Problem 6. This problem will ask you to calculate effective duration and convexity using a bino-

mial tree model.

Suppose we have the following interest rate tree:

1-year spot rate at time 0 is 6%

In each 1-year period, the 1-year spot rate can either go up by 10%, or down by 10%

The risk-neutral probabilities of the spot rate going up or down are both 50%

Assume annual compounding and annual coupon payments

(a) What is the price of a $100 par 5% coupon bond maturing in 2 years?

(b) What is the value of the bond in part (a) if it had a call option at Year 1 at a call price of

$99?

(c) Recalculate the prices for the bonds in parts (a) and (b) if the initial spot rate, S0, was 5%.

(d) Recalculate the prices for the bonds in parts (a) and (b) if the initial spot rate, S0, was 7%

(e) You have now calculated a total of 6 prices. These are the prices of the plain-vanilla bond

and the callable bond for initial 1-year spot rates of 5%, 6%, and 7%. On the same graph, plot the

prices of the two bonds against these initial spot rates.

(f) Using the 6 prices calculated in parts (a) through (d) calculate the effective durations and

convexities for the two bonds.

(g) Comment on the sign of the effective convexity calculated in part (f) and the shape of the

graphs in parts (e).