Options on dividend-paying stocks

This stock pays no dividends.. An American contingent claim paying h S t if exercised at time t does not need to be exercised before expiration, i.e., waiting until expiration to deci

Chapter 26

Options on dividend-paying stocks

26.1 American option with convex payoff function

Theorem 1.64 Consider the stock price process

where r and  are processes and r ( t )  0 ; 0  t  T; a.s This stock pays no dividends Leth ( x ) be a convex function of x  0, and assumeh (0) = 0 (E.g., h ( x ) = ( x,K ) +) An

American contingent claim paying h ( S ( t )) if exercised at time t does not need to be exercised before expiration, i.e., waiting until expiration to decide whether to exercise entails no loss of value.

Proof: For01andx0, we have

(1, ) h (0) + h ( x )

= h ( x ) :

LetT be the time of expiration of the contingent claim For0tT,

0

( T ) = exp

(

,

Z T

t r ( u ) du

)

1

andS ( T )0, so





Consider a European contingent claim payingh ( S ( T ))at timeT The value of this claim at time

1

F( t )

:

263

-6









































r r r

( x;h ( x ))

x

h

Figure 26.1: Convex payoff function

Therefore,

F( t )



1

F( t )

(by (*))



1

F( t )

(Jensen’s inequality)

= 1 ( t ) h



(S

 is a martingale)

= 1 ( t ) h ( S ( t )) :

This shows that the valueX ( t ) of the European contingent claim dominates the intrinsic value

h ( S ( t ))of the American claim In fact, except in degenerate cases, the inequality

is strict, i.e., the American claim should not be exercised prior to expiration

26.2 Dividend paying stock

Letrandbe constant, letbe a “dividend coefficient” satisfying

CHAPTER 26 Options on dividend paying stocks 265

price is given by

(

S (0)expf( r,1 2  2 ) t + B ( t )g; 0tt 1 ;

(1, ) S ( t 1 )expf( r,

1

Consider an American call on this stock At timest2 ( t 1 ;T ), it is not optimal to exercise, so the value of the call is given by the usual Black-Scholes formula

where



log x

:

At timet 1, immediately after payment of the dividend, the value of the call is

At timet 1, immediately before payment of the dividend, the value of the call is

where

w ( t 1 ;x ) = max

( x,K ) + ; v ( t 1 ; (1, ) x

:

Theorem 2.65 For0tt 1, the value of the American call isw ( t;S ( t )), where

,t) w ( t 1 ;S ( t 1 ))i

:

This function satisfies the usual Black-Scholes equation

,rw + w t + rxw x + 1 2  2 x 2 w xx = 0 ; 0tt 1 ; x0 ;

(wherew = w ( t;x )) with terminal condition

w ( t 1 ;x ) = max

( x,K ) + ; v ( t 1 ; (1, ) x )

and boundary condition

w ( t; 0) = 0 ; 0tT:

The hedging portfolio is

( t ) =

(

Proof: We only need to show that an American contingent claim with payoffw ( t 1 ;S ( t 1 ))at time

t 1need not be exercised before timet 1 According to Theorem 1.64, it suffices to prove

1 w ( t 1 ; 0) = 0,

2 w ( t 1 ;x )is convex inx

Sincev ( t 1 ; 0) = 0, we have immediately that

w ( t 1 ; 0) = max

(0,K ) + ; v ( t 1 ; (1, )0)

= 0 :

To prove thatw ( t 1 ;x )is convex inx, we need to show thatv ( t 1 ; (1, ) x )is convex isx Obviously,

( x,K ) + is convex inx, and the maximum of two convex functions is convex The proof of the convexity ofv ( t 1 ; (1, ) x )inxis left as a homework problem

26.3 Hedging at time t1

Letx = S ( t 1 )

Case I:v ( t 1 ; (1, ) x )( x,K ) +.

The option need not be exercised at timet 1 (should not be exercised if the inequality is strict) We have

( t 1 ) = w x ( t 1 ;x ) = (1, ) v x ( t 1 ; (1, ) x ) = (1, )
( t 1 +) ;

where

( t 1 +) = lim t

#t1

( t )

is the number of shares of stock held by the hedge immediately after payment of the dividend The post-dividend position can be achieved by reinvesting in stock the dividends received on the stock held in the hedge Indeed,

( t 1 +) = 1 1

,
( t 1 ) =
( t 1 ) + 1, 
( t 1 )

=
( t 1 ) + 
( t 1 ) S ( t 1 )

(1, ) S ( t 1 )

=# of shares held when dividend is paid+ dividends received

price per share when dividend is reinvested

Case II:v ( t 1 ; (1, ) x ) < ( x,K ) +.

The owner of the option should exercise before the dividend payment at timet 1and receive( x,K ) The hedge has been constructed so the seller of the option hasx,Kbefore the dividend payment

at timet 1 If the option is not exercised, its value drops fromx,Ktov ( t 1 ; (1, ) x ), and the seller

of the option can pocket the difference and continue the hedge

CHAPTER 26 Options on dividend paying stocks 265

price is given by

(

S... x,K ) + is convex inx, and the maximum of two convex functions is convex The proof of the convexity ofv ( t ; (1, ) x )inxis... )is convex inx, we need to show thatv ( t ; (1, ) x )is convex isx Obviously,

( x,K ) + is convex