valuing options

Valuing OptionsAlddon Christner C. Ang

Basfin2

Source: BMA

Outline

Simple Option Valuation Model

A Binomial Model for Valuing Options

Black-Scholes Formula

Black Scholes in Action

Option Values at a Glance

The Option Menagerie

Simple Option Valuation

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


Replicating Portfolio. 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


Since the Google call option is equal to a leveraged

position in 4/7 shares, the option delta can be computed

as follows.

Option delta, or hedge ratio, is the number of shares

needed to replicate one call.

7

4

83.250

33.143

50.32233.573

033.143

prices share possible of spread

pricesoption possible of spread DeltaOption


Risk-Neutral Valuation. 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.

The Google option can then be valued based on the

following method.

.4543 rise ofy Probabilit

.015 Return Expected

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

15.64$

015.1/11.65

)]0546(.)33.1434543[(.

}0rise ofy probabilit133.143rise ofy probabilit { ueOption val

PV

PV


The Google PUT 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


Since the Google PUT option is equal to a leveraged

position in 3/7 shares, the option delta can be computed

as follows.

429.

7

3

50.32233.573

50.1070

prices share possible of spread

pricesoption possible of spread DeltaOption


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

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.

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:

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

Binomial Pricing

The prior example can be generalized as the binomial

model and shown as follows.

where:

)(

)( upy Probabilit

du

dap

p 1downy Probabilit

yearof % as interval time

th

eu

ed

ea

h

h

rh

Binomial Pricing

Example:

Price = 36; = .40; t = 90/365; t = 30/365; Strike = 40;

r = 10%

Given this, we can compute for the following:

a = 1.0083; u = 1.1215; d = .8917; p = .5075; (1-p) = .4925

40.37

32.10

36

37.401215.136

10

UPUP

Binomial Pricing

40.37

32.10

36

37.401215.136

10

UPUP

10.328917.36

10

DPDP

Binomial Pricing

50.78 = price

40.37

32.10

25.52

45.28

36

28.62

40.37

32.10

36

1 tt PUP

Binomial Pricing

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

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

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

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.

Black-Scholes Option Pricing Model

Assumptions:

Stock prices are lognormally distributed, stock price

returns are normally distributed

Interest rate is a known constant

No dividends during the options life

No taxes, transaction costs or margin requirements

Efficient markets (movements cannot be predicted)

Options can only be exercised at expiry


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)

- volatility - annualized standard deviation of daily returns

)()()( 21 EXPVdNPdNOC


)()()( 21 EXPVdNPdNOC

t

trd EX

P

)()ln(2

1

2

tdd 12


N(d) is the probability that a normally distributed random

variable will be less than or equal to d.

N(d1) reflects the cumulative probability related to the

current value of the stock; its value shows the amount by

which the option premium increases for each unit rise in

the price of the stock.

N(d2) reflects the cumulative probability related to the

exercise price of the stock, that is the probability that the

option will be exercised.

Call Option Black-Scholes Model

Example Google:

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

P = 430 r = 3% = .4068

EX = 430 t = 180 days / 365

1952.1 d 5774.)( 1 dN

4632.5368.1)(

0925.

2

2

dN

d


Example Google:

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

P = 430 r = 3% = .4068

EX = 430 t = 180 days / 365

N(d1) = .5774 N(d2) = .4632

Also,

04.52$

015.1/)430(4632.4305774.

)()()( 21

C

C

C

O

O

EXPVdNPdNO

rteEXEXPV )(


Example:

Price = 36; = .40; t = 90/365; t = 30/365; Strike = 40;

r = 10%

70.1$

)40(3065.363794.

)()()(

)2466)(.10(.

21

C

C

rt

C

O

eO

eEXdNPdNO

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.

Implied V

ola

tility

(%)

Nasdaq (VXN)

S&P (VIX)

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

Binomial vs Black-Scholes

Binomial vs Black-Scholes

Example:

Price = 36; = .40; t = 90/365; t = 30/365; Strike = 40;

r = 10%

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.

valuing options

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