dividend-paying stocks
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Dividend-Paying Stocks. 報告人:李振綱. Outline. 5.5.1 Continuously Paying Dividend 5.5.2 Continuously Paying Dividend with Constant Coefficients 5.5.3 Lump Payments of Dividends 5.5.4 Continuously Paying Dividend with Constant Coefficients. 5.5.1 Continuously Paying Dividend. - PowerPoint PPT PresentationTRANSCRIPT
Dividend-Paying Stocks
報告人:李振綱
• 5.5.1 Continuously Paying Dividend
• 5.5.2 Continuously Paying Dividend with Constant Coefficients
• 5.5.3 Lump Payments of Dividends
• 5.5.4 Continuously Paying Dividend with Constant Coefficients
Outline
5.5.1 Continuously Paying Dividend
• Consider a stock, modeled as a generalized geometric Brownian motion, that pays dividends continuously over time at a rate per unit time. Here is a nonnegative adapted process.
• Dividends paid by a stock reduce its value, and so we shall take as our model of the stock price
• If is the number of shares held at time t, then the portfolio value satisfies
( ) , 0 ,A t t T ( )A t
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . (5.5.1)dS t t S t dt t S t dW A t S tt dt
( )t( )X t
• By Girsanov’s Theorem to change to a measure under which is a Brownian motion, so we may rewrite (5.5.2) as
The discounted portfolio value satisfies
• If we now wish to hedge a short position in a derivative security paying at time T, where is an random variable, we will need to choose the initial capital and the portfolio process , , so that .
( ) ( ) ( )[ ( ) ( ) ( )]
( ) ( ) ( ) ( ) ( )[ ( ) ( ) ( )]
( ) ( ) (
( )
( ) ( ) (
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
) ( ) ( ) (
( ) ( ) (
) )
)
(
dS t
t S t dt t S t dW t A t S t d
dX t t R t X t t St A t dt
t t A t S t dt R t X t t S t dt
t t S t dt t t S t dW t R t X t dt R
t S t
t
d
t
t
t
( )
) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )[ ( )] dW t
t
S t dt
R t X t dt t R t t S t dt t t S t dW t
R t X t dt t t S t dt dW t
R t X t dt t t S t dt dW
R t
t
t t
( ) : t market price of risk
( ) ( ) ( ) ( ) ( ) ( ) ( ).dX t R t X t dt t S t t dW t
[ ( ) ( )] ( ) ( ) ( ) ( ) ( ).d D t X t t D t S t t dW t
( )V T
P W
(0)X ( )t 0 t T ( ) ( )X T V T
( )F T measurable
• Because is a martingale under , we must have
From (5.5.1) and the definition of , we see that
Under the risk-neutral measure, the stock does not have mean rate of return , and consequently the discounted stock price is not a martingale.
is a martingale.
(P.148)
( ) ( ) ( ) ( ) | ( ) ( ) ( ) | ( ) , 0 .D t X t E D T X T F t E D T V T F t t T
( ) ( ) ( ) | ( ) , ) 0 .(D t E D T V T F TV t t t
P( ) ( )D t X t
( ) [ ] (( ) ( ) ( ) ( ) ( ))dS t S t dR t A t tt S t dW t
2
0 0( ) ( )
1( ) (0)exp ( ) ( ) [ ( )] . (5.5.7)
2
t tR uS t S u dW u uA duu
0( ) 2
0 0
1( ) ( ) exp ( ) ( ) ( )
2
tt tA u
e D t S t u dW u u du
( )W t
( )R t
5.5.2 Continuously Paying Dividend with Constant Coefficients
• For , we have
• According to the risk-neutral pricing formula, the price at time t of a European call expiring at time T with strike K is
21( ) (0)exp ( ) . (5.5.8)
2S t S W t r a t
21( ) ( ) exp ( ( ) ( )) ( ) .
2S T S t W T W t r a T t
( )( ) [ ( ( ) ) | ( )]. (5.5.9)r T tV t E e S T K F t
0 t T
where and is a standard normal r.v. under .
We define
( ) 2
2
( , )
1exp ( ( ) ( )) ( )
2
1exp , (5.5.10)
2
r T t
r
c t x
E e x W T W t r a T t K
E e x Y r a K
( ) ( )W T W tY
T t
21 1( , ) log . (5.5.11)
2
xd x r a
K
2
2
1( , ) 2 2
( , ) 2 2
1( , )
2
1 1( , ) exp -
22
1 1 1 exp
2 22
1
2
1 1 exp (
22
d x yr
d x
d x yr
a
c t x e x y r a K e dy
x y a y dy
e Ke dy
xe y
( , ) 2) ( ( , )).
d x rdy e KN d x
T t P
• We make the change of variable in the integral, which leads us to the formula
z y
2
( , )2
1( , ) ( ( , ))
2
( ( , )) ( ( , )). (5.5.12)
d x
a
za r
r
c t x xe e dz e KN d x
x N d e K xe x N d
5.5.3 Lump Payments of Dividends
• There are times and, at each time , the dividend paid is , where denotes the stock prices just prior to the dividend payment.
• We assume that each is an r.v. taking values in [0,1]. However, neither nor is a dividend payment dates(i.e., and ).
• We assume that, between dividend payment dates, the stock price follows a generalized geometric Brownian motion:
( ) ( ) ( ) (1 ) ( ). (5.5.13)j j j j j jS t S t a S t a S t
1( ) ( ) ( ) ( ) ( ) ( ), , 0,1,..., . (5.5.14)j jdS t t S t dt t S t dW t t t t j n
1 20 ... nt t t T jt
( )j ja S t ( )jS t
( )jF t measurable
0 0t 1nt T 0 0a
1 0na
ja
• Between dividend payment dates, the differential of the portfolio value corresponding to a portfolio process , , is
• At the dividend payment dates, the value of the portfolio stock holdings drops by , but the portfolio collects the dividend , and so the portfolio value does not jump. It follows that
( ) ( )j j ja t S t ( ) ( )j j ja t S t
( ) ( ) ( ) ( ) ( ) ( )[ ( ) ( )] (5.5.15)dX t R t X t dt t t S t t dt dW t
( ) ( ) ( )[ ( ) ( ) ( )]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( ) ( ) (
( ) ( )
( ) ()
dX t t R t X t t S t dt
t R t X t dt R t t S t dt
dS t
t S t dt t
R t X t dt t R t t S t dt t t S t dW t
R t X t dt t t S t
S t dW t
t R t
( )
( ) ,
( ) :
)
( )t
dt dW t
t the market price of risk
t
( )t 0 t T
5.5.4 Continuously Paying Dividend with Constant Coefficients
• We price a European call under the assumption that , , and each are constant. From(5.5.14) and the definition of , we have
Therefore,
It follows that
1( ) ( ) ( ) ( ), , 0,1,..., .j jdS t rS t dt S t W t t t t j n
2( 1) 1 1
1( ) ( ) exp ( ) ( ) ( ) . (5.5.16)
2j j j j j jS t S t W t W t r t t
21 1 1 1
1( ) (1 ) ( ) exp ( ) ( ) ( )
2j j j j j j jS t a S t W t W t r t t
左右同乘
1(1 )ja
1 21 1 1
( ) 1(1 )exp ( ) ( ) ( ) .
( ) 2j
j j j j jj
S ta W t W t r t t
S t
0,1,...,for j n
1
+1 2+11
0 j=00
( )( )( ) 1(1 ) exp ( )
(0) ( ) ( ) 2
n njn
jj j
S tS tS Ta W T r T
S S t S t
• In other words,
This is the same formula we would have for the price at time T of a geometric Brownian motion not paying dividends if the initial stock price were rather than S(0).
Therefore, the price at time zero of a European call on this dividend-paying asset, a call that expires at time T with strike price K, is obtained by replacing the initial stock price by in the classical BSM formula.
where
1
12
0
1( ) exp ( ) . (5.5.17)
2(0) (1 )
n
jj
S T W T r TS a
1
10
(0) (1 )n
jj
S a
1
10
((0) (1 ) ) ( ),j
rTn
jS a N d e KN d
12
10
1 (0) 1log log(1 ) .
2
n
jj
Sd r T a
KT
1
10
(0) (1 )n
jj
S a
Thanks for your listening !!