Exercise 1: Cox-Ingersoll-Ross Model (8 Marks) In this course, we always assume that the interest rate r is a fixed constant, however, in practice this is not the case. There are stochastic models for interest rate, for instance, one may use the OrnsteinUhlenbeck process (Assignment 4, Exercise 2) to model the interest rate. Another famous model for interest rate is the Cox-Ingersoll-Ross model: assume that the interest rate (rt) is a stochastic process that solves drt = a(b crt)dt rtdWt , r0 = r0 > 0, where a, b, c, > 0 are positive constants. (a) Find E(rt). (Hint: treat E(rt) as a function of t. Also, the solution to the ODE dR(t) dt = A BR(t) is R(t) = e Bt[R(0) A B (1 e Bt)]. ) [4 marks] (b) Show that r 2 t = r 2 0 Z t 0 (2ab 2 )rsds Z t 0 2acr2 sds Z t 0 2r 3 2 s dWs. [4 marks]