Stochastic modelling and Monte Carlo simulations in finance

Stochastic modelling of interest rates is very important for calibrating and evaluating expected payoffs of interest-rate products. Many well-known univariate linear drift stochastic models have been proposed to explain interest rate dynamics. However, by testing parametric models by comparing their implied parametric density to the same density estimated non-parametrically, Ait-Sahalia revealed all the existing univariate linear drift stochastic models could not explain well the dynamics of Euro-dollar interest rates. As a result, he proposed a new class of highly non-linear stochastic interest rate models. The original Ait-Sahalia interest rate model has been found to have considerable use for modelling time series evolution of interest rates. However, this model does not possess certain specifications to provide adequate descriptions of interest rates against unexpected empirical phenomena such as volatility 'skews' and 'smiles', jump behaviour, market regulatory lapses, economic crisis, financial clashes, political instability, among others collectively. In this thesis, we propose a modified version of this model by incorporating additional features to help collectively describe these empirical phenomena adequately. However, the proposed model does not have explicit solution. Hence, we split it into three stochastic interest rate models and construct a new implementable truncated EM scheme to approximate them numerically. Further, we study finite time strong convergence of the truncated EM solutions to the exact solutions of the three models under the local Lipschitz condition plus the Khasminskii-type condition. Moreover, we perform numerical simulations to validate the strong convergence results and justify these results within Monte Carlo frameworks to evaluate expected payoffs of some financial products.Stochastic modelling of interest rates is very important for calibrating and evaluating expected payoffs of interest-rate products. Many well-known univariate linear drift stochastic models have been proposed to explain interest rate dynamics. However, by testing parametric models by comparing their implied parametric density to the same density estimated non-parametrically, Ait-Sahalia revealed all the existing univariate linear drift stochastic models could not explain well the dynamics of Euro-dollar interest rates. As a result, he proposed a new class of highly non-linear stochastic interest rate models. The original Ait-Sahalia interest rate model has been found to have considerable use for modelling time series evolution of interest rates. However, this model does not possess certain specifications to provide adequate descriptions of interest rates against unexpected empirical phenomena such as volatility 'skews' and 'smiles', jump behaviour, market regulatory lapses, economic crisis, financial clashes, political instability, among others collectively. In this thesis, we propose a modified version of this model by incorporating additional features to help collectively describe these empirical phenomena adequately. However, the proposed model does not have explicit solution. Hence, we split it into three stochastic interest rate models and construct a new implementable truncated EM scheme to approximate them numerically. Further, we study finite time strong convergence of the truncated EM solutions to the exact solutions of the three models under the local Lipschitz condition plus the Khasminskii-type condition. Moreover, we perform numerical simulations to validate the strong convergence results and justify these results within Monte Carlo frameworks to evaluate expected payoffs of some financial products