Introduction Terminology Valuation-SimpleValuation-ActualSensitivity

IntroductionIntroduction TerminologyTerminology Valuation-SimpleValuation-Simple Valuation-ActualValuation-Actual SensitivitySensitivity

What is a financial option?What is a financial option?

It is the It is the rightright, but not the obligation, to , but not the obligation, to

buybuy (in the case of a (in the case of a call optioncall option) or ) or

sell (in the case of a put option) asell (in the case of a put option) a

stock at a stock at a certain pricecertain price through a through a certain timecertain time..

IntroductionIntroduction TerminologyTerminology Valuation-SimpleValuation-Simple Valuation-ActualValuation-Actual SensitivitySensitivity

Exercise priceExercise price is the price at which the option owner may buy is the price at which the option owner may buy the stock. Such buying is referred to as exercising the option.the stock. Such buying is referred to as exercising the option.

Expiration dateExpiration date is the date when the option expires. The is the date when the option expires. The option owner’s right to buy the stock at the exercise price option owner’s right to buy the stock at the exercise price expires on this date.expires on this date.

European optionsEuropean options can only be exercised on expiration date. can only be exercised on expiration date.American optionsAmerican options can be exercised any time through the can be exercised any time through the expiration date.expiration date.Employee stock options are usually American call options.Employee stock options are usually American call options.

Introduction TerminologyIntroduction Terminology Valuation-SimpleValuation-Simple Valuation-ActualValuation-Actual Sensitivity Sensitivity

Value of a call option at expiration is a function of the stock Value of a call option at expiration is a function of the stock price and the exercise price.price and the exercise price.

Example:Example: Option values on expiration date given an exercise price of $85. Option values on expiration date given an exercise price of $85.

Stock priceStock price 6060 7070 8080 9090 100100 110110

Option valueOption value 0 0 0 0 0 0 5 5 15 15 25 25

At any stock price less than $85, the option is worthless. If the option At any stock price less than $85, the option is worthless. If the option owner was interested in buying the stock, she could purchase the stock owner was interested in buying the stock, she could purchase the stock more cheaply from her broker!more cheaply from her broker!


Call option value on expiration date given an $85 exercise price.

Share Price

Call

op

tion

valu

e

85 105

$20


Intrinsic valueIntrinsic value is the value an option would have if it were exercised is the value an option would have if it were exercised immediately. Hence, an option may have different intrinsic values at immediately. Hence, an option may have different intrinsic values at different points in time.different points in time.

For a call option, its intrinsic value is the maximum of zero and SFor a call option, its intrinsic value is the maximum of zero and ST T - X, - X,

where Swhere STT is the stock price at time T, and X is the exercise price. is the stock price at time T, and X is the exercise price.

Fair option valueFair option value is the is the present valuepresent value of the of the option’s intrinsic valueoption’s intrinsic value..


Assume the share price of a particular stock one year from now will be Assume the share price of a particular stock one year from now will be $105.$105.

What is the fair option value of an option on this stock with an exercise What is the fair option value of an option on this stock with an exercise price of $85, expiring a year from now?price of $85, expiring a year from now?

Intrinsic value = 105 – 85 = $20.Intrinsic value = 105 – 85 = $20.

Fair option valueFair option value is the is the present valuepresent value of the of the option’s intrinsic valueoption’s intrinsic value..

Assume a discount rate of 5%.Assume a discount rate of 5%.

Fair option value = 20 / (1+.05) = $19.05.Fair option value = 20 / (1+.05) = $19.05.


If future stock price is uncertain…If future stock price is uncertain…

Assume there is an equal (20%) probability that a year from now the stock Assume there is an equal (20%) probability that a year from now the stock price would be $75, $85, $95, $105, or $115.price would be $75, $85, $95, $105, or $115.

What is the fair option value of an option on this stock with an exercise What is the fair option value of an option on this stock with an exercise price of $85, expiring a year from now?price of $85, expiring a year from now?

Stock Price at Expiration (ST)

Exercise Price (X)

Intrinsic Value Max (0, ST -X)

PV of Intrinsic Value

Probability Probability weighted PV

115 85 30 28.57 20% 5.71 105 85 20 19.05 20% 3.81 95 85 10 9.52 20% 1.90 85 85 0 0 20% 0 75 85 0 0 20% 0 Fair option value = Sum of probability weighted PV

= 5.71 + 3.81 + 1.90 + 0 + 0 = $11.42


Upper and Lower Bounds for Option PricesUpper and Lower Bounds for Option Prices

NotationNotationS: Current stock priceS: Current stock price

SSTT: Stock price at time T: Stock price at time T

T: Time to expiration for the optionT: Time to expiration for the option

X: Exercise priceX: Exercise price

r: Risk-free rate through time Tr: Risk-free rate through time T

c: Value of European call option to buy one sharec: Value of European call option to buy one share

C: Value of American call option to buy one share C: Value of American call option to buy one share

Upper BoundsUpper Bounds

The stock price is an upper bound to the option price. Else, an arbitrageur The stock price is an upper bound to the option price. Else, an arbitrageur can make a riskless profit by buying the stock and selling the call option.can make a riskless profit by buying the stock and selling the call option.

c < Sc < S

C < SC < S


Lower Bounds for Call Options on Non-dividend Paying StocksLower Bounds for Call Options on Non-dividend Paying Stocks

c > S – Xec > S – Xe-rT-rT

C > S – XeC > S – Xe-rT-rT

c = Cc = C

Consider two portfolios:Consider two portfolios:

Portfolio A: One European call option (c) plus cash equal to XePortfolio A: One European call option (c) plus cash equal to Xe-rT-rT..

Portfolio B: One share (S).Portfolio B: One share (S).

Value of portfolio A at T: Value of portfolio A at T:

Cash if invested in the risk-free interest rate grows to X.Cash if invested in the risk-free interest rate grows to X.

If SIf STT > X, call is exercised and portfolio is worth S > X, call is exercised and portfolio is worth STT..

If SIf ST T < X, call expires worthless, and portfolio is worth X.< X, call expires worthless, and portfolio is worth X.

Value of portfolio B at T: SValue of portfolio B at T: STT. Hence, portfolio A is always worth as much as, and is . Hence, portfolio A is always worth as much as, and is

sometimes worth more than, portfolio B at time T. Hence, c + Xesometimes worth more than, portfolio B at time T. Hence, c + Xe-rT -rT > S, or,> S, or,

c > S – Xec > S – Xe-rT-rT



Early exerciseEarly exercise: : It is never optimal to exercise an American call It is never optimal to exercise an American call option on a non-dividend paying stock early.option on a non-dividend paying stock early.

Consider an American call option on a non-dividend paying stock with a month to Consider an American call option on a non-dividend paying stock with a month to expiration. S=$100. X=$85.expiration. S=$100. X=$85.

The option is deep in the money and the option owner may be tempted to exercise The option is deep in the money and the option owner may be tempted to exercise it immediately.it immediately.

However, However, if the option owner wishes to hold the stock for more than one monthif the option owner wishes to hold the stock for more than one month, this , this is not the best strategy.is not the best strategy.

The better strategy is to keep the option and exercise it at the end of the month. The better strategy is to keep the option and exercise it at the end of the month. Two advantages: First, the exercise price is paid one month later. Second, there is Two advantages: First, the exercise price is paid one month later. Second, there is some chance (however remote) that the stock price will be below $85 at the end of some chance (however remote) that the stock price will be below $85 at the end of month. In this case the option owner would be glad to not have exercised the month. In this case the option owner would be glad to not have exercised the option.option.

What if the option owner wishes to hold the stock for less than one month?What if the option owner wishes to hold the stock for less than one month?



Early exerciseEarly exercise: : It is never optimal to exercise an American call It is never optimal to exercise an American call option on a non-dividend paying stock early.option on a non-dividend paying stock early.

What if the option owner wishes to hold the stock for less than one month?What if the option owner wishes to hold the stock for less than one month?

In this case the option owner is better off selling the option than owning it. The In this case the option owner is better off selling the option than owning it. The option will be bought by another investor who wants to own the stock. Such option will be bought by another investor who wants to own the stock. Such investors must exist, otherwise, the current stock price would not be $100. The investors must exist, otherwise, the current stock price would not be $100. The price obtained for the option would be greater than its intrinsic value of $15, since price obtained for the option would be greater than its intrinsic value of $15, since the exercise price does not have to be paid for a month. Hence, the exercise price does not have to be paid for a month. Hence,

C=cC=c, and, and

C > S – XeC > S – Xe-rT-rT


Upper and Lower Bounds for Option PricesUpper and Lower Bounds for Option Prices

Share Price (S)

Call

op

tion

valu

e

(C)

85 105

$20

C=S


A realistic model for future stock pricesA realistic model for future stock prices

The stock price at expiration is the key to valuing an European The stock price at expiration is the key to valuing an European option,option,

But… It is impossible to exactly predict future stock prices.But… It is impossible to exactly predict future stock prices.

However, we have models that give us a realistic probability However, we have models that give us a realistic probability distribution of future stock prices.distribution of future stock prices.

The Geometric Brownian Motion is one widely-used model of The Geometric Brownian Motion is one widely-used model of the probability distribution of future stock prices.the probability distribution of future stock prices.

If future stock prices follow the Geometric Brownian Motion, If future stock prices follow the Geometric Brownian Motion, then future stock returns will be normally distributed.then future stock returns will be normally distributed.

The well-known The well-known Black-ScholesBlack-Scholes option pricing model assumes option pricing model assumes that future stock prices follow the Geometric Brownian Motion.that future stock prices follow the Geometric Brownian Motion.


Black-Scholes Call Option ValuationBlack-Scholes Call Option Valuation

CCo o = S= SooN(dN(d11) - Xe) - Xe-rT-rTN(dN(d22))where, dwhere, d11 = [ln(S = [ln(Soo/X) + (r + /X) + (r + 22/2)T] / (/2)T] / (TT1/21/2))

dd22 = d = d11 + ( + (TT1/21/2))

where,where,

CCo o = Current call option value.= Current call option value.

SSo o = Current stock price.= Current stock price.

N(d) = probability that a random draw from a normal N(d) = probability that a random draw from a normal distribution will be less than d.distribution will be less than d.




dd22 = d = d11 + ( + (TT1/21/2))

where (continued),where (continued),X = Exercise price.X = Exercise price.e = 2.71828, the base of the natural log.e = 2.71828, the base of the natural log.r = Risk-free interest rate (annualizes continuously compounded with the r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option).same maturity as the option).T = time to maturity of the option in yearsT = time to maturity of the option in yearsln = Natural log functionln = Natural log functionStandard deviation of annualized continuously compounded rate of Standard deviation of annualized continuously compounded rate of return on the stockreturn on the stock




dd22 = d = d11 + ( + (TT1/21/2))

What is the price of a call option on stock when its share price What is the price of a call option on stock when its share price was $143.50? when, X= $11.98. was $143.50? when, X= $11.98. Assumptions: Assumptions: 2 2 = 70%, T=0.08 years, r=5%= 70%, T=0.08 years, r=5%

Using the above Black-Scholes valuation equation,Using the above Black-Scholes valuation equation,

Value of call option = $131.57Value of call option = $131.57

How does the call option value change with the assumptions: How does the call option value change with the assumptions: 2 2 ,T, r ?,T, r ?

Introduction TerminologyIntroduction Terminology Valuation-SimpleValuation-Simple Valuation-Actual Valuation-Actual SensitivitySensitivity

Strike Price Stock Price on Termination Date

Risk-free rate

Sigma Time to Maturity (in years)

Black-Scholes Option Value

Intrinsic Option Value on Termination Date

11.98 143.50 .05 .50 .08 131.57 131.52 11.98 143.50 .05 .70 .08 131.57 131.52 11.98 143.50 .05 1.00 .08 131.57 131.52 11.98 143.50 .05 1.50 .08 131.57 131.52 11.98 143.50 .05 2.00 .08 131.57 131.52 11.98 143.50 .05 2.50 .08 131.57 131.52 11.98 143.50 .05 3.00 .08 131.58 131.52 11.98 143.50 .05 3.50 .08 131.59 131.52 11.98 143.50 .05 4.00 .08 131.60 131.52

Sensitivity of Black-Scholes Option Value to

Variance of stock returns


Sensitivity of Black-Scholes Option Value to Variance stock returns

131.565131.57

131.575131.58

131.585131.59

131.595131.6

131.605

0 1 2 3 4 5

Variance of stock returns

Bla

ck-S

cho

les

Op

tio

n

Val

ue

($)



Risk-free rate

Sigma) Time to Maturity (in years)



11.98 143.50 .05 .70 .08 131.57 131.52 11.98 143.50 .05 .70 .25 131.67 131.52 11.98 143.50 .05 .70 .50 131.82 131.52 11.98 143.50 .05 .70 .75 131.96 131.52 11.98 143.50 .05 .70 1.00 132.11 131.52 11.98 143.50 .05 .70 1.25 132.25 131.52 11.98 143.50 .05 .70 1.50 132.40 131.52 11.98 143.50 .05 .70 1.75 132.55 131.52 11.98 143.50 .05 .70 2.00 132.71 131.52

Sensitivity of Black-Scholes Option Value to

Time to Maturity (in years)


Sensitivity of Black-Scholes Option Value to Time to Maturity (in years)

131.4

131.6

131.8

132132.2

132.4

132.6

132.8

0 0.5 1 1.5 2 2.5

Time to Maturity (in years)

Bla

ck-S

cho

les

Op

tio

n

Val

ue

($)



Risk-free rate

Sigma Time to Maturity (in years)



11.98 143.50 .01 .70 .08 131.53 131.52 11.98 143.50 .02 .70 .08 131.54 131.52 11.98 143.50 .03 .70 .08 131.55 131.52 11.98 143.50 .04 .70 .08 131.56 131.52 11.98 143.50 .05 .70 .08 131.57 131.52 11.98 143.50 .06 .70 .08 131.58 131.52 11.98 143.50 .07 .70 .08 131.59 131.52 11.98 143.50 .08 .70 .08 131.60 131.52 11.98 143.50 .09 .70 .08 131.61 131.52

Sensitivity of Black-Scholes Option Value to Risk-free Rate


Sensitivity of Black-Scholes Option Value to the Risk-free Rate

131.52

131.54

131.56

131.58

131.6

131.62

0 0.02 0.04 0.06 0.08 0.1

Risk-free Rate

Bla

ck-S

cho

les

Op

tio

n

Val

ue

($)

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Introduction Terminology Valuation-SimpleValuation-ActualSensitivity