BOPM FOR PUTSAND THE DIVIDEND-

ADJUSTED BOPM

Chapter 6

Binomial Stock Movement

S uSu 0 11. ($100) $110S uSu 0 11. ($100) $110

S0S0

S dSd 0 95. ($100) $95S dSd 0 95. ($100) $95

AssumptionAssumption

• Assume there is a European put option on the stock that expires at the end of the period.

• Example: X = $100.

• Assume there is a European put option on the stock that expires at the end of the period.

• Example: X = $100.

Binomial Put Movement

P0

P Max X Su u [ , ] [ , ]0 100 110 0 0

P Max X Sd d [ , ] [ , ]0 100 95 0 5

Replicating portfolioReplicating portfolio

• The replicating portfolio consist of buying Ho shares of the stock at So and investing Io dollars.

• The replicating portfolio consist of buying Ho shares of the stock at So and investing Io dollars.

H S IP0 0 0

H uS I rPf0 0 0

H dS I rPf0 0 0

Constructing the RPConstructing the RP

• The RP is formed by solving for the Ho and Bo values (Ho* and Io*) which make the two possible values of the replicating portfolio equal to the two possible values of the put (Pu and Pd).

• Mathematically, this requires solving simultaneously for the Ho and Io which satisfy the following two equations.

• The RP is formed by solving for the Ho and Bo values (Ho* and Io*) which make the two possible values of the replicating portfolio equal to the two possible values of the put (Pu and Pd).

• Mathematically, this requires solving simultaneously for the Ho and Io which satisfy the following two equations.

Solve for Ho and Io where:Solve for Ho and Io where:

• Equations:• Equations:

H uS I r P

H dS I r P

oP

f u

oP

f d

0 0

0 0

SolutionSolution

• Equations:• Equations:

HP P

uS dS

IP dS P uS

r uS dS

P u d

u d

f

00 0

00 0

0 0

*

* [ ( ) ( )]

[ ]

Note P P H Short Stock Position

I RF Investment

d uP: *

*

0

0

0

0

• Equations:• Equations:

H

I

P0

0

0 5

110 953333

0 95 5 110

105 110 9534 9206

*

*

.

[ ( ) ( )]

. [ ].

Example

Equilibrium Put PriceEquilibrium Put Price

• Law of One Price:• Law of One Price:

P H S I

Example

P

P0 0 0 0

0 3333 9206 59

* * *

*

:

. ($100) $34. $1.

Arbitrage Arbitrage

• The equilibrium price of the put is governed by arbitrage.

• If the market price of the put is above (below) the equilibrium price, then an arbitrage can be exploited by going short (long) in the put and long (short) in the replicating portfolio. For an example, see JG: 185-186.

• The equilibrium price of the put is governed by arbitrage.

• If the market price of the put is above (below) the equilibrium price, then an arbitrage can be exploited by going short (long) in the put and long (short) in the replicating portfolio. For an example, see JG: 185-186.

Alternative Equation Alternative Equation

• By substituting the expressions for Ho* and Io* into the equation for Po*, the equation for the equilibrium put price can be alternatively expressed as:

• By substituting the expressions for Ho* and Io* into the equation for Po*, the equation for the equilibrium put price can be alternatively expressed as:

Pr

pP p P

where

pr d

u d

fu d

f

0

11* [ ( ) ]

:

Put-Call Parity

• The value obtained using the binomial put model is consistent with put-call parity.

P S C PV X

P C PV X S

Example

P

( )

( )

:

..

.6 35100

105100 159

Multiple-Period Model ExampleMultiple-Period Model Example

• From Previous example for call:– u = 1.0488, d = .9747,– So = $100, n = 2,– Rf = .025,

• To price a European put with X = 100, use the same recursive procedure we used for a call option.

• From Previous example for call:– u = 1.0488, d = .9747,– So = $100, n = 2,– Rf = .025,

• To price a European put with X = 100, use the same recursive procedure we used for a call option.

110$02 SuSuu

110$02 SuSuu

S udSud 0 23$102.S udSud 0 23$102.

S d Sdd 20 $95S d Sdd 2

0 $95

S uSu 0 87$104.S uSu 0 87$104.

S dSd 0 47$97.S dSd 0 47$97.

S0 $100S0 $100

S

P

H

I

u

u

up

u

104 88

0

0

0

.

S

P

H

I

d

d

dp

d

97 47

156

6916

68 97

.

.

.

.

S

P Maxuu

uu

110

100 110 0

0

[ , ]

S

P Maxud

ud

102 23

100 102 23 0

0

.

[ . , ]

S

P Maxdd

dd

95

100 95 0

5

[ , ]

S

P

H

I

p

0

0

0

0

100

0 49

2105

2154

.

.

.

European Put Value

Put-Call Parity

• The value obtained using the binomial put model is consistent with put-call parity.

P S C PV X

P C PV X S

Example

P

( )

( )

:

.( . )

.531100

1025100 49

2

American Put Value

• Note: At Sd = 97.47, Pd = 1.56. If the put were American and priced at this value, then the put holder would find it advantageous to exercise the put: IV = 100-97.47 = 2.53.

• The BOPM for a put can be adjusted to value American puts by constraining the price of the put at each node to be the maximum of either its BOPM value or its IV:

P Max P IVta

t [ , ]

S

P Max P X uS

P Max

u

ua

u

ua

104 88

0 100 110 0

0

.

[ , ]

[ , ]

S

P Maxuu

uu

110

100 110 0

0

[ , ]

S

P Maxud

ud

102 23

100 102 23 0

0

.

[ . , ]

S

P Maxdd

dd

95

100 95 0

5

[ , ]

S

P0

0

100

0 79

.

American Put Value

S

P Max P X dS

P Max

d

da

d

da

97 47

156 100 97 47 2 53

0

.

[ , ]

[ . , . ] .

Point: Arbitrage StrategyPoint: Arbitrage Strategy

• The arbitrage strategies underlying the multi-period put model are similar to the multiple-period call model, requiring possible readjustments in subsequent periods.

• For a discussion of multiple-period arbitrage strategies, see JG, p 191.

• The arbitrage strategies underlying the multi-period put model are similar to the multiple-period call model, requiring possible readjustments in subsequent periods.

• For a discussion of multiple-period arbitrage strategies, see JG, p 191.

Dividend AdjustmentDividend Adjustment

• If a dividend is paid and the ex-dividend date occurs at the end of any of the periods, then the price of the stock will fall. The price decrease will cause the call price to fall and the put price to increase.

• The dividend may also make the early exercise of a call profitable, making an American call more valuable than a European.

• If a dividend is paid and the ex-dividend date occurs at the end of any of the periods, then the price of the stock will fall. The price decrease will cause the call price to fall and the put price to increase.

• The dividend may also make the early exercise of a call profitable, making an American call more valuable than a European.

Adjustments for Dividends and American Call Options

Adjustments for Dividends and American Call Options

– The BOPM can be adjusted for dividends by using a dividend-adjusted stock price (stock price just before ex-dividend date minus dividend) on the ex-dividend dates in calculating the option prices.

– The BOPM can be adapted to price an American call by constraining the price at each node to be the maximum of the binomial value or the IV.

– The BOPM can be adjusted for dividends by using a dividend-adjusted stock price (stock price just before ex-dividend date minus dividend) on the ex-dividend dates in calculating the option prices.

– The BOPM can be adapted to price an American call by constraining the price at each node to be the maximum of the binomial value or the IV.

Single-Period BOPM

• Assume there is an ex-dividend date at the end of the period, with the value of the dividend being D.

• Let uSo and dSo be the stock prices just before the ex-dividend date.

• As shown in the Figure, the dividend does not affect the form of the replicating portfolio, but it does lower the two possible call prices which changes H, B, and Co.

RP H uS D H D r B

H uS r B

C IV Max uS D X

u f

f

ux

0 0 0 0

0 0 0

0 0

( )

[ , ]

RP H dS D H D r B

H dS r B

C IV Max dS D X

d f

f

dx

0 0 0 0

0 0 0

0 0

( )

[ , ]

RP H S B0 0 0 0

Solve for Ho and Bo where:Solve for Ho and Bo where:

• Equations:• Equations:

H uS B r C

H dS B r C

f ux

f dx

0 0 0

0 0 0

• Assuming D = 1• Assuming D = 1

C C

H

B

C H S B

ux

dx

9 0

9 0

110 956

9 95 0 110

105 110 9554 2857

6 2857 71

0

0

0 0 0 0

,

.

( ) ( )]

. [ ].

. ($100) $54. $5.

*

*

* *

Example

Multiple-Period BOPM

• Let uSo, dSo, uuSo, udSo, and ddSo be the stock prices just before the ex-dividend date.

• Assume the dividend at end of Period 2 is $1 and the dividend at the end of Period 1 is $1.

• Assume: n = 2, u = 1.0487, d = .9747, Rf =.025.

• As shown in the Figure, H and B values reflect the ex-dividend call values.

• Note: In the formulas for H and B, the stock prices just before the ex-dividend date are used.

S

H

B

C

ux

u

u

u

104 88 1 10388

9 123

110 102 231

9 102 23 123 110

1025 110 102 2398 54

1 10388 98 54 5 34

. .

.

.( . ) . ( )

. ( . ).

( . ) . .

S

C Max

uux

uux

110 1 109

109 100 0

9

[ , ]

European Call Value

with dividends

S

C Max

udx

udx

102 23 1 10123

10123 100 0

123

. .

[ . , ]

.

S

C Max

ddx

udx

95 1 94

94 100 0

0

[ , ]

S

H

B

C

dx

d

d

d

97 47 1 96 47

123 0

102 23 9517 01

123 95 0 102 23

1025 102 23 9515 77

1701 96 47 15 77 64

. .

.

..

. ( ) ( . )

. ( . ).

. ( . ) . $0.

S

H

B

C

0

0

0

0

100

534 0 64

104 88 97 476343

534 97 47 0 64 104 88

1025 104 88 97 4759 69

6343 69 374

*

*

*

. .

. ..

. ( . ) . ( . )

. ( . . ).

. ($100) $59. .

American Call Price on Dividend-

Paying Stock

• For American call options on dividend-paying stocks, early exercise can be incorporated into the BOPM by constraining each possible call value to be the maximum of either its European value as determine by the BOPM or the IV just prior to the ex-dividend date.

• In our previous case, the $1 dividend in Period 1 would not make early exercise profitable if the call were American.

• As shown in the next Figure, if the dividend in Period 2 were $2 instead of $1, then there is an early exercise advantage at the top node in Period 1. The price of an American call in this case is 3.38, compared to a European value of $3.02.

S

C

C Max C uS X

Max

ux

u

ua

u

104 88 2 102 88

1 102 88 98 54 4 34

4 34104 88 100

4 88

0

. .

( . ) . .

[ , ]

[ . , . ]

.

S

C Max

uux

uux

110 1 109

109 100 0

9

[ , ]

American Call Value with

dividends D D: ,1 21 2

S

C Max

udx

udx

102 23 1 10123

10123 100 0

123

. .

[ . , ]

.

S

C Max

ddx

udx

95 1 94

94 100 0

0

[ , ]

S

H

B

Ca

0

0

0

0

100

4 88 0 47

104 88 97 475951

4 88 97 47 0 47 104 88

1025 104 88 97 475613

5951 13 338

*

*

*

. .

. ..

. ( . ) . ( . )

. ( . . ).

. ($100) $56. .

S

C

C Max C dS X

Max

dx

d

da

d

97 47 2 95 47

1701 95 47 15 77 0 47

0 47 97 47 100

0 47

0

. .

. ( . ) . .

[ , ]

[ . , . ]

.

European

H

B

Ce

:

. .

. ..

. ( . ) . ( . )

. ( . . ).

. ($100) $49. .

*

*

*

0

0

0

4 34 0 47

104 88 97 475223

4 34 97 47 0 47 104 88

1025 104 88 97 4749 21

5223 21 302

Dividend Adjustments for Puts

• The dividend adjustments required for European puts are similar to those for European calls.

• When applicable, the dividend is subtracted from the stock price to determine the ex-dividend put price. This price is then used to determine H, I, and Po.

• The dividend adjustment for an American put does differ from the adjustment for the American call. For the put, the price at each node is the maximum of its European value as determined by the BOPM or its IV on the ex-dividend date (see JG:210):

P Max X S D Pta

t t[ ( ), ]

Put-Call Parity With Dividends

• The value obtained using the binomial model with dividends is consistent with put-call parity.

P S C PV X PV Dtt

( ) ( )

Single Period Example

P C SX D

r

P

f

:

..

.5 71 100100 1

105190

Multiple Period Example

P C S PV D

P

tt

:

( )

.( . ) ( . )

.3 74 100100 1

1025

1

10250 85

2