1 valuing options

7/27/2019 1 Valuing Options

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Whats an Option?


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Option Positions

Long call

Long put Short call

Short put


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Long Call on eBay

Profit from buying one eBay European call option: option

price = $5, strike price = $100, option life = 2 months

30

20

10

0-5

70 80 90 100

110 120 130

Profit ($)

Terminalstock price ($)


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Short Call on eBay

Profit from writing one eBay European call option: option

price = $5, strike price = $100

-30

-20

-10

05

70 80 90 100

110 120 130

Profit ($)



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Long Put on IBM

Profit from buying an IBM European put option: option


30

20

10

0

-770605040 80 90 100

Profit ($)

Terminal

stock price ($)


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Short Put on IBM

Profit from writing an IBM European put option: option


-30

-20

-10

7

070

605040

80 90 100

Profit ($)



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Payoffs from OptionsWhat is the Option Position in Each Case?K= Strike price, ST= Price of asset at maturity

Payoff Payoff

ST STK

K

Payoff Payoff

ST STK

K


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Valuing Options


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Topics Covered

Simple Option Valuation Model

Binomial Model

Black-Scholes Model Black Scholes in Action

Option Values at a Glance


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)(

)(upyProbabilit

du

dap

Binomial Pricing

p

1downyProbabilit

yearof%asintervaltime

th

eu

ed

ea

h

h

rh


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ExamplePrice = 36 = .40 t = 90/365 t = 30/365

Strike = 40 r = 10%

Binomial Pricing

a = 1.0083

u = 1.1215

d = .8917

Pu = .5075Pd = .4925


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40.37

32.10

36

37.401215.136

10

U

PUP

Binomial Pricing


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40.37

32.10

36

37.401215.136

10

U

PUP

10.328917.36

10

DPDP

Binomial Pricing


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50.78 = price

40.37

32.10

25.52

45.28

36

28.62

40.37

32.10

36

1 tt PUP

Binomial Pricing


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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


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50.78 = price


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


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50.78 = price


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


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Option Value

Components of the Opt ion Pr ice

1 - Underlying stock price

2 - Striking or Exercise price

3 - Volatility of the stock returns (standard deviation ofannual returns)

4 - Time to option expiration

5 - Time value of money (discount rate)


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Option Value

B lack -Scho les Opt ion Pric ing Model

)()()( 21 EXPVdNPdNOC


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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

B lack-Scho les Opt ion Pric ing Model



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rteEXEXPV )(

factordiscountgcompoundincontinuous1 rt

rt

ee


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32 34 36 38 40

N(d1)=


tv

trdv

EXP )()ln( 2

1

2


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Cumulative Normal Dens i ty Funct ion

tv

trd

vEXP )()ln( 2

1

2

tvdd 12


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Call Option

3070.1 d

tv

trd

vEXP )()ln( 2

1

2

ExampleWhat is the price of a call option given the

following?

P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365

3794.6206.1)( 1 dN


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Call Option

3065.6935.1)(5056.

2

2

12

dN

d

tvdd


following?

P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365


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Call Option

70.1$

)40(3065.363794.

)()()(

)2466)(.10(.

21

C

C

rt

C

OeO

eEXdNPdNO


following?

P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365


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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


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following?

P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365Binomial price = $1.51

Black Scholes price = $1.70

The limited number of binomial outcomes produces thedifference. As the number of binomial outcomes is expanded,

the price will approach, but not necessarily equal, the Black

Scholes price.


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How estimated call price changes asnumber of binomial steps increasesNo. 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



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Numericals


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Numericals


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Numericals


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Numericals


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Numericals


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Numericals

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