In this problem we introduce a new exponential family (called the generalized Vasicek family) and we use it to estimate the term structure of interest rates. For each value of the 4-dimensional parameter θ = (θ₁, θ₂, θ₃, θ₄) we define the function f_GV (x, θ) by:

where θ₄ is assumed to be strictly positive.

1. Mimic what was done in the text in the case of the Nelson-Siegel family and comment on the meanings and roles of the parameters θ_i's. Write an S-Plus function fgv to compute the values of this function and plot the graphs of three of these functions for three values of the parameter θ which you will choose to illustrate your comments on the significance of the parameters

2. Derive an analytic formula for the price of the zero coupon bonds when the term structure of interest rates is given by an instantaneous forward curve (3.48). Follow what was done in the text for the function bns and write an S-Plus function bgv to compute the values of the zero coupon bonds derived from the instantaneous forward interest rates given by the function fgv. Plot the zero coupon curves corresponding to the three values of θ chosen above in question 1.

3. Derive an analytic formula for the yield curve, and write an S-Plus function ygv to compute the yield curve, and plot the yield curves corresponding to the three values of θ chosen above in question 1.

4. Using this new function family, estimate the term structure of interest rates (as given by the zero coupon curve, the forward curve and the yield curve) for the German bond data used in the text.