you are using vasicek model as the short rate process that is calibrated as: dr(t) = a(b r(t))dt + odz(t) where a = 0.3, b 0.05, o = 0.03 . and where z(t) is a wiener process under the risk neutral framework modelling the random market risk factor.

a. use the above model to calculate the analytical price of a 5-year zero-coupon (with a face value of $1) bond assuming that instantaneous short rate r(0) = 0.08.

b. using euler discretization (as taught in class) and monte carlo simulation technique, calculate the above 5-year zero-coupon bond price assuming that instantaneous short rate r(0) = 0.08. use appropriate time step and number of simulations to minimize the mc error.

c. calculate the swap rate (fixed rate leg) of an interest rate swap that has a maturity of 5 years and the floating leg has an annual coupon payment frequency linked to libor. [hint: note the price of a swap is given as: 1-z(n) swap rate = where z(i) is the zero coupon i3d1 price for maturity I.

d. calculate the price of a european call option on 5-year zero- coupon bond (with a face value of $1000) and a maturity of 4 years and strike price of $900 using the vasicek model that has been calibrated as above. note this solution allows for simulation of negative interest rates that is a known limitation of vasicek model. [hint: the payoff of call option is: max (z [t,t] - k,0), and you may ignore the discounting of payoff or use the 4 year zero spot rate to discount the payoff. and also note that z [t (= 4y), t (= 5y)] can be computed using exponential affine formula where the t = simulated using same steps as part b above.] 4y and r(t=4y) can be