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.
