Find the price of the Bond P (t, T) in the Vasicek model, by answering the questions below. For ease of calculations you may take all the parameters equal to 1, a = b = sigma = 1. (a) Assuming the model is specified under the Equivalent Martingale Measure (EMM), establish the formula for P(t, T) P(t, T) = E[e^integral^T_t r_s ds|F_t] (b) Let X = -integral^T_t r_s ds. Find mu_1 = E(X|r_t), the conditional mean of X given r_t. You may use without proof E(integral^T_t r_s ds|r_t) = integral^T_t E(r_s|r_t) ds. (c) Find sigma^2_1, the conditional variance of X given r_t. You may use sigma^2_1 = Cov(X, X) = Cov(integral^T_t r_s ds, integral^T_t r_u du) = integral^T_t integral^T_t Cov(r_s, r_u) ds du. (d) Using (without proof that conditional distribution of X = -integral^T_t r_s ds given r_t is Normal N (mu_1, sigma^2_1), show that P(t. T) = exp ((1 - e^-(T-t) (1/2 - r_t) - 1/2 (T - t) - 1/4 (1 - e^-(T-t))^2). (e) Give the Partial Differential Equation including the boundary condition for the bond price, and then give its solution