EXERCISE 7.1 (Parallel shifts of the yield curve) The purpose of this exercise is to find out under which assumptions the only possible shifts of the yield curve are parallel, i.e. such that d¯y

t is independent of 

where ¯y

t = yt+

t .

(a) Argue that if the yield curve only changes in the form of parallel shifts, then the zero-coupon yields at time t must have the form yT t = yT (rt, t) = rt + h(T − t)

for some function h with h(0) = 0 and that the prices of zero-coupon bonds are thereby given as BT (r, t) = e−r[T−t]−h(T−t)[T−t].

(b) Use the partial differential equation (7.3) on page 152 to show that 1 2 (r)2(T − t)2 − ˆ(r)(T − t) + h′(T − t)(T − t) + h(T − t) = 0 (*)
for all (r, t) (with t  T, of course).

(c) Using (*), show that 1 2(r)2(T − t)2 − ˆ(r)(T − t) must be independent of r. Conclude that both ˆ(r) and (r) have to be constants, so that the model is indeed Merton’s model.

(d) Describe the possible shapes of the yield curve in an arbitrage-free model in which the yield curve only moves in terms of parallel shifts. Is it possible for the yield curve to be flat in such a model?