Abstract:
The trading of financial derivatives and products in financial markets has
influenced the development of the world economy. Over the last few
decades, a rapid growth in complex financial systems, which can generate
unstable conditions in financial markets, has been observed. Therefore
models are being developed to study and examine the uncertainty
surrounding these financial systems in different circumstances. The
important milestone of this work can be traced to the Black-Scholes formula
for option pricing which was published in 1973 and revolutionized the
financial industry by introducing the no-arbitrage principle. This model
assumed that the average rates of return and volatility are constant however,
this is not realistic. Therefore, several models have been developed, based
on pragmatic studies, which generalize the Black-Scholes formula to acquire
more knowledge for these financial systems.
In this project, we focused on the following mean-reverting-theta stochastic
volatility model in finance which did not have explicit solutions.
where and for are constant and We first
developed a technique to prove the non-negativity of solutions to the model.
We then showed that the Euler–Maruyama (EM) numerical solutions will converge to the true solution in probability. We also showed that the EM
solutions can be used to compute some financial quantities related to the
Stochastic Differential Equations (SDEs) models including the option value,
for example.
