1. Option Valuation Chapter 21 21-1
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-1

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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-2

3. Time Value of Options: Call
Value of Call
Intrinsic Value
Time value X
Stock Price
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-3

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 Rate decreases
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-4

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 )
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-5

6. Allowable Range for Call
Call Value
Upper bound = S0
Lower Bound
= S0 - PV (X) - PV (D) S0
PV (X) + PV (D)
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-6

7. Binomial Option Pricing: Text Example
200 75
100 C 50
Call Option Value
X = 125
Stock Price
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-7

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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-8

9. Binomial Option Pricing: Text Example
150 75
53.70 C
2C = $53.70
C = $26.85
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-9

10. 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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-10

11. Generalising 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)
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-11

12. Generalising the Two-State Approach
121
110
104.50
100 95
90.25
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-12

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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-13

14. Expanding to Consider Three Intervals
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-14

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 / (.97)3 = 91.27
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-15

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 distribution will be less than d
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-16

17. Black-Scholes Option Valuation
X = Exercise price
e = , the base of the natural log
r = Risk-free interest rate (annualises 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 annualised cont. compounded rate of return on the stock
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-17

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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-18

19. Probabilities from Normal Distribution
Table 17.2
d N(d)
Interpolation
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-19

20. Probabilities from Normal Distribution
Table 17.2
d N(d)
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-20

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?
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-21

22. 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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-22

23. 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 =
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-23

24. 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)
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-24

25. 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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-25

26. 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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-26

27. 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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-27

28. 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 =
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-28

29. 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
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-29

30. Hedged Put Portfolio Cost to establish the hedged position:
1000 put options at $4.495 / option $ 4,495
453 shares at $90 / share ,770
Total outlay ,265
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-30

31. Profit Position on Hedged Put Portfolio
Value of put option: implied vol. = 35%
Stock Price
Put Price $ $ $4.347
Profit (loss) for each put (.148)
Value of and profit on hedged portfolio
Value of 1,000 puts $5,254 $4, $4,347
Value of 453 shares , , ,223
Total , , ,570
Profit
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-31

32. Summary Factors influencing option value
Buyer’s maximum loss is premium
When call options can be exercised?
When put options can be exercises?
Two-period two-state pricing model
Binomial model
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-32

33. Summary Black-Scholes when no dividend by stocks Hedge ratio
Portfolio Insurance
Option delta
Put-call parity usage in valuing put options
Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a Investments, by Bodie, Ariff, da Silva Rosa, Kane & Marcus
Slides prepared by Harminder Singh
21-33