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 !!