chapter 21 principles of corporate finance tenth edition valuing options slides by matthew will...
TRANSCRIPT
Chapter 21Principles of
Corporate FinanceTenth Edition
Valuing Options
Slides by
Matthew Will
McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
21-2
Topics Covered
Simple Option Valuation ModelA Binomial Model for Valuing OptionsBlack-Scholes FormulaBlack Scholes in ActionOption Values at a GlanceThe Option Menagerie
21-3
Option Valuation Methods
Google call options have an exercise price of $430
Case 1
Stock price falls to $322.50
Option value = $0
Case 2
Stock price rises to $573.33
Option value = $143.33
21-4
Option Valuation Methods
Assume you buy 4/7 of a Google share and borrow $181.58 from the bank (@1.5%).
Value of Call = 430 x (4/7) – 181.58
= $64.13
21-5
Option Valuation Methods
Since the Google call option is equal to a leveraged position in 4/7 shares, the option delta can be computed as follows.
7
4
83.250
33.143
50.32233.573
033.143
prices share possible of spread
pricesoption possible of spread DeltaOption
21-6
Option Valuation Methods
If we are risk neutral, the expected return on Google call options is 1.5%. Accordingly, we can determine the probability of a rise in the stock price as follows.
.454 rise ofy Probabilit
.015 Retrun Expected
)25(rise ofy probabilit133.33 rise ofy probabilit Retrun Expected
21-7
Option Valuation Method
The Google option can then be valued based on the following method.
13.64$
)0546(.)33.143454(.
0rise ofy probabilit133.143rise ofy probabilit ueOption val
21-8
Option Valuation Method
The Google PUTPUT option can then be valued based on the following method.
Case 1
Stock price falls to $322.50
Option value = $107.50
Case 2
Stock price rises to $573.33
Option value = $0
21-9
Option Valuation Methods
Since the Google PUTPUT option is equal to a leveraged position in 3/7 shares, the option delta can be computed as follows.
429.7
350.32233.573
50.1070
prices share possible of spread
pricesoption possible of spread DeltaOption
21-10
Option Valuation Methods
Assume you SELL 3/7 of a Google share and lend $242.09 (@1.5%).
Value of PUT = -(3/7) x 430 + 242.09
= $57.82
21-11
Binomial Pricing
Present and possible future prices of Google stock assuming that in each three-month period the price will either rise by 22.6% or fall by 18.4%. Figures in parentheses show the corresponding values of a six-month call option with an exercise price of $430.
21-12
Binomial Pricing
Now we can construct a leveraged position in delta shares that would give identical payoffs to the option:
We can now find the leveraged position in delta shares that would give identical payoffs to the option:
21-13
Binomial Pricing
Present and possible future prices of Google stock. Figures in parentheses show the corresponding values of a six-month call option with an exercise price of $430.
Option Value:PV option = PV (.569 shares)- PV($199.58)
=.569 x $430 - $199.58/1.0075 = $46.49
21-14
Binomial Pricing
)(
)( upy Probabilit
du
dap
p1downy Probabilit
yearof % as interval time
th
eu
ed
ea
h
h
rh
The prior example can be generalized as the binomial model and shown as follows.
21-15
Example
Price = 36= .40 t = 90/365 t = 30/365
Strike = 40 r = 10%
a = 1.0083
u = 1.1215
d = .8917
Pu = .5075
Pd = .4925
Binomial Pricing
21-16
40.37
32.10
36
37.401215.13610
UPUP
Binomial Pricing
21-17
40.37
32.10
36
37.401215.13610
UPUP
10.328917.3610
DPDP
Binomial Pricing
21-18
50.78 = price
40.37
32.10
25.52
45.28
36
28.62
40.37
32.10
36
1 tt PUP
Binomial Pricing
21-19
50.78 = price
10.78 = intrinsic value
40.37
.37
32.10
0
25.52
0
45.28
36
28.62
36
40.37
32.10
Binomial Pricing
21-20
50.78 = price
10.78 = intrinsic value
40.37
.37
32.10
0
25.52
0
45.28
5.60
36
28.62
40.37
32.1036
trdduu ePUPO
The greater of
Binomial Pricing
21-21
50.78 = price
10.78 = intrinsic value
40.37
.37
32.10
0
25.52
0
45.28
5.60
36
.19
28.62
0
40.37
2.91
32.10
.10
36
1.51
trdduu ePUPO
Binomial Pricing
21-22
Binomial Model
The price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Google example illustrates this fact.
21-23
Option Value
Components of the Option Price1 - Underlying stock price
2 - Striking or Exercise price
3 - Volatility of the stock returns (standard deviation of annual returns)
4 - Time to option expiration
5 - Time value of money (discount rate)
21-24
Option Value
)()()( 21 EXPVdNPdNOC
Black-Scholes Option Pricing ModelBlack-Scholes Option Pricing Model
21-25
OC- Call Option Price
P - Stock Price
N(d1) - Cumulative normal probability density function of (d1)
PV(EX) - Present Value of Strike or Exercise price
N(d2) - Cumulative normal probability density function of (d2)
r - discount rate (90 day comm paper rate or risk free rate)
t - time to maturity of option (as % of year)
v - volatility - annualized standard deviation of daily returns
)()()( 21 EXPVdNPdNOC
Black-Scholes Option Pricing Model
21-26
N(d1)=
tv
trd
vEXP )()ln( 2
1
2
Black-Scholes Option Pricing Model
21-27
Cumulative Normal Density Function
tv
trd
vEXP )()ln( 2
1
2
tvdd 12
21-28
Call Option
1952.1 d
tv
trd
vEXP )()ln( 2
1
2
5774.)( 1 dN
Example - Google
What is the price of a call option given the following?
P = 430 r = 3% v = .4068
EX = 430 t = 180 days / 365
21-29
Call Option
4632.5368.1)(
0925.
2
2
12
dN
d
tvdd
Example - Google
What is the price of a call option given the following?
P = 430 r = 3% v = .4068
EX = 430 t = 180 days / 365
21-30
Call Option
04.52$
015.1/)430(4632.4305774.
)()()( 21
C
C
rtC
O
O
eEXdNPdNO
Example - Google
What is the price of a call option given the following?
P = 430 r = 3% v = .4068
EX = 430 t = 180 days / 365
21-31
Call Option
The curved line shows how the value of the Google call option changes as the price of Google stock changes.
21-32
Call Option
3070.1 d
tv
trd
vEXP )()ln( 2
1
2
3794.6206.1)( 1 dN
Example
What is the price of a call option given the following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
21-33
Call Option
3065.6935.1)(
5056.
2
2
12
dN
d
tvdd
Example
What is the price of a call option given the following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
21-34
)()()( 21 EXPVdNPdNOC
Black-Scholes Option Pricing Model
rteEXEXPV )(
factordiscount gcompoundin continuous1
rt
rt
ee
21-35
Call Option
70.1$
)40(3065.363794.
)()()()2466)(.10(.
21
C
C
rtC
O
eO
eEXdNPdNO
Example
What is the price of a call option given the following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
21-36
Black Scholes Comparisons
Establishment Industries
Digital Organics
INPUTSStock price(P) 22 22Exercise price (EX) 25 25Interest rate, percent ® 4 4Maturity in years (t) 5 5Annual standard deviation, percent () 24 36Are these rates compounded annually (A) or continuously © ? a aequivalent continously compounded rate, percent 3.92 3.92
INTERMEDIATE CALCULATIONS:PV(EX) 20.5482 20.5482d1=log[P/PV(EX)]/……. 0.3955 0.4873d2=d1-….. -0.1411 -0.3177N(d1)= delta 0.6538 0.687N(d2) 0.4439 0.3754
OPTION VALUES:Call value = N(d1) * P - N(d2)* PV(EX) 5.26 7.4Put value = Call value + PV(EX) - S 3.81 5.95
21-37
Implied Volatility
The unobservable variable in the option price is volatility. This figure can be estimated, forecasted, or derived from the other variables used to calculate the option price, when the option price is known.
Impl
ied
Vol
atil
ity
(%)
VXN
21-38
Put - Call Parity
Put Price = Oc + EX - P - Carrying Cost + Div.
Carrying cost = r x EX x t
21-39
Valuation Variations
American Calls with no dividends European Puts with no dividends American Puts with no dividends European Calls and Puts on dividend paying stocks American Calls on dividend paying stocks
21-40
Expanding the binomial model to allow more possible price changes
Binomial vs. Black Scholes
21-41
Example
What is the price of a call option given the following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
Binomial price = $1.51
Black Scholes price = $1.70
The limited number of binomial outcomes produces the difference. As the number of binomial outcomes is expanded, the price will approach, but not necessarily equal, the Black Scholes price.
Binomial vs. Black Scholes
21-42
How estimated call price changes as number of binomial steps increases
No. of steps Estimated value
1 48.1
2 41.0
3 42.1
5 41.8
10 41.4
50 40.3
100 40.6
Black-Scholes 40.5
Binomial vs. Black Scholes
21-43
Dilution
NqN
NqEXV
exerciseafter PriceShare
shares goutstandin of #sharesnew of #1
1 FactorDilution
21-44
Web Resources
Click to access web sitesClick to access web sites
Internet connection requiredInternet connection required
www.numa.com
www.math.columbia.edu/~smirnov/options13.html
www.optionscentral.com
www.pmpublishing.com
www.schaffersresearch.com/streetools/options/option_tools.aspx?click=jumpto