Transcript BOPM FOR PUTS

BOPM FOR PUTS
AND THE DIVIDENDADJUSTED BOPM
Chapter 6
Binomial Stock Movement
S u  uS 0  1.1($100)  $110
S0
S d  dS 0  .95 ($100)  $95
Assumption
• Assume there is a European put option on
the stock that expires at the end of the
period.
• Example: X = $100.
Binomial Put Movement
Pu  M ax [ X  S u , 0 ]  [100  110 , 0 ]  0
P0
Pd  M ax [ X  S d , 0 ]  [100  95, 0 ]  5
Replicating portfolio
• The replicating portfolio consist of buying Ho shares of the
stock at So and investing Io dollars.
H 0 uS0  I 0 rf
P
H
P
0
S0  I 0
H
P
0
dS 0  I 0rf
Constructing 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.
Solve for Ho and Io where:
• Equations:
H
P
o
uS 0  I 0 r f  Pu
H
P
o
dS 0  I 0 r f  Pd
Solution
• Equations:
H
I
*
0
P*
0


Pu  Pd
uS 0  dS 0
 [ Pu ( dS 0 )  Pd ( uS 0 )]
r f [ uS 0  dS 0 ]
N ote: Pd  Pu  H 0  0  Short Stock P osition
P*
 I0
*
 0  R F Investm ent
Example
• Equations:
H
I
*
0
P*
0


0  5
110  95
  .3 3 3 3
 [ 0 ( 9 5 )  5 (1 1 0 )]
1.0 5[1 1 0  9 5 ]
 3 4 .9 2 0 6
Equilibrium Put Price
• Law of One Price:
P  H
*
0
P*
0
S0  I
*
0
E xam ple:
P0   .3333 ($100)  $34.9206  $1.59
*
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.
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:
*
0
P

1
rf
[ pPu  (1  p ) Pd ]
w here:
p 
rf  d
u  d
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
E xa m p le :
P  6 .3 5 
100
1.0 5
 1 0 0  1.5 9
Multiple-Period Model Example
• From Previous example for call:
– u = 1.0488,
– So = $100,
– Rf = .025,
d = .9747,
n = 2,
• To price a European put with X = 100, use
the same recursive procedure we used for a
call option.
S uu  u S 0  $ 110
2
S u  uS 0  $104.87
S 0  $100
S ud  udS 0  $102.23
S d  dS 0  $97.47
S dd  d S 0  $95
2
E uropean P ut V alue
S uu  110
S u  1 0 4 .8 8
Pu  0
Puu  M ax [100  110 ,0 ]
0
Hu  0
p
S0  100
Iu  0
P0  0 .4 9
H 0   .2 1 0 5
Pud  M ax [100  102 .23,0 ]
p
I 0  2 1.5 4
S ud  102 .23
S d  97 .47
0
Pd  1.56
H d   .6916
p
I d  68 .97
S dd  95
Pdd  M ax [100  95 ,0 ]
5
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
E xam ple:
P  5.31 
100
(1.025 )
2
 100  .49
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:
Pt
a
 M ax [ Pt , IV ]
A m erican P ut V alue
S uu  110
Puu  M ax [100  110 ,0 ]
S u  1 0 4 .8 8
a
 M a x [ Pu , X  u S 0 ]
a
 M a x [ 0 ,1 0 0  1 1 0 ]  0
Pu
Pu
0
S ud  102 .23
S 0  100
Pud  M ax [100  102 .23,0 ]
P0  0 .79
0
S d  97 .47
a
 M ax [ Pd , X  dS 0 ]
a
 M ax [1.56 ,100  97 .47 ]  2 .53
Pd
Pd
S dd  95
Pdd  M ax [100  95 ,0 ]
5
Point: 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.
Dividend 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.
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.
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.
R Pu  H 0 ( uS 0  D )  H 0 D  r f B 0
 H 0 uS 0  r f B 0
C u  IV  M ax [ uS 0  D  X , 0 ]
x
R P0  H 0 S 0  B 0
R Pd  H 0 ( dS 0  D )  H 0 D  r f B 0
 H 0 dS 0  r f B 0
C d  IV  M ax [ dS 0  D  X , 0 ]
x
Solve for Ho and Bo where:
• Equations:
H 0 uS 0  B 0 r f  C
x
u
H 0 dS 0  B 0 r f  C
x
d
Example
• Assuming D = 1
Cu  9, Cd  0
x
x
H 
*
0
B
*
0

9  0
110  95
 .6
9 ( 95 )  0 (110 )]
1.05[110  95 ]
 54 .2857
C 0  H 0 S 0  B 0  .6 ($100)  $54.2857  $5.71
*
*
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 $1.
• Assume: n = 2, u = 1.0487, d = .9747, Rf =.025.
• As shown in the Figure, H and B values reflect
the ex-dividend call values.
• Note: In the formulas for H and B, the stock
prices just before the ex-dividend date are used.
E u ro p e a n C a ll V a lu e
w ith d iv id e n d s
S
x
u
S uu  110  1  109
x
 1 0 4 .8 8  1  1 0 3.8 8
Hu 
Bu 
9  1.2 3
C uu  M ax [109  100 ,0 ]
x
1
1 1 0  1 0 2 .2 3
9 (1 0 2 .2 3 )  1.2 3 (1 1 0 )
1.0 2 5 (1 1 0  1 0 2 .2 3 )
9
 9 8 .5 4
C u  1(1 0 3.8 8 )  9 8 .5 4  5.3 4
S0  100
H0 
*
B0 
*
5.3 4  0 .6 4
1 0 4 .8 8  9 7 .4 7
S ud  102 .23  1  101.23
x
 .6 3 4 3
5.3 4 ( 9 7 .4 7 )  0 .6 4 (1 0 4 .8 8 )
1.0 2 5 (1 0 4 .8 8  9 7 .4 7 )
C ud  M ax [101.23  100 ,0 ]
x
 5 9 .6 9
 1.23
C 0  .6 3 4 3 ($ 1 0 0)  $ 5 9.6 9  3.7 4
*
x
Sd
 9 7 .4 7  1  9 6 .4 7
Hd 
Bd 
1.2 3  0
 1 7 .0 1
1 0 2 .2 3  9 5
1.2 3 ( 9 5 )  0 (1 0 2 .2 3 )
1.0 2 5 (1 0 2 .2 3  9 5 )
 1 5.7 7
C d  .1 7 0 1( 9 6 .4 7 )  1 5.7 7  $ 0.6 4
S dd  95  1  94
x
C ud  M ax [ 94  100 ,0 ]
x
0
American Call Price on DividendPaying 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 exdividend 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 $3.02.
A m erica n C a ll V a lu e w ith
d ivid en d s: D 1  1, D 2  2
S uu  110  1  109
x
x
Su
 1 0 4 .8 8  2  1 0 2 .8 8
C u  1(1 0 2 .8 8 )  9 8 .5 4  4 .3 4
C uu  M ax [109  100 ,0 ]
x
C u  M ax [ C u , u S 0  X ]
a
9
 M ax [ 4 .3 4 ,1 0 4 .8 8  1 0 0 ]
 4 .8 8
S0  100
4 .8 8  0 .4 7
H0 
*
1 0 4 .8 8  9 7 .4 7
B0 
*
a*
C0
S ud  102 .23  1  101.23
x
 .5 9 5 1
4 .8 8 ( 9 7 .4 7 )  0 .4 7 (1 0 4 .8 8 )
1.0 2 5 (1 0 4 .8 8  9 7 .4 7 )
C ud  M ax [101.23  100 ,0 ]
x
 5 6 .1 3
 1.23
 .5 9 5 1 ($ 1 0 0)  $ 5 6.1 3  3.3 8
x
Sd
C d  .1 7 0 1( 9 5.4 7 )  1 5.7 7  0 .4 7
E u ro p ea n :
H 
*
0
B
4 .3 4  0 .4 7
C d  M ax [ C d , d S 0  X ]
a
 .5 2 2 3
1 0 4 .8 8  9 7 .4 7
4 .3 4 ( 9 7 .4 7 )  0 .4 7 (1 0 4 .8 8 )
 M ax [ 0 .4 7 , 9 7 .4 7  1 0 0 ]
*
0

e*
 .5 2 2 3 ($ 1 0 0)  $ 4 9.2 1  3.0 2
C0
 9 7 .4 7  2  9 5.4 7
1.0 2 5 (1 0 4 .8 8  9 7 .4 7 )
 4 9 .2 1
 0 .4 7
S dd  95  1  94
x
C ud  M ax [ 94  100 ,0 ]
x
0
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):
Pt
a
M ax [ X  ( S t  D ), Pt ]
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 ( D )
t
t
Single  P eriod E xam ple :
P C  S 
X D
P C  S 
rf
P  5.71  100 
M ultiple  P eriod E xam ple :

P V ( Dt )
t
100  1
1.05
 1.90
P  3.7 4  1 0 0 
100  1
(1.0 2 5 )
2

1
(1.0 2 5 )
 0 .8 5