Chapter 22 PrinciplesPrinciples

ofof

CorporateCorporate

FinanceFinance

Ninth Edition

Valuing Options

Slides by

Matthew Will

Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved

McGraw Hill/Irwin

22- 2

Topics Covered

Simple Option Valuation ModelBinomial ModelBlack-Scholes FormulaBlack Scholes in ActionOption Values at a GlanceThe Option Menagerie

22- 3

Option Valuation Methods

Genentech call options have an exercise price of $80.

Case 1

Stock price falls to $60

Option value = $0

Case 2

Stock price rises to $106.67

Option value = $26.67

22- 4

Option Valuation Methods

Assume you borrow 4/7 of the value of the Genentech exercise price ($33.45).

Value of Call = 80 x (4/7) – 33.45 = $12.26

22- 5

Option Valuation Methods

Since the Genentech call option is equal to a leveraged position in 4/7 shares, the option delta can be computed as follows.

7

4

67.46

67.26

6067.106

067.26

prices share possible of spread

pricesoption possible of spread DeltaOption

22- 6

Option Valuation Methods

If we are risk neutral, the expected return on Genentech call options is 2.5%. Accordingly, we can determine the probability of a rise in the stock price as follows.

.471 rise ofy Probabilit

.025 Retrun Expected

)25(rise ofy probabilit133.33 rise ofy probabilit Retrun Expected

22- 7

Option Valuation Method

The Genentech option can then be valued based on the following method.

57.12$

)0529(.)67.26471(.

0rise ofy probabilit167.26rise ofy probabilit ueOption val

22- 8

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.

22- 9

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

22- 10

40.37

32.10

36

37.401215.13610

UPUP

Binomial Pricing

22- 11

40.37

32.10

36

37.401215.13610

UPUP

10.328917.3610

DPDP

Binomial Pricing

22- 12

50.78 = price

40.37

32.10

25.52

45.28

36

28.62

40.37

32.10

36

1 tt PUP

Binomial Pricing

22- 13

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

22- 14

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

22- 15

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

22- 16

Binomial Model

The price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Genentech example illustrates this fact.

22- 17

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)

22- 18

Option Value

)()()( 21 EXPVdNPdNOC

Black-Scholes Option Pricing ModelBlack-Scholes Option Pricing Model

22- 19

OC- Call Option Price

P - Stock Price

N(d1) - Cumulative normal density function of (d1)

PV(EX) - Present Value of Strike or Exercise price

N(d2) - Cumulative normal 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

22- 20

)()()( 21 EXPVdNPdNOC

Black-Scholes Option Pricing Model

rteEXEXPV )(

factordiscount gcompoundin continuous1

rt

rt

ee

22- 21

N(d1)=

tv

trd

vEXP )()ln( 2

1

2

Black-Scholes Option Pricing Model

22- 22

Cumulative Normal Density Function

tv

trd

vEXP )()ln( 2

1

2

tvdd 12

22- 23

Call Option

2297.1 d

tv

trd

vEXP )()ln( 2

1

2

5908.)( 1 dN

Example - Genentech

What is the price of a call option given the following?

P = 80 r = 5% v = .4068

EX = 80 t = 180 days / 365

22- 24

Call Option

4769.5231.1)(

0580.

2

2

12

dN

d

tvdd

Example - Genentech

What is the price of a call option given the following?

P = 80 r = 5% v = .4068

EX = 80 t = 180 days / 365

22- 25

Call Option

05.10$

)80(4769.805908.

)()()()5)(.05(.

21

C

C

rtC

O

eO

eEXdNPdNO

Example - Genentech

What is the price of a call option given the following?

P = 80 r = 5% v = .4068

EX = 80 t = 180 days / 365

22- 26

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

22- 27

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

22- 28

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

22- 29

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

22- 30

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

(%)

22- 31

Put - Call Parity

Put Price = Oc + EX - P - Carrying Cost + Div.

Carrying cost = r x EX x t

22- 32

Put - Call ParityExample

ABC is selling at $41 a share. A six month May 40 Call is selling for $4.00. If a May $ .50 dividend is expected and r=10%, what is the put price?

OP = OC + EX - P - Carrying Cost + Div.

OP = 4 + 40 - 41 - (.10x 40 x .50) + .50

OP = 3 - 2 + .5

Op = $1.50

22- 33

Expanding the binomial model to allow more possible price changes

1 step 2 steps 4 steps

(2 outcomes) (3 outcomes) (5 outcomes)

etc. etc.

Binomial vs. Black Scholes

22- 34

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

22- 35

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

22- 36

Dilution

NqN

NqEXV

exerciseafter PriceShare

shares goutstandin of #sharesnew of #1

1 FactorDilution

22- 37

Web Resources

www.numa.com

www.fintools.net/options/optcalc.html

www.optionscentral.com

www.pcquote.com/

www.pmpublishing.com

www.schaffersresearch.com/stock/calculator.asp

Click to access web sitesClick to access web sites

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