Lecture 16
Components of the Option Price
1 - 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)
Black-Scholes Option Pricing Model
OC  N (d1 )  P  N (d 2 )  PV ( EX )
OC  N (d1 )  P  N (d 2 )  PV ( EX )
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
OC  N (d1 )  P  N (d 2 )  PV ( EX )
PV ( EX )  EX  e
e
 rt
 rt

1
 rt  continuous compoundin g discount factor
e
Black-Scholes Option Pricing Model
d1 
ln(
P
EX
)  (r 
v t
v2
2
)t
N(d1)=
d1 
ln(
P
EX
)  (r 
v t
d 2  d1  v t
v2
2
)t
Cumulative Normal Density Function
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
d1 
ln(
d1  .2297
P
EX
)  (r 
v t
v2
2
)t
N (d1 )  .5908
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
d 2  d1  v t
d 2  .0580
N (d 2 )  1  .5231  .4769
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

 .5908  80  .4769  (80)e
OC  N (d1 )  P   N (d 2 )  ( EX )e  rt
OC
OC  $10.05

 (.05)(.5 )

Example
What is the price of a call option given the following?
P = 36
r = 10%
v = .40
EX = 40
t = 90 days / 365
d1 
ln(
d1  .3070
P
EX
)  (r 
v t
v2
2
)t
N (d1 )  1  .6206  .3794
.3070 = .3
= .00
= .007
Example
What is the price of a call option given the following?
P = 36
r = 10%
v = .40
EX = 40
t = 90 days / 365
d 2  d1  v t
d 2  .5056
N ( d 2 )  1  .6935  .3065
Example
What is the price of a call option given the following?
P = 36
r = 10%
v = .40
EX = 40
t = 90 days / 365
OC
OC
 rt



 N ( d1 )  P  N ( d 2 )  ( EX )e 
 .3794  36  .3065  ( 40)e (.10)(.2466) 
OC  $1.70
Example
What is the price of a call option given the following?
P = 36
r = 10%
v = .40
EX = 40
t = 90 days / 365
$ 1.70
36
40
41.70
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365
41
(d1) =
.422
ln
+ ( .1 +
) 30/365
40
2
.42
(d1) = .3335
30/365
N(d1) =.6306
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365
41
(d1) =
.422
ln
+ ( .1 +
) 30/365
40
2
.42
(d1) = .3335
30/365
N(d1) =.6306
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365
(d2) = d1 - v
(d2) = .2131
N(d2) = .5844
t
= .3335 - .42 (.0907)
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365
OC = Ps[N(d1)] - S[N(d2)]e-rt
OC = 41[.6306] - 40[.5844]e - (.10)(.0822)
OC = $ 2.67
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365
$ 1.70
40
41 41.70
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365

Intrinsic Value = 41-40 = 1

Time Premium = 2.67 + 40 - 41 = 1.67

Profit to Date = 2.67 - 1.70 = .94

Due to price shifting faster than decay in
time premium


Q: How do we lock in a profit?
A: Sell the Call
$ 1.70
40 41


Q: How do we lock in a profit?
A: Sell the Call
$ 2.67
$ 1.70
40 41


Q: How do we lock in a profit?
A: Sell the Call
$ 2.67
$ 0.97
$ 1.70
40 41


Q: How do we lock in a profit?
A: Sell the Call
$ 2.67
$ 0.97
$ 1.70
40 41
Black-Scholes
Op = EX[N(-d2)]e-rt - Ps[N(-d1)]
Put-Call Parity (general concept)
Put Price = Oc + EX - P - Carrying Cost + D
Carrying cost = r x EX x t
Call + EXe-rt = Put + Ps
Put = Call + EXe-rt - Ps
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365
N(-d1) = .3694
N(-d2)= .4156
Black-Scholes
Op = EX[N(-d2)]e-rt - Ps[N(-d1)]
Op = 40[.4156]e-.10(.0822) - 41[.3694]
Op = 1.34
Example
What is the price of a call option given the following?
P = 41
r = 10%
v = .42
EX = 40
t = 30 days / 365
Put-Call Parity
Put = Call + EXe-rt - Ps
Put = 2.67 + 40e-.10(.0822) - 41
Put = 42.34 - 41 = 1.34
Put-Call Parity & American Puts
Ps - EX < Call - Put < Ps - EXe-rt
Call + EX - Ps > Put > EXe-rt - Ps + call
Example - American Call
2.67 + 40 - 41 > Put > 2.67 + 40e-.10(.0822) - 41
1.67 > Put > 1.34
With Dividends, simply add the PV of dividends