Chapter 21, Problem 27PS

a)

Summary Introduction

To determine: Risk (beta) share of Company G’s stock

Explanation of Solution

The stock beta of Company G’s stock is 1.28.

b)

Summary Introduction

To determine: One-year call option on Company G.

Explanation of Solution

Calculation of value of call option:

$\begin{array}{c}{\text{d}}_1=\log \frac{\left[\frac{\text{P}}{\text{PV}\left(\text{EX}\right)}\right]}{\sigma {\left(t\right)}^{0.5}}+\frac{\sigma {\left(t\right)}^{0.5}}{2}\\ =\log \frac{\left[\frac{\$400}{\left(\frac{\$400}{1.035}\right)}\right]}{\left(0.3156\times {\left(1\right)}^{0.5}\right)}+\left(0.3156\times \frac{{\left(1\right)}^{0.5}}{2}\right)\\ =0.2668\end{array}$

The value of ${\text{d}}_1$ is 0.2668.

$\begin{array}{c}{\text{d}}_2={\text{d}}_1-\sigma {\left(t\right)}^{0.5}\\ =0.2668-0.3156\times {\left(1\right)}^{0.5}\\ =-0.0488\end{array}$

The value of ${\text{d}}_2$ is –0.0488.

Therefore,

   $\begin{array}{c}\text{N}\left({\text{d}}_{\text{1}}\right)=0.6052\\ \text{N}\left({\text{d}}_{\text{2}}\right)=0.4805\end{array}$

  $\begin{array}{c}\text{Value}\text{}\text{of}\text{call}\text{option}=\left[\text{N}\left({\text{d}}_{\text{1}}\right)\times \text{P}\right]-\left[\text{N}\left({\text{d}}_{\text{2}}\right)\times \text{PV}\left(\text{EX}\right)\right]\\ =\left[0.6052\times \$400\right]-\left[0.4805\times \left(\frac{\$400}{1.035}\right)\right]\\ =\$56.38\end{array}$

Hence, the value of call option is $56.38.

Calculation of option beta:

$\begin{array}{c}\text{Option}\text{beta}=\frac{\left(\$242\times 1.28-\$186\times 0\right)}{-\$186+\$242}\\ =5.53\end{array}$

Therefore, the option beta is 5.53.

c)

Summary Introduction

To determine: One year put option.

Explanation of Solution

Calculation of one year put option:

From the put-call parity relationship,

$\begin{array}{l}\text{Value}\text{of}\text{call}+\text{present}\text{value}\text{of}\text{exercise}\text{price}=\text{Put}\text{value}+\text{share}\text{price}\\ \text{Value}\text{of}\text{put}=\text{Value}\text{of}\text{call}+\text{PV}\left(\text{EX}\right)-\text{share}\text{price}\end{array}$

Therefore, to replicate the put pay-offs, person X would buy a call with $400 as an exercise price and invest the present value of exercise price and sell the short stock.

The risk can be as follows,

$\begin{array}{c}\text{Risk}=\frac{\left[\left(\$56.38\times 5.53\right)+\left(\frac{\$400}{1.035}\times 0\right)-\left(\$400\times 1.28\right)\right]}{\left[\$56.38+\left(\frac{\$400}{1.035}\right)-\$400\right]}\\ =-4.67\end{array}$

Therefore risk is -4.67.

d)

Summary Introduction

To determine: One share stock plus one put option.

Explanation of Solution

Calculation of one share plus one option:

It is related to previous question, except the stock positions cancel out, which leaves us with an exercise price of $400 investment in the present value of exercise price.

The risk can be as follows,

$\begin{array}{c}\text{Risk}=\frac{\left[\left(\$56.38\times 5.53\right)+\left(\frac{\$400}{1.035}\times 0\right)\right]}{\left[\$56.38+\left(\frac{\$400}{1.035}\right)\right]}\\ =0.704\end{array}$

Therefore, risk is 0.704 which partially hedged the position.

e)

Summary Introduction

To determine: One share stock plus one put option minus one call option.

Explanation of Solution

Calculation of one share plus one option minus one call option:

It is related to previous question, except the all call positions cancel out, which leaves us with an investment in the present value of exercise price.

The risk can be as follows,

$\begin{array}{c}\text{Risk}=\frac{\left[\left(\frac{\$400}{1.035}\times 0\right)\right]}{\left[\left(\frac{\$400}{1.035}\right)\right]}\\ =0.00\end{array}$

Therefore, risk is 0.00 which reduced our position to risk-free loan.