Fractional Pricing Models: Transformations to a Heat Equation and Lie Symmetries

Abstract

The study of fractional partial differential equations is often plagued with complicated models and solution processes. In this paper, we tackle how to simplify a specific parabolic model to facilitate its analysis and solution process. That is, we investigate a general time-fractional pricing equation, and propose new transformations to reduce the underlying model to a different but equivalent problem that is less challenging. Our procedure leads to a conversion of the model to a fractional 1 + 1 heat transfer equation, and more importantly, all the transformations are invertible. A significant result which emerges is that we prove such transformations yield solutions under the Riemann–Liouville and Caputo derivatives. Furthermore, Lie point symmetries are necessary to construct solutions to the model that incorporate the behaviour of the underlying financial assets. In addition, various graphical explorations exemplify our results.

Lie Symmetries and the Invariant Solutions of the Fractional Black–Scholes Equation under Time-Dependent Parameters

“…This study is the first of its kind. In [17], a general fractional bond equation was considered that had no connections to the B-S model. Moreover, the model in [17] had constant coefficients.…”

“…In [17], a general fractional bond equation was considered that had no connections to the B-S model. Moreover, the model in [17] had constant coefficients. In this study, the model possesses functions dependent on time and it is a famous version of the B-S equation.…”

“…Furthermore, in [17] we chose to transform the model to a fractional-order equation for further analysis, whereas here, we showcase how it is possible to transform a different model into an integer-order model. Hence, this paper has a novel approach with original transformations for a model that has not been studied before under these contexts, and of course, all solutions presented are new.…”