, among others. After we present corresponding (1+1) parabolic linear PDE for BDT-SDE, we use theoretical framework about the invariant approaches for the (1+1) linear PDE being done in the literature. We show that it is not possible to reduce BDT-PDE into the first and second Lie canonical forms. On the other hand, we success to find transformations for reducing it to the third Lie canonical form. After that, we obtain analytical solution of BDT-PDE by using these transformations. Moreover, we conclude that it can be reduced to the fourth Lie canonical form but, to the best of our knowledge, its analytical solution in this form is hard to find yet.