
Requirement 1
To determine:
To determine the implied standard deviation of the call at price of $8.
Introduction:
Put option is the option which provides the option holder with the right to sell the assets or to exercise the contract but it does not give any obligation. The put option is exercised at a price which is specific or predetermined price before or on the expiration date.
Call option is the option which provides the right to the option holder to purchase the assets or right to exercise the contract. However it does not give the obligation. This option is exercised at a specific or predetermined price on the expiration date or before it.

Answer to Problem 28PS
The Standard deviation is 0.3213
Explanation of Solution
Given Information:
The spreadsheet containing the data is available in the given website.
Underlying stock sells for $100 a share and no dividend is paid. 5% is Risk free rate.
Value of call option is $8.
Performing the what-if analysis by setting the call option value to $8, and changing the standard deviation accordingly, we get the following data:
INPUTS | OUTPUTS | |||
Standard deviation (annual) | 0.3213 | d1 | 0.0089 | |
Expiration (in years) | 0.5 | d2 | -0.2183 | |
Risk-free rate (annual) | 0.05 | N(d1) | 0.5036 | |
Stock Price | 100 | N(d2) | 0.4136 | |
Exercise price | 105 | B/S call value | 8.0000 | |
Dividend yield (annual) | 0 | B/S put value | 10.4076 |
The Standard deviation for call price of $8 is 0.3213
Requirement 2
To determine:
To determine the impact on the volatility when the option sells at $9
Introduction:
Put option is the option which provides the option holder with the right to sell the assets or to exercise the contract but it does not give any obligation. The put option is exercised at a price which is specific or predetermined price before or on the expiration date.
Call option is the option which provides the right to the option holder to purchase the assets or right to exercise the contract. However it does not give the obligation. This option is exercised at a specific or predetermined price on the expiration date or before it.

Answer to Problem 28PS
Implied volatility increases when the option sells at $9
Explanation of Solution
Given Information:
The spreadsheet containing the data is available in the given website.
Underlying stock sells for $100 a share and no dividend is paid. 5% is Risk free rate.
Value of call option is $9.
Using the spreadsheet and setting the value to $9, we get
INPUTS | OUTPUTS | |||
Standard deviation (annual) | 0.3568 | d1 | 0.0318 | |
Expiration (in years) | 0.5 | d2 | -0.2204 | |
Risk-free rate (annual) | 0.05 | N(d1) | 0.5127 | |
Stock Price | 100 | N(d2) | 0.4128 | |
Exercise price | 105 | B/S call value | 9.0000 | |
Dividend yield (annual) | 0 | B/S put value | 11.4075 |
This shows an increase in the standard deviation which is now 0.3568
With increase in the value of the option, the volatility increases. So there is an increase in the implied volatility
Requirement 3
To determine:
To determine the effect on implied volatility when the option price is $8 but expiration time is only four months
Introduction:
Put option is the option which provides the option holder with the right to sell the assets or to exercise the contract but it does not give any obligation. The put option is exercised at a price which is specific or predetermined price before or on the expiration date.
Call option is the option which provides the right to the option holder to purchase the assets or right to exercise the contract. However it does not give the obligation. This option is exercised at a specific or predetermined price on the expiration date or before it.

Answer to Problem 28PS
Implied volatility increases when the expiration time is reduced while keeping the option price same.
Explanation of Solution
Given Information:
The spreadsheet containing the data is available in the given website.
Underlying stock sells for $100 a share and no dividend is paid. 5% is Risk free rate.
When the expiration time is set to 4 months, while option price is at $8, the values are as below:
INPUTS | OUTPUTS | |||
Standard deviation (annual) | 0.4087 | d1 | -0.0182 | |
Expiration (in years) | 0.33333 | d2 | -0.2541 | |
Risk-free rate (annual) | 0.05 | N(d1) | 0.4927 | |
Stock Price | 100 | N(d2) | 0.3997 | |
Exercise price | 105 | B/S call value | 8.0000 | |
Dividend yield (annual) | 0 | B/S put value | 11.2645 |
This shows that implied volatility rises to 0.4087 when the option price is kept at $8 and maturity time is lowered to 4 months.
With shorter maturity, the value of the option generally decreases. To keep the value of the option the same, then the implied volatility has to increase in this scenario.
Requirement 4
To determine:
To determine the effect on implied volatility when the option price is $8 but exercise price is lowered to $100.
Introduction:
Put option is the option which provides the option holder with the right to sell the assets or to exercise the contract but it does not give any obligation. The put option is exercised at a price which is specific or predetermined price before or on the expiration date.
Call option is the option which provides the right to the option holder to purchase the assets or right to exercise the contract. However it does not give the obligation. This option is exercised at a specific or predetermined price on the expiration date or before it.

Answer to Problem 28PS
Implied volatility decreases.
Explanation of Solution
Given Information:
The spreadsheet containing the data is available in the given website.
Underlying stock sells for $100 a share and no dividend is paid. 5% is Risk free rate.
When the exercise price is lowered to $100, while option price is at $8, the values are as below:
INPUTS | OUTPUTS | |||
Standard deviation (annual) | 0.2406 | d1 | 0.2320 | |
Expiration (in years) | 0.5 | d2 | 0.0619 | |
Risk-free rate (annual) | 0.05 | N(d1) | 0.5917 | |
Stock Price | 100 | N(d2) | 0.5247 | |
Exercise price | 100 | B/S call value | 8.0010 | |
Dividend yield (annual) | 0 | B/S put value | 5.5320 |
This shows that implied volatility reduces to 0.2406 when the option price is kept at $8 and exercise price is lowered to $100.
With lower exercise price, the value of call increases. But here to keep the value of the option the same, then the implied volatility has to decrease in this scenario.
Requirement 5
To determine:
To determine the effect on implied volatility when the option price is $8 but stock price is lowered to $98.
Introduction:
Put option is the option which provides the option holder with the right to sell the assets or to exercise the contract but it does not give any obligation. The put option is exercised at a price which is specific or predetermined price before or on the expiration date.
Call option is the option which provides the right to the option holder to purchase the assets or right to exercise the contract. However it does not give the obligation. This option is exercised at a specific or predetermined price on the expiration date or before it.

Answer to Problem 28PS
Implied volatility increases when the stock price is decreased.
Explanation of Solution
Given Information:
The spreadsheet containing the data is available in the given website.
Underlying stock sells for $100 a share and no dividend is paid. 5% is Risk free rate.
When the stock price is lowered to $98, while option price is at $8, the values are as below:
INPUTS | OUTPUTS | |||
Standard deviation (annual) | 0.3566 | d1 | -0.0484 | |
Expiration (in years) | 0.5 | d2 | -0.3006 | |
Risk-free rate (annual) | 0.05 | N(d1) | 0.4807 | |
Stock Price | 98 | N(d2) | 0.3819 | |
Exercise price | 105 | B/S call value | 8.0001 | |
Dividend yield (annual) | 0 | B/S put value | 12.4077 |
This shows that implied volatility increases to 0.35666 when the option price is kept at $8 and stock price is lowered to $98.
With lower stock price, the value of call decreases. But here to keep the value of the option the same, then the implied volatility has to increase in this scenario.
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Chapter 21 Solutions
Investments
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