1. Chapter 16
Option Valuation

2. Option Values
Intrinsic value - payoff that could be made if the option was immediately exercised
Call: stock price - exercise price
Put: exercise price - stock price
Time value - the difference between the option price and the intrinsic value 2

3. Time Value of Options: Call
Value of Call
Intrinsic Value
Time value X
Stock Price 3

4. Factors Influencing Option Values: Calls
Factor Effect on value
Stock price increases
Exercise price decreases
Volatility of stock price increases
Time to expiration increases
Interest rate increases
Dividend yield decreases 4

5. A Simple Binomial Model
A stock price is currently $20
In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20
Stock Price = $18

6. A Call Option
A 3-month call option on the stock has a strike price of 21.
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0

7. Setting Up a Riskless Portfolio
Consider the Portfolio: long D shares short 1 call option
Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
22D – 1
18D

8. Valuing the Portfolio (Risk-Free Rate is 12%)
The riskless portfolio is:
long 0.25 shares short 1 call option
The value of the portfolio in 3 months is 22´0.25 – 1 = 4.50
The value of the portfolio today is e – 0.12´0.25 =

9. Valuing the Option The portfolio that is
long shares short 1 option
is worth 4.367
The value of the shares is (= 0.25´20 )
The value of the option is therefore (= – )

10. Example:
Suppose the stock now sells at $100, and the price will either double to $200 or fall in half to $50 by the year-end. A call option on the stock might specify an exercise price of $125 and a time to expiration of one year. The interest rate is 8%. What is the option price today?

11. Black-Scholes Option Valuation
Co = Soe-dTN(d1) - Xe-rTN(d2)
d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2)
d2 = d1 - (s T1/2)
where
Co = Current call option value.
So = Current stock price
N(d) = probability that a random draw from a normal dist. will be less than d. 9

12. Black-Scholes Option Valuation
X = Exercise price.
d = Annual dividend yield of underlying stock
e = , the base of the nat. log.
r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option.
T = time to maturity of the option in years.
ln = Natural log function
s = Standard deviation of annualized cont. compounded rate of return on the stock 10

13. Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter)
s = d = 0
d1 = [ln(100/95)+(.10-0+(.5 2/2))]/( /2)
= .43
d2 = ((.5)( .251/2)
= .18 11

14. Probabilities from Normal Dist.
Table 17.2
d N(d)
Interpolation 12

15. Probabilities from Normal Dist.
Table 17.2
d N(d) 13

16. Call Option Value Co = Soe-dTN(d1) - Xe-rTN(d2)
Co = 100 X e- .10 X .25 X .5714
Co = 13.70
Implied Volatility
Using Black-Scholes and the actual price of the option, solve for volatility.
Is the implied volatility consistent with the stock? 14

17. Put Option Value: Black-Scholes
P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)]
Using the sample data
P = $95e(-.10X.25)( ) - $100 ( )
P = $6.35

18. Put Option Valuation: Using Put-Call Parity
P = C + PV (X) - So
= C + Xe-rT - So
Using the example data
C = X = 95 S = 100
r = .10 T = .25
P = e -.10 X
P = 15

19. Exercise in class
The stock price of Ajax Inc. is currently $105. The stock price a year from now will be either $130 or $90 with equal probabilities. The interest rate at which investors can borrow is 10%. Using the binomial OPM, the value of a call option with an exercise price of $110 and an expiration date one year from now should be worth __________ today.
A) $11.60
B) $15.00
C) $20.00
D) $40.00
The stock price of Bravo Corp. is currently $100. The stock price a year from now will be either $160 or $60 with equal probabilities. The interest rate at which investors invest in riskless assets at is 6%. Using the binomial OPM, the value of a put option with an exercise price of $135 and an expiration date one year from now should be worth __________ today.
A) $34.09
B) $37.50
C) $38.21
D) $45.45
Answer: A Difficulty: Hard
Answer: C Difficulty: Hard