(4) Find the optimal portfolio for this individual. Problem 3 Consider a money market account process defined as follows: dB. = re Bidt, Bo = 1. where re is the instantaneous interest rate at time t. Assume that the process of zero-coupon bond prices P(t, T) is given as: dP(t, T) = m(t, T) P(t, T)dt + s(t, T)P(t, T)du. The bond gives a payoff of 1 when it matures, so that P(T, T) = 1. (1) Explain why in a world which is forward risk neutral with respect to B, it holds that m(t, T) = r. (2) A variable 0 follows a martingale if its process has the form: de = adz, i.e. it has a drift of zero. Use Ito's lemma to show that 0 = P/B is a martingale process in a world that is forward risk neutral with respect to B. (3) Use the result from (2) to show that P(t, T) = EQ[e-fi reds], where EQ denotes the expected value under the risk-neutral probability measure. Now assume that rt follows the Vasicek model under the risk-neutral measure: drt = a(b - rt)dt + odzt with solution given as CoTT = ree -a ( T-t ) + 6 ( 1 - e - a (T-") ) to / e-a(T-") dz.. When rt follows the Vasicek model, zero-coupon bond prices are exponential linear in rt, P(t, T) = eA(t, T)-B(t, I). where A(t, T) and B(t, T) are deterministic functions of t. In particular: B(t, T) = 1 - e-a(T-1) a and A (t , T ) = [B(t, I) - (T - t)](a26 - 202) 02 B2(t, T) a2 4a (4) The instantaneous forward rate is given by f(t, T) = -olgt:1. Show that if o = 0, E[rr] = f(t, T). What does the result imply? (5) Suppose that the initial observed spot interest rate is 5%, a = 0.1, b = 0.05 and o = 0.03 in the Vasicek model. Calculate the probability of a negative spot interest rate in 2 years. Explain how to alleviate this