Options on dividend-paying stocks
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Chapter 26
Options on dividend-paying stocks
26.1 American option with convex payoff function
Theorem 1.64 Consider the stock price process
dS t=rtStdt + tS t dB t;
where
r
and
are processes and
rt 0; 0 t T;
a.s. This stock pays no dividends.
Let
hx
be a convex function of
x 0
, and assume
h0 = 0
.(E.g.,
hx=x,K
+
). An
American contingent claim paying
hS 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: For
0 1
and
x 0
,wehave
hx=h1 , 0 + x
1 , h0 + hx
= hx:
Let
T
be the time of expiration of the contingent claim. For
0 t T
,
0
t
T
= exp
,
Z
T
t
ru du
1
and
S T 0
,so
h
t
T
S T
t
T
hS T :
(*)
Consider a European contingent claim paying
hS T
at time
T
. The value of this claim at time
t 2 0;T
is
X t=tIE
1
T
hST
F t
:
263
264
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r
r
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x; hx
x
h
hx
hx
Figure 26.1: Convex payoff function
Therefore,
X t
t
=
1
t
IE
t
T
hS T
F t
1
t
IE
h
t
T
S T
F t
(by (*))
1
t
h
t IE
S T
T
F t
(Jensen’s inequality)
=
1
t
h
t
S t
t
(
S
is a martingale)
=
1
t
hS t:
This shows that the value
X t
of the European contingent claim dominates the intrinsic value
hS t
of the American claim. In fact, except in degenerate cases, the inequality
X t hS t; 0 t T;
is strict, i.e., the American claim should not be exercised prior to expiration.
26.2 Dividend paying stock
Let
r
and
be constant, let
be a “dividend coefficient” satisfying
0 1:
CHAPTER 26. Options on dividend paying stocks
265
Let
T0
be an expiration time, and let
t
1
2 0;T
be the time of dividend payment. The stock
price is given by
S t=
S0 expfr ,
1
2
2
t + Btg; 0 t t
1
;
1 , S t
1
expfr ,
1
2
2
t , t
1
+Bt,Bt
1
g; t
1
tT:
Consider an American call on this stock. At times
t 2 t
1
;T
, it is not optimal to exercise, so the
value of the call is given by the usual Black-Scholes formula
v t; x= xN d
+
T , t; x , Ke
,rT ,t
N d
,
T , t; x; t
1
tT;
where
d
T , t; x=
1
p
T ,t
log
x
K
+T ,tr
2
=2
:
At time
t
1
, immediately after payment of the dividend, the value of the call is
v t
1
; 1 , S t
1
:
At time
t
1
, immediately before payment of the dividend, the value of the call is
wt
1
;St
1
;
where
wt
1
;x = max
x , K
+
;vt
1
;1 , x
:
Theorem 2.65 For
0 t t
1
, the value of the American call is
wt; S t
,where
wt; x=IE
t;x
h
e
,rt
1
,t
wt
1
;St
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;
(where
w = wt; x
) with terminal condition
wt
1
;x = max
x , K
+
;vt
1
;1 , x
;x0;
and boundary condition
wt; 0 = 0; 0 t T:
The hedging portfolio is
t=
w
x
t; S t; 0 t t
1
;
v
x
t; S t; t
1
tT:
Proof: We only need to show that an American contingent claim with payoff
wt
1
;St
1
at time
t
1
need not be exercised before time
t
1
. According to Theorem 1.64, it suffices to prove
1.
wt
1
; 0 = 0
,
266
2.
wt
1
;x
is convex in
x
.
Since
v t
1
; 0 = 0
,wehaveimmediatelythat
wt
1
; 0 = max
0 , K
+
;vt
1
;1 , 0
=0:
To prove that
wt
1
;x
is convex in
x
, we need to show that
v t
1
; 1 , x
is convex is
x
. Obviously,
x , K
+
is convex in
x
, and the maximum of two convex functions is convex. The proof of the
convexity of
v t
1
; 1 , x
in
x
is left as a homework problem.
26.3 Hedging at time
t
1
Let
x = S t
1
.
Case I:
v t
1
; 1 , x x , K
+
.
The option need not be exercised at time
t
1
(should not be exercised if the inequality is strict). We
have
wt
1
;x= vt
1
;1 , x;
t
1
=w
x
t
1
;x= 1,v
x
t
1
;1 , x=1,t
1
+;
where
t
1
+ = lim
tt
1
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 time
t
1
and receive
x , K
.
The hedge has been constructed so the seller of the option has
x , K
before the dividend payment
at time
t
1
. If the option is not exercised, its value drops from
x , K
to
v t
1
; 1 , x
, and the seller
of the option can pocket the difference and continue the hedge.
