Chapter 21, Problem 15PS

Option risk*

1. a. In Section 21-3, we calculated the risk (beta) of a six-month call option on Amazon stock with an exercise price of $900. Now repeat the exercise for a similar option with an exercise price of $750. Does the risk rise or fall as the exercise price is reduced?
2. b. Now calculate the risk of a one-year call on Amazon stock with an exercise price of $750. Does the risk rise or fall as the maturity of the option lengthens?

Summary Introduction

To determine: Value of option when the option beta at an exercise price of $750.

Explanation of Solution

Given information:

Current stock selling price (P) is $900

Exercise price (EX) is $750

Standard deviation (σ) is 0.25784

Risk free rate (rf) is 0.01 annually,

t = 0.5

Stock beta is 1.5 and risk free loan beta is 0

Calculation of value of option:

       $\begin{array}{c}{\text{d}}_1=\log \frac{\left[\frac{\text{P}}{\text{PV}\left(\text{EX}\right)}\right]}{{\sigma }_t{}^{0.5}}+\frac{{\sigma }_t{}^{0.5}}{2}\\ =\log \frac{\left[\frac{\$900}{\left(\frac{\$750}{{1.01}^{0.5}}\right)}\right]}{\left(0.25784\times {0.5}^{0.5}\right)}+\left(0.25784\times \frac{{0.5}^{0.5}}{2}\right)\\ =1.1185\end{array}$

The value of ${\text{d}}_1$ is 1.1185

$\begin{array}{c}{\text{d}}_2={\text{d}}_1-{\sigma }_t{}^{0.5}\\ =1.1185-0.25784\times {0.5}^{0.5}\\ =0.9361\end{array}$

The value of ${\text{d}}_2$ is 0.9361

Therefore,

   $\begin{array}{c}\text{N}\left({\text{d}}_{\text{1}}\right)=0.8683\\ \text{N}\left({\text{d}}_{\text{2}}\right)=0.8254\end{array}$

  $\begin{array}{c}\text{Value}\text{}\text{of}\text{call}\text{option}=\left[\text{N}\left({\text{d}}_{\text{1}}\right)\times \text{P}\right]-\left[\text{N}\left({\text{d}}_{\text{2}}\right)\times \text{PV}\left(\text{EX}\right)\right]\\ =\left[0.8683\times \$900\right]-\left[0.8254\times \left(\frac{\$750}{{1.01}^{0.5}}\right)\right]\\ =\$165.51\end{array}$

Hence, the value of call option is $165.51

Person X has investing $781.48 and he borrows $615.98,

Calculation of option beta:

$\begin{array}{c}\text{Option}\text{beta}=\frac{\left(\$781.48\times 1.5-\$615.98\times 0\right)}{\$781.48-\$615.98}\\ =\frac{\$1,172.22}{\$165.5}\\ =\$7.082\end{array}$

Therefore, the option beta is $7.082

The lower exercise price decreases the beta of call option ($10.95 to $7.08)

Summary Introduction

To determine: Value of option when the option beta at an exercise price of $750 and time period is 1 year.

Explanation of Solution

Given information:

Current stock selling price (P) is $900

Exercise price (EX) is $750

Standard deviation (σ) is 0.25784

Risk free rate (rf) is 0.01 annually,

t = 1

Stock beta is 1.5 and risk free loan beta is 0

Calculation of value of option:

       $\begin{array}{c}{\text{d}}_1=\log \frac{\left[\frac{\text{P}}{\text{PV}\left(\text{EX}\right)}\right]}{{\sigma }_t{}^{0.5}}+\frac{{\sigma }_t{}^{0.5}}{2}\\ =\log \frac{\left[\frac{\$900}{\left(\frac{\$750}{{1.01}^{0.5}}\right)}\right]}{\left(0.25784\times 1^{0.5}\right)}+\left(0.25784\times \frac{1^{0.5}}{2}\right)\\ =0.8746\end{array}$

The value of ${\text{d}}_1$ is 0.8746

 $\begin{array}{c}{\text{d}}_2={\text{d}}_1-{\sigma }_t{}^{0.5}\\ =1.1185-0.25784\times 1^{0.5}\\ =0.6168\end{array}$

The value of ${\text{d}}_2$ is 0.6168

Therefore,

  $\begin{array}{c}\text{N}\left({\text{d}}_{\text{1}}\right)=0.8091\\ \text{N}\left({\text{d}}_{\text{2}}\right)=0.7313\end{array}$

  $\begin{array}{c}\text{Value}\text{}\text{of}\text{call}\text{option}=\left[\text{N}\left({\text{d}}_{\text{1}}\right)\times \text{P}\right]-\left[\text{N}\left({\text{d}}_{\text{2}}\right)\times \text{PV}\left(\text{EX}\right)\right]\\ =\left[0.8091\times \$900\right]-\left[0.7313\times \left(\frac{\$750}{{1.01}^1}\right)\right]\\ =\$185.15\end{array}$

Hence, the value of call option is $185.15

Person X has investing $728.20 and he borrows $543.05,

Calculation of option beta:

$\begin{array}{c}\text{Option}\text{beta}=\frac{\left(\$728.20\times 1.5-\$543.05\times 0\right)}{\$728.20-\$543.05}\\ =\frac{\$1,092.3}{\$185.15}\\ =\$5.899\end{array}$

Therefore, the option beta is $5.8995

The risk also decreases from $7.08 to $5.90 as the maturity is extended.