Chapter 21, Problem 19PS

a)

Summary Introduction

To determine: Value of option when the option beta at an exercise price of $530.

Explanation of Solution

Calculation of value of option:

       $\begin{array}{c}{\text{d}}_1=\log \frac{\left[\frac{\text{P}}{\text{PV}\left(\text{EX}\right)}\right]}{{\sigma }_t{}^{0.5}}+\frac{{\sigma }_t{}^{0.5}}{2}\\ =\log \frac{\left[\frac{\$530}{\left(\frac{\$530}{{1.01}^{0.5}}\right)}\right]}{\left(0.3156\times {0.5}^{0.5}\right)}+\left(0.3156\times \frac{{0.5}^{0.5}}{2}\right)\\ =0.1562\end{array}$

The value of ${\text{d}}_1$ is 0.1562.

$\begin{array}{c}{\text{d}}_2={\text{d}}_1-{\sigma }_t{}^{0.5}\\ =0.1562-0.3156\times {0.5}^{0.5}\\ =0.0670\end{array}$

The value of ${\text{d}}_2$ is 0.0670.

Therefore,

   $\begin{array}{c}\text{N}\left({\text{d}}_{\text{1}}\right)=0.5621\\ \text{N}\left({\text{d}}_{\text{2}}\right)=0.4733\end{array}$

  $\begin{array}{c}\text{Value}\text{}\text{of}\text{call}\text{option}=\left[\text{N}\left({\text{d}}_{\text{1}}\right)\times \text{P}\right]-\left[\text{N}\left({\text{d}}_{\text{2}}\right)\times \text{PV}\left(\text{EX}\right)\right]\\ =\left[0.5621\times \$530\right]-\left[0.4733\times \left(\frac{\$530}{1.01}\right)\right]\\ =\$49.52\end{array}$

Hence, the value of call option is $49.52.

Person X has investing $297.89 and he borrows $248.36

Calculation of option beta:

$\begin{array}{c}\text{Option}\text{beta}=\frac{\left(\left[\$297.89\times 1.15\right]-\left[\$248.36\times 0\right]\right)}{\$297.89-\$248.36}\\ =\$6.92\end{array}$

Therefore, the option beta is $6.92

Compute option beta when exercise price is $450:

Calculation of value of option:

       $\begin{array}{c}{\text{d}}_1=\log \frac{\left[\frac{\text{P}}{\text{PV}\left(\text{EX}\right)}\right]}{{\sigma }_t{}^{0.5}}+\frac{{\sigma }_t{}^{0.5}}{2}\\ =\log \frac{\left[\frac{\$530}{\left(\frac{\$450}{{1.01}^{0.5}}\right)}\right]}{\left(0.3156\times {0.5}^{0.5}\right)}+\left(0.3156\times \frac{{0.5}^{0.5}}{2}\right)\\ =0.8894\end{array}$

The value of ${\text{d}}_1$ is 08894.

$\begin{array}{c}{\text{d}}_2={\text{d}}_1-{\sigma }_t{}^{0.5}\\ =0.8894-\left(0.3156\times {0.5}^{0.5}\right)\\ =0.6662\end{array}$

The value of ${\text{d}}_2$ is 0.662.

Therefore,

   $\begin{array}{c}\text{N}\left({\text{d}}_{\text{1}}\right)=0.8131\\ \text{N}\left({\text{d}}_{\text{2}}\right)=0.7474\end{array}$

  $\begin{array}{c}\text{Value}\text{}\text{of}\text{call}\text{option}=\left[\text{N}\left({\text{d}}_{\text{1}}\right)\times \text{P}\right]-\left[\text{N}\left({\text{d}}_{\text{2}}\right)\times \text{PV}\left(\text{EX}\right)\right]\\ =\left[0.8131\times \$530\right]-\left[0.7474\times \left(\frac{\$450}{1.01}\right)\right]\\ =\$97.96\end{array}$

Hence, the value of call option is $97.96.

Calculation of option beta:

$\begin{array}{c}\text{Option}\text{beta}=\frac{\left(\left[\$430.95\times 1.15\right]-\left[\$430.95\times 0\right]\right)}{\$430.95-\$332.99}\\ =5.06\end{array}$

Therefore, the option beta is 5.06

Thus, option beta decreases because of the decrease in exercise price and indicates a decrease in risk.

b)

Summary Introduction

To determine: Value of option when the option beta at an exercise price of $530 and time period is 1 year.

Explanation of Solution

Calculation of value of option:

       $\begin{array}{c}{\text{d}}_1=\log \frac{\left[\frac{\text{P}}{\text{PV}\left(\text{EX}\right)}\right]}{{\sigma }_t{}^{0.5}}+\frac{{\sigma }_t{}^{0.5}}{2}\\ =\log \frac{\left[\frac{\$530}{\left(\frac{\$530}{1.0201}\right)}\right]}{\left(0.3156\times 1^{0.5}\right)}+\left(0.3156\times \frac{1^{0.5}}{2}\right)\\ =0.2209\end{array}$

The value of ${\text{d}}_1$ is 0.2209.

 $\begin{array}{c}{\text{d}}_2={\text{d}}_1-{\sigma }_t{}^{0.5}\\ =0.2209-0.3156\times 1^{0.5}\\ =-0.0947\end{array}$

The value of ${\text{d}}_2$ is -0.0947.

Therefore,

  $\begin{array}{c}\text{N}\left({\text{d}}_{\text{1}}\right)=0.5874\\ \text{N}\left({\text{d}}_{\text{2}}\right)=0.4623\end{array}$

  $\begin{array}{c}\text{Value}\text{}\text{of}\text{call}\text{option}=\left[\text{N}\left({\text{d}}_{\text{1}}\right)\times \text{P}\right]-\left[\text{N}\left({\text{d}}_{\text{2}}\right)\times \text{PV}\left(\text{EX}\right)\right]\\ =\left[0.5874\times \$530\right]-\left[0.4623\times \left(\frac{\$530}{1.0201}\right)\right]\\ =\$71.15\end{array}$

Hence, the value of call option is $71.15.

Calculation of option beta:

$\begin{array}{c}\text{Option}\text{beta}=\frac{\left(\$311.32\times 1.15-\$240.17\times 0\right)}{\$311.32-\$240.17}\\ =\$5.03\end{array}$

Therefore, the option beta is $5.8995

The risk also decreases as the maturity is extended.