A Short-term Interest Rate Extended Merton's Model Influenced by A Risk Market Factor

In the context of the interest rate derivatives, a short-rate model is a mathematical model that can predict the random movement of the interest rates. In the present paper we introduce a short-term interest rate Extended Merton's model for which the movement of the interest rate is given by a stochastic differential equation. For this model we consider the zero-coupon bond's price which is determined by using the apparatus of the stochastic differential equations and the partial differential equations. We use the diffusion equation to calculate the bond's price for this model in the case of a risk market factor and without a risk market factor. Numerical experiments and graphics are presented to determine the zero-coupon bond's price. Results obtained by a Monte Carlo method for evaluation the zero-coupon bond's price in case with the risk market factor is a constant, demonstrate that this stochastic method could be applied in more complicated cases when the risk market factor depends on the time. The results also show that the zero coupon-bond's price depends on the risk market factor.