1. CHAPTER 21 Option Valuation Investments Cover image Slides by
Richard D. Johnson
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved

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. Figure 21.1 Call Option Value before Expiration

4. Table 21.1 Determinants of Call Option Values

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. Figure 21.2 Range of Possible Call Option Values

7. Figure 21.3 Call Option Value as a Function of the Current Stock Price

8. Figure 21.4 Put Option Values as a Function of the Current Stock Price

9. Binomial Option Pricing: Text Example
120 10
100 C 90
Call Option Value
X = 110
Stock Price

10. Binomial Option Pricing: Text Example30
Alternative Portfolio
Buy 1 share of stock at $100
Borrow $81.82 (10% Rate)
Net outlay $18.18
Payoff
Value of Stock
Repay loan
Net Payoff
18.18
Payoff Structure
is exactly 3 times
the Call

11. Binomial Option Pricing: Text Example30 30
18.18 C
3C = $18.18
C = $6.06

12. Replication of Payoffs and Option Values
Alternative Portfolio - one share of stock and 3 calls written (X = 110)
Portfolio is perfectly hedged
Stock Value
Call Obligation
Net payoff
Hence C = or C = 6.06

13. 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).

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

15. 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.

16. Expanding to Consider Three Intervals

17. 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

18. Figure 21.5 Probability Distributions

19. 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.

20. Black-Scholes Option Valuation
X = Exercise price
e = , the base of the natural 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

21. Figure 21.6 A Standard Normal Curve

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

23. Probabilities from Normal Dist
Table 21.2
d N(d)
Interpolation

24. Probabilities from Normal Dist.
Table 21.2
d N(d)

25. Table 21.2 Cumulative Normal Distribution

26. 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?

27. Spreadsheet 21.1 Spreadsheet to Calculate Black-Scholes Option Values

28. Figure 21.7 Using Goal Seek to Find Implied Volatility

29. Figure 21.8 Implied Volatility of the S&P 500

30. Black-Scholes Model with 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)

31. Put Value Using Black-Scholes
P = Xe-rT [1-N(d2)] - S0 [1-N(d1)]
Using the sample call data
S = 100 r = .10 X = 95 g = .5 T = .25
95e-10x.25( )-100( ) = 6.35

32. 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 =

33. 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.

34. Figure 21.9 Call Option Value and Hedge Ratio

35. Portfolio Insurance
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.

36. Figure 21.10 Profit on a Protective Put Strategy

37. Figure 21.11 Hedge Ratios Change as the Stock Price Fluctuates

38. Figure 21.12 S&P 500 Cash-to-Futures Spread in Points at 15 Minute Intervals

39. Hedging On Mispriced Options
Option value is positively related to volatility:
If an investor believes that the volatility that is implied in an option’s price is too low, a profitable trade is possible.
Profit must be hedged against a decline in the value of the stock.
Performance depends on option price relative to the implied volatility.

40. The appropriate hedge will depend on the delta.
Hedging and Delta
The appropriate hedge will depend on the delta.
Recall the delta is the change in the value of the option relative to the change in the value of the stock.
Change in the value of the option
Change of the value of the stock
Delta =

41. Mispriced Option: Text Example
Implied volatility = 33%
Investor believes volatility should = 35%
Option maturity = 60 days
Put price P = $4.495
Exercise price and stock price = $90
Risk-free rate r = 4%
Delta = -.453

42. Table 21.3 Profit on a Hedged Put Portfolio

43. Table 21.4 Profits on Delta-Neutral Options Portfolio

44. Figure 21.13 Implied Volatility of the S&P 500 Index as a Function of Exercise Price