lecture 16. option value components of the option price 1 - underlying stock price 2 - striking or...
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DerivativesLecture 16
Option Value
Components of the Option Price1 - Underlying stock price2 - Striking or Exercise price3 - Volatility of the stock returns (standard deviation of
annual returns)4 - Time to option expiration5 - Time value of money (discount rate)
Option Value
)()()( 21 EXPVdNPdNOC
Black-Scholes Option Pricing Model
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
)()()( 21 EXPVdNPdNOC
Black-Scholes Option Pricing Model
rteEXEXPV )(
factordiscount gcompoundin continuous1
rt
rt
ee
N(d1)=
tv
trd
vEXP )()ln( 2
1
2
Black-Scholes Option Pricing Model
Cumulative Normal Density Function
tv
trd
vEXP )()ln( 2
1
2
tvdd 12
Cumulative Normal Density Function
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
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
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
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
.3070 = .3
= .00
= .007
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
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
Call Option
$ 1.70
36 40 41.70
Example
What is the price of a call option given the following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
Call Option
(d1) =
ln + ( .1 + ) 30/36541
40
.422
2
.42 30/365
(d1) = .3335 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
Call Option
(d1) =
ln + ( .1 + ) 30/36541
40
.422
2
.42 30/365
(d1) = .3335 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) = .2131
N(d2) = .5844
(d2) = d1 - v t = .3335 - .42 (.0907)
Call OptionExample
What is the price of a call option given the following?
P = 41 r = 10% v = .42
EX = 40 t = 30 days / 365
Call Option
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
Call Option
$ 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
Call Option
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
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
Call Option Q: How do we lock in a profit? A: Sell the Call
$ 1.70
40 41
$ 2.67
Call Option Q: How do we lock in a profit? A: Sell the Call
$ 1.70
40 41
$ 2.67
$ 0.97
Call Option Q: How do we lock in a profit? A: Sell the Call
Call Option
$ 1.70
40 41
$ 2.67
$ 0.97
Q: How do we lock in a profit? A: Sell the Call
Put OptionBlack-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
Put Option
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 ParityPut = Call + EXe-rt - Ps
Put = 2.67 + 40e-.10(.0822) - 41
Put = 42.34 - 41 = 1.34
Put OptionExample
What is the price of a call option given the following?
P = 41 r = 10% v = .42
EX = 40 t = 30 days / 365
Put OptionPut-Call Parity & American PutsPs - 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