Extension Of Cox-Ingersoll-Ross Model And Its Euler-Maruyama Method Approximation

Stochastic differential equations are more and more applicable in financial economics, and they are often used to describe the dynamics of asset's price. However, most of stochastic differential equations in finance do not have explicit solutions. Hence, numerical methods is a very useful method, if we can control the error bound of its numerical simulations, then this method can describe some financial quantities. As one of the most important numerical methods, Monte Carlo simulations have received more and more attention on the valuation of financial quantities. In this paper, we mainly consider the Euler-Maruyama method of Monte Carlo simulations.Ait-Sahalia in his seminal paper proposed a highly nonlinear model in finance which is a nonlinear stochastic differential equation, where both the drift and diffusion coefficients do not obey the classical linear growth condition. Of course, this nonlinear stochastic differential equation do not have explicit solution either. In addition to the usual external disturbance, financial assets are also often subject to the impact of unexpected events, more and more empirical evidence shows that the jump-diffusion process is more ideal than the general stochastic differential equations to model an asset's price, the interest rate or stochastic volatility. Consequently, by the Ait-Sahalia's model, we in this paper concentrate on investigating the Euler-Maruyama method for a highly nonlinear model with jump and deriving explicitly comparable error bounds over finite time intervals. These error bounds imply strong convergence as the time step tends to zero.For considering the stochastic volatility, many papers introduce jump into the underlying price process. In this paper, we also consider the highly nonlinear model with jump as stochastic volatility, then study the Euler-Maruyama method and the strong convergence of error bounds under stochastic volatility with correlated jumps. Finally, from those convergence results, we show that the Euler-Maruyama method can be applied to compute some financial quantities, for example, bonds, path-dependent option and so on.