This problem will ask you to calculate effective duration and convexity using a binomial 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, S$0,$ was $5\%.$

$($d$)$ Recalculate the prices for the bonds in parts $($a$)$ and $($b$)$ if the initial spot rate, S$0,$ was $7\%.$

$2$

$($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$)$