1
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Chapter 17
Option Valuation
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
2
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Option Values
• _______ value – Call: stock price - exercise price
– Put: exercise price - stock price
• ______ value -
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
3
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Time Value of Options: Call
Option
value
Value of Call
_______Value
Time value
X
Irwin / McGraw-Hill
Stock Price
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
4
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Factors Influencing Option Values:
Calls
Factor
Effect on value
Stock price
Volatility of stock price
Time to expiration
Interest rate
Dividend Rate
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
5
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Binomial Option Pricing:
Text Example
____
100
___
C
___
Stock Price
Irwin / McGraw-Hill
___
Call Option Value
X = 125
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
6
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Binomial Option Pricing:
Text Example
Alternative Portfolio
Buy ___ share of stock at $100
Borrow $_____ (8% Rate) 53.70
Net outlay $53.70
Payoff
Value of Stock
Repay loan
Net Payoff
Irwin / McGraw-Hill
150
0
Payoff Structure
is exactly 2 times
the Call
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
7
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Binomial Option Pricing:
Text Example
____
53.70
75
C
0
0
2C = $53.70
C = $____
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
8
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Another View of Replication of
Payoffs and Option Values
Alternative Portfolio - _____ share of
stock and ____ calls written (X = 125)
Portfolio is perfectly hedged
Stock Value
Call Obligation
Net payoff
Hence
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
9
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Black-Scholes Option Valuation
Co = Soe-dTN(d1) - Xe-rTN(d2)
d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2)
d2 = d1 - (s 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.
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
10
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Black-Scholes Option Valuation
X = Exercise price.
d = Annual dividend yield of underlying stock
e = 2.71828, the base of the nat. log.
r = Risk-free interest rate (annualized
continuously compounded with the same
maturity as the option.
T = time to maturity of the option in years.
ln = Natural log function
s = Standard deviation of annualized cont.
compounded rate of return on the stock
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
11
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Call Option Example
So = ____
X = ____
r = .10
T = .25 (quarter)
s = .50
d = 0
d1 = [ln(100/95)+(.10-0+(.5 2/2))]/(.5 .251/2)
= ____
d2 = .43 - ((.5)( .251/2)
= ____
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
12
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Probabilities from Normal Dist.
N (.43) = .6664
Table 17.2
d
N(d)
.42
.6628
.43
Interpolation
.44
.6700
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
13
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Probabilities from Normal Dist.
N (.18) = .5714
Table 17.2
d
N(d)
.16
.5636
.18
.5714
.20
.5793
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
14
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Call Option Value
Co = Soe-dTN(d1) - Xe-rTN(d2)
Co = 100 X .6664 - 95 e- .10 X .25 X .5714
Co = 13.70
Implied Volatility
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
15
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Put Option Value: Black-Scholes
P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)]
Using the sample data
P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664)
P = $_____
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
16
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Put Option Valuation: Using Put-Call
Parity
P = C + PV (X) - So
= C + Xe-rT - So
Using the example data
C = ____
X = ___ S = ____
r = .10
T = .25
P = 13.70 + 95 e -.10 X .25 - 100
P = ____
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
17
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Using the Black-Scholes Formula
Hedging: Hedge ratio or delta
Call = N (d1)
Put = N (d1) - 1
Option Elasticity
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
18
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Portfolio Insurance - Protecting
Against Declines in Stock Value
• Buying Puts
• Limitations
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
