Exercise . (Slope of the yield curve at zero maturity) Consider a time-homogeneous affine model in which the yield curve at time t is given by y¯

τ

t = a(τ )

τ

+

b(τ )

τ

r, compare (.).The slope of the yield curve at zero maturity is the limit limτ→

∂ y¯τ

t

∂τ .

(a) Show by differentiation and an application of l’Hôpital’s rule that the slope is lim

τ→

∂ y¯τ

t

∂τ = 



a

() +





b

()r.

(b) Differentiate the ordinary differential equations (.) and (.) to find expressions for b

(τ ) and a

(τ ).

(c) Conclude that the slope is given by (ϕˆ − ˆκr)/.