4. Consider a Heath-Jarrow-Morton (HJM) model for forward rates f(t,T) which, for all positive values of T, te [0, T) is given by df(t, T) = a(t,T)dt + ot, T)dW(t), where alt, T) and o(t,T) are scalar deterministic functions of t and T, while W denotes a Wiener process and furthermore f(0,T) = f*(0,T), where the function f*(0,T) is given. (a) (i) What is at,T) in terms of the function ot, S), under the assumption that the market is arbitrage free? (ii) What is it in the special case that o(t, T) = ye-a(T-t), where y > 0, a > 0 are parameters? (b) How are the unit bond prices p(t, T) obtained for this model? Show that they are lognormally distributed. What are the mean and the variance of log(p(t, T)) in terms of the functions f*(0, S) and o(s, S)? = 4. Consider a Heath-Jarrow-Morton (HJM) model for forward rates f(t,T) which, for all positive values of T, te [0, T) is given by df(t, T) = a(t,T)dt + ot, T)dW(t), where alt, T) and o(t,T) are scalar deterministic functions of t and T, while W denotes a Wiener process and furthermore f(0,T) = f*(0,T), where the function f*(0,T) is given. (a) (i) What is at,T) in terms of the function ot, S), under the assumption that the market is arbitrage free? (ii) What is it in the special case that o(t, T) = ye-a(T-t), where y > 0, a > 0 are parameters? (b) How are the unit bond prices p(t, T) obtained for this model? Show that they are lognormally distributed. What are the mean and the variance of log(p(t, T)) in terms of the functions f*(0, S) and o(s, S)? =