chapter 21 principles of corporate finance tenth edition valuing options slides by matthew will...

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