Chapter 24, Problem 3QP

Calculating Payoffs [LO1] Use the option quote information shown here to answer the questions that follow. The stock is currently selling for S40.

a. Suppose you buy 10 contracts of the February 38 call option. How much will you pay, ignoring commissions?

b. In part (a), suppose that Macrosoft stock is selling for S43 per share on the expiration date. How much is your options investment worth? What if the terminal stock price is $39? Explain.

c. Suppose you buy 10 contracts of the August 38 put option. What is your maximum gain? On the expiration date, Macrosoft is selling for $32 per share. How much is your options investment worth? What is your net gain?

d. In part (c), suppose you sell 10 of the August 38 put contracts. What is your net gain or loss if Macrosoft is selling for $34 at expiration? For $41? What is the break-even price—that is, the terminal stock price that results in a zero profit?

Summary Introduction

To find: The payment that has to be made by Person X by not taking the commission.

Introduction:

The right of an individual to purchase an asset at the price that is fixed and at a specific period is the call option. The right that an individual has to sell the asset at the price that is fixed and during the specific time is a put option.

Answer to Problem 3QP

The payment that Person X makes is $2,350.

Explanation of Solution

Given information:

The current selling price of the stock is $40, the option quote information are provided below:

The option of the M Corporation has different expiration months and the strike price at every month of expiration is $38.

Information of the call option and put option:

• The call option that expires in February has a volume of 85 and the last value is 2.35
• The call option that expires in March has a volume of 61 and the last value is 3.15
• The call option that expires in May has a volume of 22 and the last value is 4.87
• The call option that expires in August has a volume of 3 and the last value is 6.15
• The put option that expires in February has a volume of 37 and the last value is 0.24
• The put option that expires in March has a volume of 22 and the last value is 0.93
• The put option that expires in May has a volume of 11 and the last value is 2.44
• The put option that expires in August has a volume of 3 and the last value is 3.56

Computation of the total cost:

Each contract for the above options are for 100 shares, and the total cost is calculated by multiplying 10 contracts with 100 shares and the last value of the call in February is calculated with a strike price is 38.

$\begin{array}{c}\text{Total cost}=10\left(100\text{ shares for a contract}\right)\left(\$2.35\right)\\ =\$2,350\end{array}$

Hence, the total cost is $2,350.

Summary Introduction

To find: The worth of the option investment

Introduction:

The right of an individual to purchase an asset at the price that is fixed and at a specific period is the call option. The right that an individual has to sell the asset at the price that is fixed and during the specific time is a put option.

Answer to Problem 3QP

The worth of the option investment is $5,000 and $1,000.

Explanation of Solution

Given information:

In part (a) if the stock of M Incorporation is selling at $43 per share and the terminal price of the stock is $39.

Computation of the payoff:

If the price of the stock at expiration is $43 the payoff is calculated by multiplying the 10 contracts with the 100 shares and the difference between the strike price and the expiration price.

$\begin{array}{c}\text{Payoff}=10\left(100\right)\left(\$43-38\right)\\ =\$5,000\end{array}$

Hence, the worth of the option investment with the expiration price $43 is $5,000.

If the price of the stock at expiration is $39, the payoff is calculated by multiplying 10 contracts with 100 shares and the difference between the strike price and the expiration price.

$\begin{array}{c}\text{Payoff}=10\left(100\right)\left(\$39-38\right)\\ =\$1,000\end{array}$

Hence, the worth of the option investment with the expiration price $39 is $1,000.

Summary Introduction

To determine: The maximum gain on expiration. The worth of the option investment and the net gain of Person X.

Introduction:

The right of an individual to purchase an asset at the price that is fixed and at a specific period is the call option. The right that an individual has to sell the asset at the price that is fixed and during the specific time is a put option.

Answer to Problem 3QP

The worth of the option investment is $6,000 and the net gain is $2,440. The maximum gain at expiration is $34,440.

Explanation of Solution

Given information:

Person X purchases ten contracts of the put option on August with a strike price of 38. The selling price by Company M is $32 per share.

Computation of the cost of the put option:

The cost of the put option is computed by multiplying number of contract, number of per share, and last call option for the month of August.

$\begin{array}{c}\text{Cost}=\text{10}\times \text{100}\times \text{\$3}\text{.56}\\ =\text{\$3,560}\end{array}$

Hence, the cost of the put option is $3,560.

Formula to calculate the maximum gain of the put option:

$\text{Maximum gain}=\left(\begin{array}{c}\text{Number of contract}\\ \times \text{ Number of shares}\\ \times \text{Strike price}\end{array}\right)-\text{Cost of put option}$

Computation of the maximum gain of the put option:

$\begin{array}{c}\text{Maximum gain}=\left(\begin{array}{c}\text{Number of contract}\\ \times \text{ Number of shares}\\ \times \text{Strike price}\end{array}\right)-\text{Cost of put option}\\ =\left(10\times 100\times \$38\right)-\$3,560\\ =\$38,000-\$3,560\\ =\$34,440\end{array}$

Hence, the maximum gain is $34,440.

Formula to calculate the worth of the investment option:

$\text{Position value}=\left[\begin{array}{c}\text{Number of contract }\\ \times \text{Number of shares}\\ \times \left(\text{Strike price}-\text{Stock price}\right)\end{array}\right]$

Note: The price of the stock is $32

Computation of the worth of the investment:

$\begin{array}{c}\text{Position value}=\left[\begin{array}{c}\text{Number of contract }\\ \times \text{Number of shares}\\ \times \left(\text{Strike price}-\text{Stock price}\right)\end{array}\right]\\ =\left[10\times 100\times \left(\$38-\$32\right)\right]\\ =1000\times \$6\\ =\$6,000\end{array}$

Hence, the worth of the investment is $6,000.

Formula to calculate the net gain:

$\text{Net gain}=\text{?}\left(\begin{array}{l}\text{Net worth on option investment}\\ -\text{Cost of put option}\end{array}\right)$

Computation of the net gain:

$\begin{array}{c}\text{Net gain}=\text{?}\left(\begin{array}{l}\text{Net worth on option investment}\\ -\text{Cost of put option}\end{array}\right)\\ =\$6,000-\$3,560\\ =\$2,440\end{array}$

Hence, the net gain is $2,440.

Summary Introduction

To find: The net gain or loss and the break-even price.

Introduction:

The right of an individual to purchase an asset at the price that is fixed and at a specific period is the call option. The right that an individual has to sell the asset at the price that is fixed and during the specific time is a put option.

Answer to Problem 3QP

The net loss is - $440, the net gain is $6,560, and the break-even price is $34.44.

Explanation of Solution

Given information:

In part (c), Person X sells 10 contracts of the August put options and the selling price of the company at expiration is $34, the other price is $41. The break-even price is the terminal price of the stock that results in a profit as zero. The strike price is $38.

Formula to calculate the net gain or loss if the selling price at expiry is $34:

$\text{Net gain or loss}=\left[\begin{array}{l}\text{Cost of put option}\\ -\left(\text{Number of contract}\times \text{Number of shares}\right)\\ \times \left(\text{Strike price}-\text{Stock price}\right)\end{array}\right]$

Computation of the net gain or loss if the selling price at expiry is $34:

$\begin{array}{c}\text{Net gain or loss}=\left[\begin{array}{l}\text{Cost of put option}\\ -\left(\text{Number of contract}\times \text{Number of shares}\right)\\ \times \left(\text{Strike price}-\text{Stock price}\right)\end{array}\right]\\ =\$3,560-\left(10\times 100\right)\times \left(\$38-\$34\right)\\ =\$3,560-\left(1,000\times \$4\right)\\ =-\$440\end{array}$

Hence, the net loss is -$440.

Computation of the net gain or loss if the selling price at expiry is $41:

$\begin{array}{c}\text{Net gain or loss}=\left[\begin{array}{l}\text{Cost of put option}\\ -\left(\text{Number of contract}\times \text{Number of shares}\right)\\ \times \left(\text{Strike price}-\text{Stock price}\right)\end{array}\right]\\ =\$3,560-\left(10\times 100\right)\times \left(\$38-\$41\right)\\ =\$3,560-\left(1,000\times \left(-\$3\right)\right)\\ =\$6,560\end{array}$

Formula to calculate the break-even price (the terminal price of the stock):

$\text{Net gain}=\left[\begin{array}{l}\text{Number of contract}\\ \times \text{Number of shares}\\ \left(\text{Strike price}-\text{Breakeven price}\right)\end{array}\right]$

Note: In the above formula, the cost of the put option is the net gain, thus the initial cost is recovered.

Computation of the break-even price (the terminal price of the stock):

$\begin{array}{c}\text{Net gain}=\left[\begin{array}{l}\text{Number of contract}\\ \times \text{Number of shares}\\ \left(\text{Strike price}-\text{Breakeven price}\right)\end{array}\right]\\ \$3,560=10\times 100\left(\$38-\text{Breakeven price}\right)\\ \frac{\$3,560}{1,000}=\$38-\text{Breakeven price}\\ \$3.56=\$38-\text{Breakeven price}\end{array}$

$\begin{array}{c}\text{Breakeven price}=\$38-\$3.56\\ =\$34.44\end{array}$

Hence, the break-even price is $34.44.