Chapter 21, Problem 28PS

Summary Introduction

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

Summary Introduction

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

Summary Introduction

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.

Summary Introduction

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.

Summary Introduction

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.