Black Scholes OPM
Essentials of
Black - Scholes option
Investment
200113008
200512248
200512326
200612401
권용현
윤희욱
주윤하
이정협
Option valuation : intro
Factor influencing option price
Assumptions of B-S model
Black-Scholes OPM
Put-call parity relationship
Put value by B-S OPM
AGENDA
B-S OPM in Korea
Black Scholes OPM
Option Valuation
Essentials of
Investment
Intrinsic value
profit that could be made if the option was
immediately exercised.
Call: stock price -exercise price
Put: exercise price -stock price
Time value
the difference between the option price and
the intrinsic value.
Black Scholes OPM
Option Valuation
Volatility value
Essentials of
The entire value of an out-of-the-money (forward)
option. With basis value added, the sum is the
premium over intrinsic value for an in-the-money
option
Investment
Adjusted intrinsic value
the stock price minus the present value of the
exercise prices,
So-PV(X)
Black Scholes OPM
Option Valuation
Call Option Value before expiration
Option Value
Essentials of
Investment
Value of call option
Time Value
Value of option if now at
expiration = intrinsic value
x
S0
Out of the money
In the money
Black Scholes OPM
Factors Influencing Option Values: Calls
Essentials of
Investment
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Dividend Rate
Effect on value
increases
decreases
increases
increases
increases
decreases
Black Scholes OPM
Assumptions
1.
Essentials of
Investment
2.
3.
4.
5.
6.
The stock pays no dividends during the
option's life
European exercise terms are used
Markets are efficient
No commissions are charged
Interest rates remain constant and
known
Returns are lognormally distributed
Black Scholes OPM
Black-Scholes Option Valuation
Co = SoN(d1) - Xe-rTN(d2)
d1= [ln(So/X) + (r + 2/2)T] / (T1/2)
Essentials of
Investment
d2=d1 - (T1/2)
Black Scholes OPM
Black-Scholes Option Valuation
C = Current call option value.
S = Current stock price
N(d) = probability that a random draw from a normal dist.
will be less than d.
Essentials of
Investment
X = Exercise price
e = 2.71828, the base of the natural log
r = Risk-free interest rate (annualizes continuously
compounded with the same maturity as the option)
T = time to maturity of the option in years
ln= Natural log function
Standard deviation of annualized cont. compounded
rate of return on the stock
Black Scholes OPM
Call Option Example
So= 100
r = .10
σ = .50
X = 95
T = .25 (quarter)
Essentials of
Investment
d1= [ln(100/95) + (.10+(.5 2/2))] / (.5 .251/2)
= .43
d2= .43 + ((.5)( .251/2)
= .18
Black Scholes OPM
Probabilities from Normal Dist
Essentials of
Investment
N (.43) = .6664
Table 15.2
d
.42
.43
.44
N (.18) = .5714
Table 15.2
d
.16
.18
.20
N(d)
.6628
.6664 Interpolation
.6700
N(d)
.5636
.5714
.5793
Black Scholes OPM
Call Option Value
Co = SoN(d1) -Xe-rTN(d2)
Essentials of
Investment
Co = 100 X .6664 -95 e-.10 X .25 X .5714
Co = 13.70
Black Scholes OPM
The Put-Call Parity Relationship
Put prices can be derived from the prices of calls.
Essentials of
Investment
Prices of European put and call options are
linked together in an equation known as the putcall parity relationship.
Black Scholes OPM
The Put-Call Parity Relationship
C – P = S – Xe-rT
Essentials of
Investment
C = Current call option value
P = Current put option value
S0 = Current stock price
X = Exercise price
e = The base of the natural log function
r = Risk-free interest rate
T = Time remaining until expiration of option
Black Scholes OPM
Put Option Valuation:
Using Put-Call Parity
Essentials of
Investment
P = C + PV (X) - So
= C + Xe-rT - So
Using the example data
C = 13.70
X = 95 S = 100
r = .10
T = .25
P = 13.70 + 95 e -.10 X .25 -100
P = 6.35
Black Scholes OPM
Put Option Valuation
Essentials of
Investment
We can use the put-call parity relationship to
value put options once we know the call
option value.
Sometimes, it’s easier to work with a put
option formula directly.
Black Scholes OPM
Put Option Valuation
P = Xe-rT [1-N(d2)] -S0 [1-N(d1)]
Essentials of
Investment
P = Current put option value
X = Exercise price
e = The base of the natural log function
r = Risk-free interest rate
T = Time remaining until expiration of option
S0 = Current stock price
σ = Standard deviation of annualized
cont. compounded rate of return on the
stock
Black Scholes OPM
Put Value Using Black-Scholes
P = Xe-rT [1-N(d2)] -S0[1-N(d1)]
Essentials of
Investment
Using the sample call data
S = 100 r = .10 X = 95 g = .5 T = .25
95e-10x.25(1-.5714)-100(1-.6664) = 6.35
Black Scholes OPM
Essentials of
Investment
Thanks