1

Chapter

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

Option valuation7

Group Members

M.ZEESHAN ANWAR

MUSHTAQ HASSAN

RIZWAN ASHRAF

SHAHID IQBAL

3

Financial Options and Their Valuation

• Financial options

• Valuation to expiration with one period

• Binomial option pricing of a hedged volatility

• Black-Scholes Option Pricing Model

4

What is a financial option?

“Keep your option open is sound business advice ,and we

are out of option is sure sign of trouble”

An option is an agreement/contract which gives its

holder the right, but not the obligation, to buy (or sell) an

asset at some predetermined price within a specified

period of time.

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Options Contracts: Preliminaries

• Calls versus Puts– Call options gives the holder the right, but not the

obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. When exercising a call option, you “call in” the asset.

– Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone.

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Options Contracts: Preliminaries

• Exercising the Option– The act of buying or selling the underlying asset through the option

contract.

• Strike Price or Exercise Price– Refers to the fixed price in the option contract at which the holder can

buy or sell the underlying asset.

• Expiry– The maturity date of the option is referred to as the expiration date, or

the expiry.

• European versus American options– European options can be exercised only at expiry.– American options can be exercised at any time up to expiry.

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Options Contracts: Preliminaries

• In-the-Money– The exercise price is less than the spot price of the

underlying asset.

• At-the-Money– The exercise price is equal to the spot price of the

underlying asset.

• Out-of-the-Money– The exercise price is more than the spot price of the

underlying asset.

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Options Contracts: Preliminaries

• Intrinsic Value– The difference between the exercise price of the option and

the spot price of the underlying asset.

• Speculative Value– The difference between the option premium and the intrinsic

value of the option.

OPTION PREMIUM= INTRINSIC VALUE+SPECULATIVE VALUE

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Call Options

• Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today.

• When exercising a call option, you “call in” the asset.

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Basic Call Option Pricing Relationshipsat Expiry

• At expiry, an American call option is worth the same as a European option with the same characteristics.– If the call is in-the-money, it is worth ST – E.

– If the call is out-of-the-money, it is worthless:

Vo= Max[ST – E, 0]

Where

ST is the value of the stock at expiry (time T)

E is the exercise price.

C is the value of the call option at expiry

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22-12

CALL OPTION PAYOFFS

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Op

tion

pay

offs

($) Buy

a ca

ll

Exercise price = $50

50

12

22-13

CALL OPTION PAYOFFS

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Op

tion

pay

offs

($)

Sell a call

Exercise price = $50

50

13

22-14

CALL OPTION PROFITS

Exercise price = $50; option premium = $10

Sell a call

Buy a call

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Op

tion

pay

offs

($)

50–10

10

14

“Stock options are Zero sum Game”

• Example: You sell 50 option contracts. You receive $16250 up front, with strike price $20,you will be $16250 ahead.

• You will have to sell something for less than its worth, so will lose the difference.

• If the stock price is $25 you will have to sell 50x100=5000 shares at $20 per share, so you will be out $25-20=$5 per share, or $25000 total and net loss is $8750.

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Exercise price = $20.

Ending stock

Price

$15

17

20

23

25

30

Net profit to option seller

$16250

16250

16250

-1250

-8750

-33750

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Call Premium Diagram

5 10 15 20 25 30 35 40 45 50

Stock Price

Option value

30

25

20

15

10

5

Market price

Exercise value

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Notations for option valuation

S1 = Stock price at expiration(In one

period)

S0 = Stock price today

C1 = Value of the call option on the

expiration date

C0 = Value of the call option today

E = Exercise price on the option

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Case 1 : If the strike price (S1) ends up below the exercise price (E) on the expiration date, then the call option (C1) is worth zero . In other words:

C1= 0 if S1 ≤ E

Or equivalently: C1= 0 if S1-E ≤ 0

Case 2 : If the option finishes in the money then S1 › E ,and the value of the option at expiration is equal to the difference:

C1= S1-E if S1 › E

C1= S1-E if S1 › 0

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OPTION VALUATION WITH ONE PERIOD

• We assume a European option with unknown value of

stock at expiration date. We assume that we are able to

formulate probabilistic belief about its value one period

hence. The 450 line represents the theoretical value of

the option. It simply the current stock price less the

exercise price of the option. When the price of the stock

is less than the exercise price of the option, the option

has a zero theoretical value; when more, it has a

theoretical value on the line.

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MARKET VERSUS THEORETICAL VALUE

• Suppose the current market price of ABC Corporation`s stock is $10, which is equal to the exercise price. Theoretically, the option has no value; however, if there is some probability that the price of the stock will exceed $10 before expiration. Suppose further the that the option has 30 days to expiration and that there is .3 probability that the stock will have a market price of $5 per share at the end of 30 days, .4 that it will be $10, and .3 that it will be $15. The expected value of the option at the end of 30 days is thus

• 0(.3) +0(.4) + ($15-$10) (.3) =$1.50

21

Chapter

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

BINOMIAL OPTION PRICING OF HEDGED

RATIOBy

MUSHTAQ Hassan

7.2

Hedged Position

• Tow related financial assets– Stock– Option on that Stock

• In this way prices of one financial assets off set by opposite price movements.

• To maintain the risk free position

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• Return on option and stock, opportunity cost is

important to maintain hedged position.

• The opportunity cost is equal to risk free rate of

return to establishing hedged position.

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Binomial Option

Maps probabilities as a branching process.

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Problem for solutionCurrent value = 50

Probability of Occurrence

2/3 for increase by 20%

1/3 for decrease by 10%

Calculate

(a) Stock Value at the End of Period

(b) Expected Value of Stock Value at the End of Period

(c) Option Value at the End of Period

(d) Expected Value of Option Value at the End of Period

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uVs = One value higher than current value

dVs = One value lower than current value

Vs = Current value

u = One plus percentage increase in value from

beginning to end

d = One minus percentage decrease in value from

beginning to end

q = Probability of upward movement of stock

1 – q = Probability of downward movement of stock27

Delta Option

• A hedged position ascertained by long position and short position.

• This is also called Hedged Ratio of Stock to Options.

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Delta Option = Spread of possible option prices Spread of possible stock prices

Where

Spread of possible option prices meansuVo – dVo

Spread of possible stock prices meansuVs – dVs

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Stock Prices at end of Period

Value of long Position in Stock

(Out flow)

Value of Short Position in Option

(Inflow)

Value of Combined

Hedged Position

60 2(60) = 120 -3(10) = -30 120 – 3 = 90

45 2(45) = 90 -3(0) = 0 90 – 0 = 90

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Determination of Option Value at Beginning Period

Equation to solve for Vo B

[Long position – Short Position(Vo B)]1.05=Value of Hedged Position

31

Chapter

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

Black scholes option pricing model

By

Rizwan Ashraf

7.3

OBJECTIVE

Our main objective is to find the current price of a derivative.

• Derivatives are securities that do not convey ownership, but rather a promise to convey

ownership.

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The concepts behind black-scholes

• The option price and the stock price depend on the same underlying source of uncertainty

• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty

• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

• This leads to the Black-Scholes differential equation

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BSOPM

• The Black-Scholes OPM:

1 2

2

1

2 1

( ) ( )

ln( / ) ( / 2)

rtC S N d Ke N d

S K R td

t

d d t

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Black-Scholes Option Pricing Model (cont’d)

• Variable definitions:►C = theoretical call premium►S = current stock price► t = time in years until option expiration►K = option striking price►R = risk-free interest rate► = standard deviation of stock returns►N(x) = cumulative standard normal distribution ► functions►ln = natural logarithm►e = base of natural logarithm (2.7183)

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Assumptions of the Model

The stock pays no dividends during the

option’s life

European exercise terms

Markets are efficient

No commissions

Constant interest rates

Lognormal returns

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The Stock Pays no Dividends During the Option’s Life

• The OPM assumes that the underlying security pays

no dividends

• If you apply the BSOPM to two securities, one with no

dividends and the other with a dividend yield, the

model will predict the same call premium

• Valuing securities with different dividend yields using

the OPM will result in the same price

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European Exercise Terms

• The OPM assumes that the option is European

• Not a major consideration since very few calls are ever exercised prior to expiration

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Markets Are Efficient

• The OPM assumes markets are informational efficient

–People cannot predict the direction of the market or of an individual stock

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No Commissions

• The OPM assumes market participants do not have to pay any commissions to buy or sell

• Commissions paid by individual can significantly affect the true cost of an option

– Trading fee differentials cause slightly different effective option prices for different market participants

41

Constant Interest Rates

• The OPM assumes that the interest rate R in the model is known and constant

• It is common use to use the discount rate on a U.S. Treasury bill that has a maturity approximately equal to the remaining life of the option

– This interest rate can change

42

Lognormal Returns

• The OPM assumes that the logarithms of returns of the underlying security are normally distributed

• A reasonable assumption for most assets on which options are available

43

Black-Scholes Option Pricing Model

Example

Stock ABC currently trades for $30. A call option on ABC stock has a striking price of $25 and expires in three months. The current risk-free rate is 5%, and ABC stock has a standard deviation of 0.45.

According to the Black-Scholes OPM, but should be the call option premium for this option?

44

• S = CURRENT STOCK PRICE = $30• K = STRIKE PRICE = $25• t = time = 3 month• R =5%=0.05• =standard deviation=0.45

45

Black-Scholes Option Pricing Model (cont’d)

Example (cont’d)

Solution: We must first determine d1 and d2:

2

1

2

ln( / ) ( / 2)

ln(30 / 25) 0.05 (0.45 / 2) 0.25

0.45 0.250.1823 0.0378

0.9780.225

S K R td

t

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Black-Scholes Option Pricing Model (cont’d)

Example (cont’d)

Solution (cont’d):

2 1

0.978 (0.45) 0.25

0.978 0.225

0.753

d d t

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Black-Scholes Option Pricing Model (cont’d)

Example (cont’d)

Solution (cont’d): The next step is to find the normal probability values for d1 and d2. Using Excel’s NORMSDIST function yields:

1

2

( ) 0.836

( ) 0.774

N d

N d

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Using Excel’s NORMSDIST Function

• The Excel portion below shows the input and the result of the function:

49

Black-Scholes Option Pricing Model (cont’d)

Example (cont’d)

Solution (cont’d): The final step is to calculate the option premium:

1 2

(0.05)(0.25)

( ) ( )

$30 0.836 $25 0.774

$25.08 $19.11

$5.97

rtC S N d Ke N d

e

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Insights Into the Black-Scholes Model

• Divide the OPM into two parts:

1 2( ) ( )rtC S N d Ke N d

Part A Part B

51

Insights Into the Black-Scholes Model (cont’d)

• Part A is the expected benefit from acquiring the stock:

– S is the current stock price and the discounted

value of the expected stock price at any future point

– N(d1) is a pseudo-probability

• It is the probability of the option being in the

money at expiration, adjusted for the depth the

option is in the money

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Insights Into the Black-Scholes Model (cont’d)

• Part B is the present value of the exercise price on the

expiration day:

– N(d2) is the actual probability the option will be in the money on

expiration day

53

Insights Into the Black-Scholes Model (cont’d)

• The value of a call option is the difference between the expected benefit from acquiring the stock and paying the exercise price on expiration day

54

Fischer Black

Born: 1938 Died: 1995 1959 -- earned bachelor's degree in physics 1964 -- earned PhD. from Harvard in applied math 1971 -- joined faculty of University of Chicago Graduate School of

Business 1973 -- Published "The Pricing of Options and Corporate Liabilities“ 19XX -- Left the University of Chicago to teach at MIT 1984 -- left MIT to work for Goldman Sachs & Co.

Myron Scholes Born: 1941 1973 -- Published "The Pricing of Options and Corporate Liabilities“ Currently works in the derivatives trading group at Salomon Brothers.

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Chapter

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

Other measures or parameters of

sensitivityBy

SHAHID IQBAL

7.4

Other parameters measuring the risk

•Gamma ┌

• Theta θ

• Rho p

• Vega

57

Option Gamma

• The gamma of an option indicates how the delta of an option will change relative to a 1 point move in the underlying asset.

• The Gamma shows the option delta's sensitivity to market price changes.

58

59

Other parameters measuring the risk

• Theta: measure of option price sensitiveness to a change in time to expiration.

• Rho:measure of option price sensitiveness to a change in the interest rate

• Vega: of an option indicates how much, theoretically at least, the price of the option will change as the volatility of the underlying asset changes.

60

Parameters measuring the risk

• Gamma(stock price, strike price)

• Theta(time until to expiration)

• Rho (risk free rate)

• Vega(volitality)

61

 volatility

•  volatility is a measure for variation of price of a financial instrument over time.

• Volatility can be measure by using the standard deviation or variance. Commonly the higher the volatility the riskier the security.

62

Implied volatility

• Implied volatility tells a trader what level of volatility to expect from the asset given the current share price and current option price.

63

Debt & Other Options

• Debt option may be on the actual debt instrument or on an interest- rate future contract.

• Debt option provides a means for protection against adverse- rate movements.

64

Foreign currency options

• Fx options(foreign-exchange option)

• is written on the number of units of a foreign currency that a U.S dollar will buy.

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Thanks for your listening!!

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