Transcript OPTION PRICING MODEL

BLACK-SCHOLES
OPTION PRICING MODEL
• Chapters 7 and 8
BOPM and the B-S OPM
– The BOPM for large n is a practical, realistic
model.
– As n gets large, the BOPM converges to the BS OPM.
• That is, for large n the equilibrium value of a call
derived from the BOPM is approximately the
same as that obtained by the B-S OPM.
– The math used in the B-S OPM is complex but
the model is simpler to use than the BOPM.
B-S OPM Formula
• B-S Equation:
X
C  S0 N ( d1 )  RT N ( d 2 )
e
2
ln( S0 / X )  ( R .5 )T
d1 
 T
*
0
d 2  d1  
T
Terms:
– T = time to expiration, expressed as a
proportion of the year.
– R = continuously compounded annual RF rate.
– R = ln(1+Rs), Rs = simple annual rate.
–  = annualized standard deviation of the
– logarithmic return.
– N(d) = cumulative normal probabilities.
N(d) term
• N(d) is the probability that deviations less
than d will occur in the standard normal
distribution. The probability can be looked
up in standard normal probability table (see
JG, p.217) or by using the following:
N(d) term
n( d )  1.5[1.196854( d )
.115194( d ) 2 .000344( d ) 3
.019527( d ) 4 ] 4
N ( d )  n( d ); d  0
N ( d )  1  n( d ); d  0
B-S Features
• Model specifies the correct relations
between the call price and the explanatory
variables:
C  f ( S , X , T , R,  )
*
0





Arbitrage Portfolio
• The B-S equation is equal to the value of
the replicating portfolio:
C0*  H0* S0  B0*
where:
H0*  N (d1 )
X
B  RT N (d 2 )
e
*
0
Arbitrage Portfolio
• The replicating portfolio in our example consist
of buying .4066 shares of stock, partially
financed by borrowing $15.42:
C0*  H0* S0  B0*  .4066($45)  $15.42  $2.88
where:
H0*  N (d1 )  .4066
X
$50
B  RT N (d 2 )  (.06)(.25) (.3131)  $15.42
e
e
*
0
Arbitrage Portfolio
• If the price of the call were $3.00, then an
arbitrageur should go short in the overpriced call
and long in the replicating portfolio, buying
.4066 shares of stock at $45 and borrowing
$15.42.
• Since the B-S is a continuous model, the
arbitrageur would need to adjust the position
frequently (every day) until it was profitable to
close. For an example, see JG: 222-223.
Dividend Adjustments: PseudoAmerican Model
• The B-S model can be adjusted for dividends
using the pseudo-American model. The model
selects the maximum of two B-S-determined
values:
C0A  Max[C ( Sd , t * , X  D), C ( Sd , T , X )]
Where:
Sd  S0 
D
Rt *
e
t *  ex  dividend .. time.
Dividend Adjustments: Continuous
Dividend-Adjustment Model
• The B-S model can be adjusted for dividends using the
continuous dividend-adjustment model.
• In this model, you substitute the following dividendadjusted stock price for the current stock price in the B-S
formula:
S0
Sd  T
e
where:   annual dividend yield
DA

S0
Black-Scholes Put Model
X
P   S0 (1  N (d1 ))  RT N (d 2 )
e
Example: ABC 50 Put:
*
0
P0*   (.5934) $45  $50 e  (.06)(.25) (.6869)  $7.13
B-S Put Model’s Features
• The model specifies the correct relations
between the put price and the explanatory
variables:
*
0
P
F
I
 fG
, X , T , R, J
S
H
K





• Note: Unlike the call model, the put model is unbound.
Arbitrage Portfolio
• The B-S put equation is equal to the value
of the replicating portfolio:
P0*  H0* S0  I 0*
where:
H0*   [1  N (d1 )]
I
*
0
X
 RT [1  N (d 2 )
e
Arbitrage Portfolio
• The replicating portfolio in our example consist
of selling .5934 shares of stock short at $45 and
investing $33.83 in a RF security:
P0*  H0* S0  I 0*  .5934($45)  $33.83  $7.13
where:
H0*  1[1  N (d1 )]  1[1  .4066]  .5934
X
$50
B  RT [1  N (d 2 )]  (.06)(.25) (.6869)  $33.83
e
e
*
0
Dividend Adjustments
• B-S put model can be adjusted for dividends by using the
continuous dividend-adjustment model where Sd  S0 eT is
substituted for So. A pseudo-American model can also be used.
This model for puts is similar to calls, selecting the maximum
of two B-S-determined values:
P0a  Max[ P( Sd , t * , X ), P( Sd , T , X )]
D
where: Sd  S0  Rt *
e
Note: X is used instead of X  D
Barone-Adesi and Whaley Model
• The pseudo-American model estimates the value of an American
put in reference to an ex-dividend date. When dividends are not
paid (and as a result, we do not have a specific reference date)
the model cannot be applied.
• This is not a problem with applying the pseudo model to calls,
since the advantage of early exercise applies only when an exdividend date exist.
• As we saw with the BOPM for puts, early exercise can
sometimes be profitable, even when there is not a dividend.
• A model that addresses this problem and can be used to price
American puts, as well as calls, is the Barone-Adesi Whaley
(BAW) model. See JG: 246-248.
Estimating the B-S Model:
Implied Variance
– The only variable to estimate in the B-S OPM
(or equivalently, the BOPM with large n) is the
variance. This can be estimated using historical
averages or an implied variance technique.
– The implied variance is the variance which
makes the OPM call value equal to the market
value. The software program provided each
student calculates the implied variance.
Estimating the B-S Model:
Implied Variance
• For at-the-money options, the implied
variance can be estimated using the
following formula:
.5(C0  P0 ) 2 / T
 
T
X (1  R)
B-S Empirical Study
• Black-Scholes Study (1972): Black and Scholes
conducted an efficient market study in which
they simulated arbitrage positions formed when
calls were mispriced (C* not = to Cm).
• They found some abnormal returns before
commission costs, but found they disappeared
after commission costs.
• Galai found similar results.
MacBeth-Merville Studies
• MacBeth and Merville compared the prices
obtained from the B-S OPM to observed
market prices. They found:
– the B-S model tended to underprice in-themoney calls and overprice out-of-the money
calls.
– the B-S model was good at pricing on-themoney calls with some time to expiration.
Bhattacharya Studies
• Bhattacharya (1980) examined arbitrage
portfolios formed when calls were
mispriced, but assumed the positions were
closed at the OPM values and not market
prices.
• Found: B-S OPM was correctly specified.
General Conclusion
• Empirical studies provide general support
for the B-S OPM as a valid pricing model,
especially for near-the-money options.
• The overall consensus is that the B-S OPM
is a useful model.
• Today, the OPM may be the most widely
used model in the field of finance.
Uses of the B-S Model
• Identification of mispriced options
• Generating profit tables and graphs for different
time periods, not just expiration.
• Evaluation of time spreads.
• Estimating option characteristics:
– Expected Return, Variance, and Beta
– Option’s Price sensitivity to changes in S, T, R, and
variability.
Expected Return and Risk
• Recall, the value of a call is equal to the value of
the RP. The expected return, standard deviation,
and beta on a call can therefore be defined as the
expected return, standard deviation, and beta on a
portfolio consisting of the stock and risk-free
security (short):
E ( Rc )  ws E ( Rs )  wR Rf
 ( Rc )  ws ( Rs )
 c  ws  s
Expected Return and Risk
• In term of the OPM, the total investment in the RP
is equal to the call price, the investment in the
stock is equal HoSo, and the investment in the RF
security is -B. Thus:
H0 S0
B0
E ( Rc ) 
E ( Rs ) 
Rf
C0
C0
H0 S0
 ( Rc ) 
 ( Rs )
C0
H0 S 0
c 
s
C0
Expected Return and Risk
for Puts
P
0 0
H S
I0
E ( Rp ) 
E ( Rs ) 
Rf
P0
P0
 ( Rp ) 
F
H S I
 (R )
G
J
HP K
p
0 0
2
s
0
p
0 0
H S
p 
s
P0
2
Delta, Gamma, and Theta
• Delta is a measure of an option’s price
sensitivity to a small change in the stock
price.
Call:  
–
–
–
–
–
C
P
 N (d1 ) Put:  
 N ( d1 )  1
S
S
Delta is N(d1) for calls and ranges from 0 to 1.
Delta is N(d1) - 1 for puts and ranges from -1 to 0.
Delta for the call in the example is .4066
Delta for the put in the example is .5934.
Delta changes with time and stock prices changes.
Delta, Gamma, and Theta
• Theta is the change in the price of an option with
respect to a change in the time to expiration.
–
–
–
–
C
P
Call:   
Put:  
Theta is a measure of the option’s
time decay. T
T
Theta is usually defined as the negative of the partial of the option price
with respect to T.
Interpretation: An option with a theta of 7 would find for a 1% decrease in
the time to expiration (2.5 days), the option would lose 7% in value.
For formulas for estimating theta, see JG: 258-259.
Delta, Gamma, and Theta
• Gamma measures the change in the option’s
delta for a small change in the price of the
stock. It is the second derivative of the option
with respect to a change in the stock price.
 2 C d
Call:   2 
S
dS
2 P d
Put:   2 
S
dS
• For formulas for estimating gamma, see JG:
258-259.
Position Delta, Gamma, and
Theta
• The description of call and put options in terms
of their delta, gamma, and theta values can be
extended to option positions.
• For example, consider an investor who
purchases n1 calls at C1 and n2 calls on another
call option on the same stock at a price of C2.
• The value of the portfolio (V) is
V  n1C1  n2 C2
Position Delta, Gamma, and
Theta
• The call prices are a function of S, T, variability,
and Rf. Taking the partial derivative of V with
respect to S yields the position delta:
F
I
F
I
G
J
G
J
HK HK
V
C1
C2
 n1
 n2
S
S
S
 p  n11  n2 2
Position Delta, Gamma, and
Theta
• The position delta measure the change in the position’s
value in response to a small change in the stock price.
• By setting the position delta equal to zero and solving
for n1 in terms of n2 a neutral position delta can be
constructed with a value invariant to small changes in
the stock price.
0  n1 1  n2  2
n1
F
 I
  G Jn
H K
F
N (d ) I
 G
n
J
HN (d ) K
2
2
1
n1
1 2
2
1 1
Position Delta, Gamma, and
Theta
• The position theta is obtained by taking the
partial derivative of V with respect to T:
F
I
F
I
G
J
G
J
HK HK
V
C1
C2
 n1
 n2
T
T
T
 p  n1 1  n2 2
Position Delta, Gamma, and Theta
• The position gamma is obtained by taking the
derivative of the position delta respect to S:
 p
1
 2
p  n11  n2 2
S
F
I
F
I
 n G J n G J
HS K HS K
1
2
• Strategy: For a neutral position delta with a positive position
gamma, the value of the position will decrease for small changes
in the stock price and increase for large increases or decreases in
the stock price.
• Strategy:For a neutral position delta with a negative position
gamma, the value of the position will increase for small changes in
the stock price and decrease for large increases or decreases in the
stock price.