Part $2$: Callable Bonds

Now, let's consider the pricing of callable bonds. We will compare this with non$-$callable bonds.

Using the tree from Part $1,$ calculate the value of a $1\mathrm{.}5-$year bond with a coupon rate of $3\%($coupons paid semi$-$annually$)$ that is callable at $$100$ at each coupon date $($starting at $t=0\mathrm{.}5$ just after the coupon is paid. Recall that a callable bond price is equal to an equivalent non$-$callable bond price minus the value of the call.

Add $0\mathrm{.}001$ to all of the interest rates in the tree from Part $1.$ Calculate the price of the callable bond in $(1)$ using this tree.

Subtract $0\mathrm{.}001$ from all of the interest rates in the tree from Part $1.$ Calculate the price of the callable bond in $(1)$ using this tree.

Using the callable bond prices in $(1),(2),$ and $(3),$ calculate the modified duration of the callable bond using

$MD$~~$-\frac{B(y+y)-B(y-y)}{2y}\frac{1}{B(y)}$

Calculate the modified duration of the non$-$callable version of the $1\mathrm{.}5-$year bond. You should already have the prices you need for this in $(1),(2),$ and $(3).$