1. 1

2. Chapter
7 Option valuation
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

3. Group Members
M.ZEESHAN ANWAR
MUSHTAQ HASSAN
RIZWAN ASHRAF
SHAHID IQBAL
3

4. 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

5. 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.
5

6. 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.
6

7. 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.
7

8. 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.
8

9. 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
9

10. 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.
10

11. Basic Call Option Pricing Relationships
at 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
11

12. 22-12
CALL OPTION PAYOFFS
60
Option payoffs ($)
40
20
20 40 60 80 100 120
50
Stock price ($)
–20
–40 Exercise price = $50
12

13. 22-13
CALL OPTION PAYOFFS
60
Option payoffs ($)
40
20
20 40 60 80 100 120
50
Stock price ($)
–20
Exercise price = $50
–40
13

14. 22-14
CALL OPTION PROFITS
60
Option payoffs ($)
40 Buy a call
20
10
20 40 50 60 80 100 120
–10 Stock price ($)
–20
Exercise price = $50;
–40 Sell a call
option premium = $10
14

15. “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.
15

16. Exercise price = $20.
Ending stock Net profit to option
Price seller
$15 $16250
17 16250
20 16250
23 -1250
25 -8750
30 -33750
16

17. Call Premium Diagram
Option
value
30
25
20
15
10 Market price
5
Exercise value
5 10 15 20 25 30 35 40 45 50
Stock Price
17

18. 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
18

19. 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
19

20. 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.
20

21. 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

22. Chapter
7.2 BINOMIAL OPTION PRICING
OF HEDGED RATIO
By
MUSHTAQ Hassan
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

23. 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
23

24. • 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.
24

25. Binomial Option
Maps probabilities as a
branching process.
25

26. Problem for solution
Current 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
26

27. 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 stock
27

28. Delta Option
• A hedged position ascertained by long position
and short position.
• This is also called Hedged Ratio of Stock to
Options.
28

29. Delta Option = Spread of possible option prices
Spread of possible stock prices
Where
Spread of possible option prices means
uVo – dVo
Spread of possible stock prices means
uVs – dVs
29

30. Stock Prices at Value of long Value of Short Value of
end of Period Position in Stock Position in Combined
(Out flow) Option (Inflow) Hedged Position
60 2(60) = 120 -3(10) = -30 120 – 3 = 90
45 2(45) = 90 -3(0) = 0 90 – 0 = 90
30

31. 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

32. Chapter
Black scholes option
pricing model
7.3
By
Rizwan Ashraf
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

33. 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.
33

34. 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
34

35. BSOPM
• The Black-Scholes OPM:
rt
C S N (d1 ) Ke N (d 2 )
2
ln ( S / K ) R ( / 2) t
d1
t
d2 d1 t
35

36. 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)
36

37. 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
37

38. 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
38

39. 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
39

40. Markets Are Efficient
• The OPM assumes markets are
informational efficient
– People cannot predict the direction
of the market or of an individual
stock
40

41. 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

42. 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

43. 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

44. 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

45. • S = CURRENT STOCK PRICE = $30
• K = STRIKE PRICE = $25
• t = time = 3 month
• R =5%=0.05
• =standard deviation=0.45
45

46. Black-Scholes
Option Pricing Model (cont’d)
Example (cont’d)
Solution: We must first determine d1 and d2:
2
ln( S / K ) R ( / 2) t
d1
t
2
ln(30 / 25) 0.05 (0.45 / 2) 0.25
0.45 0.25
0.1823 0.0378
0.978
0.225
46

47. Black-Scholes
Option Pricing Model (cont’d)
Example (cont’d)
Solution (cont’d):
d2 d1 t
0.978 (0.45) 0.25
0.978 0.225
0.753
47

48. 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:
N (d1 ) 0 .8 3 6
N (d 2 ) 0 .7 7 4
48

49. Using Excel’s
NORMSDIST Function
• The Excel portion below shows the input and the result of
the function:
49

50. Black-Scholes
Option Pricing Model (cont’d)
Example (cont’d)
Solution (cont’d): The final step is to calculate the
option premium:
rt
C S N (d1 ) Ke N (d 2 )
( 0.05 )( 0.25 )
$30 0.836 $25 e 0.774
$25.08 $19.11
$5.97
50

51. Insights Into the
Black-Scholes Model
• Divide the OPM into two parts:
rt
C S N (d1 ) Ke N (d 2 )
Part A Part B
51

52. 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
52

53. 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

54. 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

55. 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.
55

56. Chapter
Other measures or
7.4 parameters of
sensitivity
By
SHAHID IQBAL
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

57. Other parameters measuring the risk
• Gamma ┌
• Theta θ
• Rho p
• Vega
57

58. 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. 59

60. 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

61. Parameters measuring the risk
• Gamma(stock price, strike price)
• Theta(time until to expiration)
• Rho (risk free rate)
• Vega(volitality)
61

62. 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

63. 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

64. 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

65. 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.
65

66. Thanks for your listening!!
66