Analysis of option pricing within the scope of fractional calculus

[摘要] This works opens new and promising avenues of investigation in pricing mechanisms of financial derivatives. A notorious problem when pricing an option with the Black Scholes model is the poor long-term prediction, and the failure to capture the large jumps that often occur over small time intervals. This is mostly due to the fact that in his classic version, the Black Scholes model assumption for the change in price of the underlying asset is that prices are subjected to a Brownian motion type of process. As Gaussian Markovian processes proved incapable of satisfactorily give account of those occurrences, various ways of incorporating memory as well as models with jumps were considered as a remediation. Based on the fact that diffusion equations with fractional derivatives have been efficient in describing some very complex anomalous diffusion systems, fractional operators were introduced in mathematical finance. The rationale of our exploratory analysis is to exploit memory properties, non-locality, non-singularity, 'globalness', of fractional differentiation operators, to develop and analyse a new class of Time Fractional Black Scholes Equations (TFBSEs). Some recent fractional operators like, the Caputo-Fabrizio, the Atangana-Baleanu in the sense of Riemann and the Atangana-Baleanu in the sense of Caputo, show crossover properties and behaviours of some basic measures and indicators. Some related statistics that are not scale invariant, but crossing over from ordinary to sub-diffusion properties, with waiting time distributions, moving from Gaussian to non-Gaussian distributions, etc.In the first part of this PhD, we generalise a double barrier knock out Black Scholes diffusion equation to five Fractional Partial Differential Equations (FPDEs) that we will analyse. The fractional differential operator is successively defined in the sense of Caputo, Riemann Liouville, Caputo-Fabrizio, Atangana-Baleanu in the Riemann sense and Atangana-Baleanu in the sense of Caputo. To the best of our knowledge no single appearance of any of the last three equations can be found in the literature. With the given boundary conditions, we establish the existence and uniqueness of solutions to the five equations. We develop six new numerical scheme solutions to the TFBSEs on one side, five semi-analytical solutions to our new TFBSEs using Laplace transform and Sumudu transform, on the other side. We assess the convergence of the numerical scheme solutions derived with Caputo and Riemann Liouville fractional derivative operator, to compare with the very scarce similar literature. With the obtained numerical scheme solution from the Caputo-Fabrizio TFBSE, we proceed to price a double barrier knock out call option with specific parameter values, and look at the price behaviours if they seem to reflect some of the properties we ambition to capture.Additionally, due to the fact with FPDEs there is usually a very cumbersome and tricky to handle summation term that appears in the numerical scheme solutions, making the stability analysis a considerable challenge, we develop a complete novel method to tackle Partial Differential Equations (PDEs) and FPDEs. The new method causes the incriminated summation term to disappear, when it is used with some fractional differential operators. The stability and error analysis of the new method are also presented. The method is conceived from a skilful combination of higher order accuracy Adam-Bashforth method and Laplace transform. To illustrate the potential of the method, we present some general applications on PDE and FPDEs. We then use the method to derive a numerical scheme solution the TFBSE with ABC fractional derivative.In the last part of the thesis we comment the results we obtained, conclude our analysis and share our outlook on fractional operators and Black Scholes models in option pricing.

[发布机构] University of the Free State