Robust time spectral methods for solving fractional differential equations in finance

Abstract

In this work, we construct numerical methods to solve a wide range of problems in finance. This includes the valuation under affine jump diffusion processes, chaotic and hyperchaotic systems, and pricing fractional cryptocurrency models. These problems are of extreme importance in the area of finance. With today’s rapid economic growth one has to get a reliable method to solve chaotic problems which are found in economic systems while allowing synchronization. Moreover, the internet of things is changing the appearance of money. In the last decade, a new form of financial assets known as cryptocurrencies or cryptoassets have emerged. These assets rely on a decentralized distributed ledger called the blockchain where transactions are settled in real time. Their transparency and simplicity have attracted the main stream economy players, i.e, banks, financial institutions and governments to name these only. Therefore it is very important to propose new mathematical models that help to understand their dynamics. In this thesis we propose a model based on fractional differential equations. Modeling these problems in most cases leads to solving systems of nonlinear ordinary or fractional differential equations. These equations are known for their stiffness, i.e., very sensitive to initial conditions generating chaos and of multiple fractional order. For these reason we design numerical methods involving Chebyshev polynomials. The work is done from the frequency space rather than the physical space as most spectral methods do. The method is tested for valuing assets under jump diffusion processes, chaotic and hyperchaotic finance systems, and also adapted for asset price valuation under fraction Cryptocurrency. In all cases the methods prove to be very accurate, reliable and practically easy for the financial manager.