Post on 25-Nov-2015
description
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
Simple Option Valuation
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
= $64.13
Simple Option Valuation
Since the Google call option is equal to a leveraged
position in 4/7 shares, the option delta can be computed
as follows.
Option delta, or hedge ratio, is the number of shares
needed to replicate one call.
7
4
83.250
33.143
50.32233.573
033.143
prices share possible of spread
pricesoption possible of spread DeltaOption
Simple Option Valuation
Risk-Neutral Valuation. If we are risk neutral, the
expected return on Google call options is 1.5%.
Accordingly, we can determine the probability of a rise in
the stock price as follows.
The Google option can then be valued based on the
following method.
.4543 rise ofy Probabilit
.015 Return Expected
)25(rise ofy probabilit133.33 rise ofy probabilit Return Expected
15.64$
015.1/11.65
)]0546(.)33.1434543[(.
}0rise ofy probabilit133.143rise ofy probabilit { ueOption val
PV
PV
Simple Option Valuation
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
Simple Option Valuation
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
Simple Option Valuation
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
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
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
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
Black-Scholes Option Pricing Model
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
Black-Scholes Option Pricing Model
)()()( 21 EXPVdNPdNOC
t
trd EX
P
)()ln(2
1
2
tdd 12
Black-Scholes Option Pricing Model
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
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
N(d1) = .5774 N(d2) = .4632
Also,
04.52$
015.1/)430(4632.4305774.
)()()( 21
C
C
C
O
O
EXPVdNPdNO
rteEXEXPV )(
Call Option Black-Scholes Model
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.