Chapter 23, Problem 15P

a)

Summary Introduction

To compute: The number of put option bonds are required to hedge portfolio of bond.

Explanation of Solution

The computation of number of put option bond is as follows:

$\begin{array}{c}\text{Number of put option bond}\left({\text{N}}_{\text{p}}\right)=\frac{\text{D}\times \text{P}}{\left|\text{δ}\right|\times \text{D}\times \text{B}}\\ =\frac{\text{5 }\times \text{ \$100,000,000}}{\text{0}\text{.625 }\times \text{ 10}\text{.1 }\times \text{\$96,157}}\\ =\text{823}\text{.74}\approx \text{824 Contracts}\end{array}$

b)

Summary Introduction

To compute: The anticipated gain or loss in the hedge on the put option.

Explanation of Solution

The computation of gain or loss is as follows:

A $100,000 20year, 8% bond vending at $96,157 indicates 8.4% yield.

$\begin{array}{c}\text{Put option hedge}\left(\Delta \text{P}\right)={\text{N}}_{\text{P}}\times {\Delta }_{\text{P}}\\ =824\times \left(-0.625\right)\times \left(-10.1\right)\times -10.1\times \$96,157\times \frac{0.01}{1.084}\\ =\$4,614,028\text{ Gain}\text{.}\end{array}$

Hence, the gain from the put option hedge is $4,614,028.

c)

Summary Introduction

To discuss: The anticipated change in the market value of portfolio of bond.

Explanation of Solution

The computation of change in market value of portfolio of bond is as follows:

$\begin{array}{c}\text{Change in bond }\left(\Delta \text{B}\right)=-\text{D}\times \text{1Million}\times \frac{\text{0}\text{.01}}{\left(1+\text{Cupon rate}\right)}\\ =-5\times \$100,000,000\times \frac{0.01}{1.08}\\ =\$4,629,630\end{array}$

Hence, the change in bond is $4,629,630.

d)

Summary Introduction

To discuss: Whether the payoff of hedge set off cost of placing the hedge.

Explanation of Solution

The computation of payoff of hedge is as follows:

$\begin{array}{c}\text{Pay off of hedge}\left(\Delta \text{P}\right)=\left(\begin{array}{l}\text{Number of contracts }\times \text{Premium}=\text{Number of contracts}\\ \times \text{Delta}\times \text{Duration}\times \text{Market value}\times \frac{\Delta \text{R }}{1.084}\end{array}\right)\\ =\left(\begin{array}{l}824\times 3,250=824\times \left(-0.625\right)\times \\ \left(-10.1\right)\times \$96,157\times \frac{\Delta \text{R }}{1.084}\end{array}\right)\\ \text{For solving }\Delta \text{R }=\frac{\left(\text{Premium}\times \text{1}\text{.084}\right)}{\left(\text{Delta}\times \text{Duration}\times \text{Market value}\right)}\\ =\frac{\left(\$3,250\times 1.084\right)}{\left(0.625\times 10.1\times \$96,157\right)}\\ =0.005804\text{ or 0}\text{.58\%}\text{.}\end{array}$

Hence the payoff of hedge is 0.58%.

e)

Summary Introduction

To discuss: Whether the gain on the portfolio bonds set off cost of placing the hedge.

Explanation of Solution

The computation of gain from the bond portfolio is as follows:

$\begin{array}{c}\text{Gain from the bond}\left(\Delta \text{B}\right)=\left(\begin{array}{l}\text{Premium }\times \text{Number of contracts}\\ =-\text{Bond duration}\times \frac{\Delta \text{R }}{1.084}\end{array}\right)\\ =\left(\begin{array}{l}3,250\times 824=-5\times \\ \$100,000,000\times \frac{\Delta \text{R }}{1.084}\end{array}\right)\\ \text{For solving }\Delta \text{R }=\left(\frac{\text{Premium }\times \text{Number of contracts}\times \text{1}\text{.08}}{-\text{Bond duration}\times \text{1Million}}\right)\\ =-0.0057845\text{ Or}-\text{0}\text{.58\%}\end{array}$

Hence the gain from bond is -0.58%.