Study of Positivity Preserving Numerical Methods

The Cox-Ingersoll-Ross (CIR) interest rate model is one of the most celebrated models in financial industry. The CIR interest rate model always has been the focus of study in mathematical finance litrature [2] as well as in the financial industry. It is also the area of interest in numerical science litrature [3 4 6 8 9 10 13 14 15 16 20] because of its non-explicit analytical solution. Significant research in this area led to the development and implementation of numerical schemes to simulate the CIR interest rate model, which concentrate on the accuracy and time efficiency of the methods. In this work, we have shown the efficency of the existing FIS-α numerical scheme [1] to simulate the CIR model in context of accuracy and the average CPU time to simulate one path of the CIR interest rate model. We have also compared the CIR model discretization with the FIS-α and the other existing methods with respect to accuracy and the average CPU time to simulate the CIR model. We have also studied the behaviour of the methods with larger time step size. The logarithmically transformed CIR interest rate model has been discretized and its result has been studied. The logarithmic co-ordinate transformation on the CIR interest rate model ensures numerical positivity as mentioned in [3]. In all simulations result of this project report the MATLAB Version 7.7.071(R2008b) has been used. In order to simulate the FIS-α(log), an in-built function neamed “fsolve” has been used. The “randn” has been used to generate the Gaussian random numbers. To calculate time of exeucution of a program, Matlab “tic” and “toc” have been used which give time in seconds. In present work we have developed a new scheme, named as the “Mixed Method”, to simulate the CIR interest rate model which ensures non-negative numerical solution of the CIR model with varying parameters of the model.