Chapter 21
Options Valuation
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

Time Value of Options: Call
Option
value
Value of Call
Intrinsic Value
Time value
X
Stock Price
Factors Influencing Option Values:
Calls
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Dividend Rate
Effect on value
increases
decreases
increases
increases
increases
decreases
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 + R f )T
C > S0 - PV ( X ) - PV ( D )
Allowable Range for Call
Call Value
Lower Bound
= S0 - PV (X) - PV (D)
S0
PV (X) + PV (D)
Binomial Option Pricing:
Text Example
200
100
75
C
50
Stock Price
0
Call Option Value
X = 125
Binomial Option Pricing:
Text Example
Alternative Portfolio
Buy 1 share of stock at $100
Borrow $46.30 (8% Rate)
Net outlay $53.70
Payoff
Value of Stock 50 200
Repay loan
- 50 -50
Net Payoff
0 150
150
53.70
0
Payoff Structure
is exactly 2 times
the Call
Binomial Option Pricing:
Text Example
150
53.70
C
0
2C = $53.70
C = $26.85
75
0
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
50
200
Call Obligation
0
-150
Net payoff
50
50
Hence 100 - 2C = 46.30 or C = 26.85
Generalizing the
Two-State Approach
Assume that we can break the year into two sixmonth 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)
Generalizing the
Two-State Approach
121
110
104.50
100
95
90.25
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

Expanding to
Consider Three Intervals
S+++
S++
S++-
S+
S+-
S
S+-S-
S-S---
Possible Outcomes with
Three Intervals
Event
Probability
Stock Price
3 up
1/8
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
1/8
100 (.97)3
= 91.27
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.
Black-Scholes Option Valuation
X = Exercise price.
e = 2.71828, 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
Call Option Example
So = 100
X = 95
r = .10
T = .25 (quarter)
= .50
d1 = [ln(100/95) + (.10+(5 2/2))] / (5 .251/2)
= .43
d2 = .43 + ((5.251/2)
= .18
Probabilities from Normal Dist
N (.43) = .6664
Table 17.2
d
N(d)
.42
.6628
.43
.6664 Interpolation
.44
.6700
Probabilities from Normal Dist.
N (.18) = .5714
Table 17.2
d
N(d)
.16
.5636
.18
.5714
.20
.5793
Call Option Value
Co = SoN(d1) - Xe-rTN(d2)
Co = 100 X .6664 - 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?
Put Option Valuation:
Using Put-Call Parity
P = C + PV (X) - So
= C + Xe-rT - So
Using the example data
C = 13.70X = 95 S = 100
r = .10
T = .25
P = 13.70 + 95 e -.10 X .25 - 100
P = 6.35
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)
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
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