1. Chapter 18
Option Valuation

2. Chapter Summary
Objective: To discuss factors that affect option prices and to present quantitative option pricing models.
Factors influencing option values
Black-Scholes option valuation
Using the Black-Scholes formula
Binomial Option Pricing

3. 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

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

5. 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

6. 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 )

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

8. Summary Reminder
Objective: To discuss factors that affect option prices and to present quantitative option pricing models.
Factors influencing option values
Black-Scholes option valuation
Using the Black-Scholes formula
Binomial Option Pricing

9. 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

10. Black-Scholes Option Valuation (cont’d)
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 continuously compounded rate of return on the stock

11. Call Option Example
So = 100 X = 95
r = T = .25 (quarter)
= .50

12. Probabilities from Normal Distribution
Table 18.2
d N(d)
Interpolation

13. Probabilities from Normal Distribution
Table 18.2
d N(d)

14. Call Option Value Co = SoN(d1) - Xe-rTN(d2)
Co = 100 x – (95 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?

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

16. 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 =

17. 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)

18. Summary Reminder
Objective: To discuss factors that affect option prices and to present quantitative option pricing models.
Factors influencing option values
Black-Scholes option valuation
Using the Black-Scholes formula
Binomial Option Pricing

19. 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

20. 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

21. Hedging Bets 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

22. 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.

23. 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

24. Hedged Put Portfolio Cost to establish the hedged position
1000 put options at $4.495 / option $ 4,495
453 shares at $90 / share 40,770
Total outlay ,265

25. Profit Position on Hedged Put Portfolio
Value of put as function of stock price:
implied volatility = 35%
Stock Price
Put Price $ $ $4.347
Profit/loss per put (.148)
Value of and profit on hedged portfolio
Stock Price
Value of 1,000 puts $ 5, $ 4, $ 4,347
Value of 453 shares 40, , ,223
Total , , ,570
Profit

26. Summary Reminder
Objective: To discuss factors that affect option prices and to present quantitative option pricing models.
Factors influencing option values
Black-Scholes option valuation
Using the Black-Scholes formula
Binomial Option Pricing

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

28. Binomial Option Pricing: Text Example
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
150
Payoff Structure
is exactly 2 times
the Call

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

30. 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

31. 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)

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

33. 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

34. Expanding to Consider Three Intervals

35. Possible Outcomes with Three Intervals
Event Probability Stock Price
3 up /8 100 (1.05)3 =115.76
2 up 1 down 3/8 100 (1.05)2 (.97) =106.94
1 up 2 down 3/8 100 (1.05) (.97)2 = 98.79
3 down /8 100 (.97)3 = 91.27

36. Multinomial Option Pricing
Incomplete markets
If the stock return has more than two possible outcomes it is not possible to replicate the option with a portfolio containing the stock and the riskless asset
Markets are incomplete when there are fewer assets than there are states of the world (here possible stock outcomes)
No single option price can be then derived by arbitrage methods alone
Only upper and lower bounds exist on option prices, within which the true option price lies
An appropriate pair of such bounds converges to the Black-Scholes price at the limit