1. Chapter 21
Options Valuation

2. Option Values
Intrinsic value - profit 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

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

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 Rate decreases

5. Restrictions on Option Value: Call
Value cannot be negative
Value cannot exceed the stock value
Value of the call must be greater than the value of levered equity
C > S0 - ( X + D ) / ( 1 + Rf )T
C > S0 - PV ( X ) - PV ( D )

6. Allowable Range for Call
Call Value
Upper bound = S0
Lower Bound
= S0 - PV (X) - PV (D) S0
PV (X) + PV (D)

7. Binomial Option Pricing: Text Example
200 C 75
Call Option Value
X = 125
100 50
Stock Price

8. Binomial Option Pricing: Text Example
150
Alternative Portfolio
Buy 1 share of stock at $100
Borrow $ (8% Rate)
Net outlay $53.70
Payoff
Value of Stock
Repay loan
Net Payoff
53.70
Payoff Structure
is exactly 2 times
the Call

9. Binomial Option Pricing: Text Example
150 C 75
53.70
2C = $53.70
C = $26.85

10. Another View of Replication of Payoffs and Option Values
Alternative Portfolio - one share of stock and 2 calls written (X = 125)
Portfolio is perfectly hedged
Stock Value
Call Obligation
Net payoff
Hence C = or C = 26.85

11. Generalizing the Two-State Approach
Assume that we can break the year into two six-month segments
In each six-month segment the stock could increase by 10% or decrease by 5%
Assume the stock is initially selling at 100
Possible outcomes
Increase by 10% twice
Decrease by 5% twice
Increase once and decrease once (2 paths)

12. Generalizing the Two-State Approach
100
110
121 95
90.25
104.50

13. Expanding to Consider Three Intervals
Assume that we can break the year into three intervals
For each interval the stock could increase by 5% or decrease by 3%
Assume the stock is initially selling at 100

14. Expanding to Consider Three Intervals

15. Possible Outcomes with Three Intervals
Event Probability Stock Price
3 up 1/ (1.05)3 =115.76
2 up 1 down 3/ (1.05)2 (.97) =106.94
1 up 2 down 3/ (1.05) (.97)2 = 98.79
3 down 1/ (.97)3 = 91.27

16. Black-Scholes Option Valuation
Co = SoN(d1) - Xe-rTN(d2)
d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)
d2 = d1 + (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.

17. Black-Scholes Option Valuation
X = Exercise price.
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
Standard deviation of annualized cont. compounded rate of return on the stock

18. Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter) = .50
= .50
d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2)
= .43
d2 = ((5.251/2)
= .18

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

20. Probabilities from Normal Dist.
Table 17.2
d N(d)

21. Call Option Value Co = SoN(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?

22. 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 =

23. Adjusting the Black-Scholes Model for Dividends
The call option formula applies to stocks that pay dividends
One approach is to replace the stock price with a dividend adjusted stock price
Replace S0 with S0 - PV (Dividends)

24. Using the Black-Scholes Formula
Hedging: Hedge ratio or delta
The number of stocks required to hedge against the price risk of holding one option
Call = N (d1)
Put = N (d1) - 1
Option Elasticity
Percentage change in the option’s value given a 1% change in the value of the underlying stock

25. Portfolio Insurance - Protecting Against Declines in Stock Value
Buying Puts - results in downside protection with unlimited upside potential
Limitations
Tracking errors if indexes are used for the puts
Maturity of puts may be too short
Hedge ratios or deltas change as stock values change