Partial Differential Equations are of paramount importance to the applied mathematician. Applications of PDEs in economics and physics have lead to major design enhancements in the fields of finance and engineering by maximizing efficiency and modeling phenomena. Moreover, in the development of theories, PDEs finds its place in electrodynamics, diffusion, quantum mechanics, and of personal interest to myself, financial predictions.
Initially, I researched the mathematics used in Fourier Expansions to represent approximations of an arbitrary function in an effort to reduce 2D and 3D file sizes. The use of Fourier analysis was prevalent throughout this area of expertise. Thus, in order to take a step towards other 'real-life' application of Fourier Transforms, I decided that solving partial differential equations would be an ideal entry point. Despite the complexity of the topic, a thorough exploration of its components, alongside a personal interest in the new knowledge has allowed me to gain a deeper insight into the nature of stochastic calculus and its significance in a variety of financial topics.
The two applications of PDEs and Fourier Transforms I will be investigating are related to European call and put options. I decided to investigate the nature of the financial market as I thought it would be interesting to see the use of PDEs when analyzing visible real life phenomenal. I will derive a partial differential equation that aims to simulate the value of an option for an underlying asset. This will allow me to gain insight into the behavior of the markets growth and a feel for the power of Fourier Transforms when extracting data through mathematical analysis. Moreover, in an effort to validate the mathematics used in this investigation, the solution will be applied to an asset currently trading on the market, Tesla’s stock ($TSLA). After exploring crucial aspects in the prediction of option values, I will apply knowledge to the importance of deducing the optimal sell and buy conditions. Progressively this investigation will take a turn towards the theoretical aspects of Fourier Analysis, deducing important axioms and rules in the field of solving PDEs, and their uses or limitations.
The research question I aim to address: ”How can Fourier Transforms be applied to the process of solving partial differential equations and applications in finance?”, with hopes of highlighting the mathematics related to PDEs used by economists and getting one step closer to understanding the mathematical relationships in Fourier Analysis.