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BLACK-SCHOLES OPTION PRICING MODEL • Chapters 7 and 8

BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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Page 1: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

BLACK-SCHOLESOPTION PRICING MODEL

BLACK-SCHOLESOPTION PRICING MODEL

• Chapters 7 and 8• Chapters 7 and 8

Page 2: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

BOPM and the B-S OPMBOPM and the B-S OPM

– The BOPM for large n is a practical, realistic model.

– As n gets large, the BOPM converges to the B-S 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.

– The BOPM for large n is a practical, realistic model.

– As n gets large, the BOPM converges to the B-S 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.

Page 3: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

B-S OPM FormulaB-S OPM Formula

• B-S Equation:• B-S Equation:

C S N dX

eN d

dS X R T

T

d d T

RT0 0 1 2

10

2

2 1

5

* ( ) ( )

ln( / ) ( . )

C S N dX

eN d

dS X R T

T

d d T

RT0 0 1 2

10

2

2 1

5

* ( ) ( )

ln( / ) ( . )

Page 4: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Terms: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.

– 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.

Page 5: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

N(d) termN(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) 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:

Page 6: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

N(d) termN(d) term

n d d

d d

d

N d n d d

N d n d d

( ) . [ . ( )

. ( ) . ( )

. ( ) ]

( ) ( );

( ) ( );

1 5 1 196854

115194 000344

019527

0

1 0

2 3

4 4

n d d

d d

d

N d n d d

N d n d d

( ) . [ . ( )

. ( ) . ( )

. ( ) ]

( ) ( );

( ) ( );

1 5 1 196854

115194 000344

019527

0

1 0

2 3

4 4

Page 7: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

B-S FeaturesB-S Features

• Model specifies the correct relations between the call price and the explanatory variables:

• Model specifies the correct relations between the call price and the explanatory variables:

C f S X T R0* ( , , , , )

C f S X T R0

* ( , , , , )

Page 8: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Arbitrage Portfolio

• The B-S equation is equal to the value of the replicating portfolio:

C H S B

where

H N d

BX

eN d

RT

0 0 0 0

0 1

0 2

* * *

*

*

:

( )

( )

Page 9: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Arbitrage Portfolio• The replicating portfolio in our example consist

of buying .4066 shares of stock, partially financed by borrowing $15.42:

C H S B

where

H N d

BX

eN d

eRT

0 0 0 0

0 1

0 2 06 25

4066 42 88

4066

3131 42

* * *

*

*(. )(. )

. ($45) $15. $2.

:

( ) .

( )$50

(. ) $15.

Page 10: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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.

Page 11: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Dividend Adjustments: Pseudo-American Model

Dividend Adjustments: Pseudo-American 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:

• 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:

C Max C S t X D C S T X

Where

S SD

e

t ex dividend time

Ad d

d Rt

0

0

[ ( , , ), ( , , )]

:

.. .

*

*

*

C Max C S t X D C S T X

Where

S SD

e

t ex dividend time

Ad d

d Rt

0

0

[ ( , , ), ( , , )]

:

.. .

*

*

*

Page 12: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Dividend Adjustments: Continuous Dividend-Adjustment Model

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 dividend-adjusted stock price for the current stock price in the B-S formula:

• The B-S model can be adjusted for dividends using the continuous dividend-adjustment model.

• In this model, you substitute the following dividend-adjusted stock price for the current stock price in the B-S formula:

SS

ewhere annual dividend yield

D

S

d T

A

0

0

:

Page 13: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Black-Scholes Put Model

P S N dX

eN d

Example ABC Put

P e

RT0 0 1 2

006 25

1

50

5934 6869 13

*

* (. )(. )

( ( )) ( )

: :

(. ) $45 $50 (. ) $7.

Page 14: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

B-S Put Model’s FeaturesB-S Put Model’s Features

• The model specifies the correct relations between the put price and the explanatory variables:

• Note: Unlike the call model, the put model is unbound.

• The model specifies the correct relations between the put price and the explanatory variables:

• Note: Unlike the call model, the put model is unbound.

P f S X T R0* , , , ,FHG

IKJ

Page 15: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Arbitrage Portfolio

• The B-S put equation is equal to the value of the replicating portfolio:

P H S I

where

H N d

IX

eN d

RT

0 0 0 0

0 1

0 2

1

1

* * *

*

*

:

[ ( )]

[ ( )

Page 16: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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:

P H S I

where

H N d

BX

eN d

eRT

0 0 0 0

0 1

0 2 06 25

5934 83 13

1 1 1 1 4066 5934

1 6869 83

* * *

*

*(. )(. )

. ($45) $33. $7.

:

[ ( )] [ . ] .

[ ( )]$50

(. ) $33.

Page 17: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Dividend AdjustmentsDividend Adjustments

• B-S put model can be adjusted for dividends by using the continuous dividend-adjustment model where 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:

• B-S put model can be adjusted for dividends by using the continuous dividend-adjustment model where 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:

P Max P S t X P S T X

where S SD

e

ad d

d Rt

0

0

[ ( , , ), ( , , )]

:

*

*

Note X is used instead of X D:

S S edT

0

Page 18: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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 ex-dividend 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.

Page 19: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Estimating the B-S Model: Implied Variance

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.

– 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.

Page 20: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Estimating the B-S Model: Implied Variance

Estimating the B-S Model: Implied Variance

• For at-the-money options, the implied variance can be estimated using the following formula:

• For at-the-money options, the implied variance can be estimated using the following formula:

. ( ) /

( )

5 2

10 0C P T

X R T

Page 21: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

B-S Empirical StudyB-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.

• 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.

Page 22: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

MacBeth-Merville StudiesMacBeth-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-the-

money calls and overprice out-of-the money calls.

– the B-S model was good at pricing on-the-money calls with some time to expiration.

• 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-the-

money calls and overprice out-of-the money calls.

– the B-S model was good at pricing on-the-money calls with some time to expiration.

Page 23: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Bhattacharya StudiesBhattacharya 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.

• 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.

Page 24: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

General ConclusionGeneral 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.

• 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.

Page 25: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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.

Page 26: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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 R w E R w Rc s s R f( ) ( )

( ) ( )R w Rc s s c s sw

Page 27: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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:

E RH S

CE R

B

CRc s f( ) ( ) 0 0

0

0

0

( ) ( )RH S

CRc s 0 0

0

c s

H S

C 0 0

0

Page 28: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Expected Return and Riskfor Puts

E RH S

PE R

I

PRp

P

s f( ) ( ) 0 0

0

0

0

( ) ( )RH S

PRp

p

sFHG

IKJ

0 0

0

2

2

p

p

s

H S

P 0 0

0

Page 29: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Delta, Gamma, and Theta

• Delta is a measure of an option’s price sensitivity to a small change in the stock price.

– 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.

CallC

SN d: ( )

1 PutP

SN d: ( )

1 1

Page 30: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Delta, Gamma, and Theta

• Theta is the change in the price of an option with respect to a change in the time to expiration.

– Theta is a measure of the option’s time decay.

– 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.

CallC

T:

PutP

T:

Page 31: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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.

• For formulas for estimating gamma, see JG: 258-259.

CallC

S

d

dS:

2

2 PutP

S

d

dS:

2

2

Page 32: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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 n C n C 1 1 2 2

Page 33: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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:

FHGIKJ

FHGIKJ

V

Sn

C

Sn

C

S11

22

p n n 1 1 2 2

Page 34: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

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 1 1 2 2

12

12

11 2

1 12

FHGIKJ

FHG

IKJ

n n

n n

nN d

N dn

( )

( )

Page 35: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Position Delta, Gamma, and Theta

• The position theta is obtained by taking the partial derivative of V with respect to T:

FHGIKJ

FHGIKJ

V

Tn

C

Tn

C

T11

22

p n n 1 1 2 2

Page 36: BLACK-SCHOLES OPTION PRICING MODEL Chapters 7 and 8

Position Delta, Gamma, and Theta

• The position gamma is obtained by taking the derivative of the position delta respect to S:

• 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.

FHGIKJ

FHGIKJ

p

Sn

Sn

S11

22 p n n 1 1 2 2