Chapter 21, Problem 28PS

Option risk Calculate and compare the risk (betas) of the following investments: (a) a share of Amazon stock; (b) a one-year call option on Amazon; (c) a one-year put option; (d) a portfolio consisting of a share of Amazon stock and a one-year put option; (e) a portfolio consisting of a share of Amazon stock, a one-year put option, and the sale of a one-year call. In each case, assume that the exercise price of the option is $900, which is also the current price of Amazon stock.

Summary Introduction

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

Explanation of Solution

The stock beta of Company A’s stock is 1.5

Summary Introduction

To determine: One year call option on Company A.

Explanation of Solution

Given information:

Current stock selling price (P) is $900

Exercise price (EX) is $900

Standard deviation (σ) is 0.25784

Risk free rate (rf) is 0.01 annually,

t = 1

Stock beta is 1.5

Calculation of value of call option:

$\begin{array}{c}{\text{d}}_1=\log \frac{\left[\frac{\text{P}}{\text{PV}\left(\text{EX}\right)}\right]}{\sigma {\left(t\right)}^{0.5}}+\frac{\sigma {\left(t\right)}^{0.5}}{2}\\ =\log \frac{\left[\frac{\$900}{\left(\frac{\$900}{{1.01}^{0.5}}\right)}\right]}{\left(0.25784\times {\left(1\right)}^{0.5}\right)}+\left(0.25784\times \frac{{\left(1\right)}^{0.5}}{2}\right)\\ =0.1675\end{array}$

The value of ${\text{d}}_1$ is 0.1675

$\begin{array}{c}{\text{d}}_2={\text{d}}_1-\sigma {\left(t\right)}^{0.5}\\ =0.1675-0.25784\times {\left(1\right)}^{0.5}\\ =-0.0903\end{array}$

The value of ${\text{d}}_2$ is –0.0903

Therefore,

   $\begin{array}{c}\text{N}\left({\text{d}}_{\text{1}}\right)=0.5665\\ \text{N}\left({\text{d}}_{\text{2}}\right)=0.4640\end{array}$

  $\begin{array}{c}\text{Value}\text{}\text{of}\text{call}\text{option}=\left[\text{N}\left({\text{d}}_{\text{1}}\right)\times \text{P}\right]-\left[\text{N}\left({\text{d}}_{\text{2}}\right)\times \text{PV}\left(\text{EX}\right)\right]\\ =\left[0.5665\times \$900\right]-\left[0.4640\times \left(\frac{\$900}{{1.01}^1}\right)\right]\\ =\$96.39\end{array}$

Hence, the value of call option is $96.39

Person X has investing $509.86 and he borrows $413.48,

Calculation of option beta:

$\begin{array}{c}\text{Option}\text{beta}=\frac{\left(\$509.86\times 1.5-\$413.48\times 0\right)}{\$509.86-\$413.48}\\ =\$7.93\end{array}$

Therefore, the option beta is $7.93

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,

$\begin{array}{c}\text{Risk}=\frac{\left[\left(96.39\times 7.93\right)+\left(\frac{\$900}{1.01}\times 0\right)-\left(900\times 1.5\right)\right]}{\left[96.39+\left(\frac{\$900}{1.01}\right)-900\right]}\\ =-6.7\end{array}$

Therefore risk is -6.7

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 $900 investment in the present value of exercise price.

The risk can be as follows,

$\begin{array}{c}\text{Risk}=\frac{\left[\left(96.39\times 7.93\right)+\left(\frac{\$900}{1.01}\times 0\right)\right]}{\left[96.39+\left(\frac{\$900}{1.01}\right)\right]}\\ =0.77\end{array}$

Therefore, risk is 0.77 which partially hedged the position.

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,

$\begin{array}{c}\text{Risk}=\frac{\left[\left(\frac{\$900}{1.001}\times 0\right)\right]}{\left[\left(\frac{\$900}{1.001}\right)\right]}\\ =0.00\end{array}$

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