Properties of American Derivative Securities
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (134.06 KB, 6 trang )
Chapter 6
Properties of American Derivative
Securities
6.1 The properties
Definition 6.1 An American derivative security is a sequence of non-negative random variables
fG
k
g
n
k=0
such that each
G
k
is
F
k
-measurable. The owner of an American derivative security can
exercise at any time
k
, and if he does, he receives the payment
G
k
.
(a) The value
V
k
of the security at time
k
is
V
k
= max
1 + r
k
f
IE 1 + r
,
G
jF
k
;
where the maximum is over all stopping times
satisfying
k
almost surely.
(b) The discounted value process
f1 + r
,k
V
k
g
n
k=0
is the smallest supermartingale whichsatisfies
V
k
G
k
; 8k;
almost surely.
(c) Any stopping time
which satisfies
V
0
=
f
IE 1 + r
,
G
is an optimal exercise time. In particular
4
= minfk; V
k
= G
k
g
is an optimal exercise time.
(d) The hedging portfolio is given by
k
!
1
;::: ;!
k
=
V
k+1
!
1
;::: ;!
k
;H, V
k+1
!
1
;::: ;!
k
;T
S
k+1
!
1
;::: ;!
k
;H, S
k+1
!
1
;::: ;!
k
;T
;k =0;1;::: ;n , 1:
85
86
(e) Suppose for some
k
and
!
,wehave
V
k
!=G
k
!
. Then the owner of the derivative security
should exercise it. If he does not, then the seller of the security can immediately consume
V
k
! ,
1
1+ r
f
IEV
k+1
jF
k
!
and still maintain the hedge.
6.2 Proofs of the Properties
Let
fG
k
g
n
k=0
be a sequence of non-negative random variables such that each
G
k
is
F
k
-measurable.
Define
T
k
to be the set of all stopping times
satisfying
k n
almost surely. Define also
V
k
4
=1+r
k
max
2T
k
f
IE 1 + r
,
G
jF
k
:
Lemma 2.18
V
k
G
k
for every
k
.
Proof: Take
2 T
k
to be the constant
k
.
Lemma 2.19 The process
f1 + r
,k
V
k
g
n
k=0
is a supermartingale.
Proof: Let
attain the maximum in the definition of
V
k+1
, i.e.,
1 + r
,k+1
V
k+1
=
f
IE
h
1 + r
,
G
jF
k+1
i
:
Because
is also in
T
k
,wehave
f
IE 1 + r
,k+1
V
k+1
jF
k
=
f
IE
h
f
IE 1 + r
,
G
jF
k+1
jF
k
i
=
f
IE 1 + r
,
G
jF
k
max
2T
k
f
IE 1 + r
,
G
jF
k
= 1 + r
,k
V
k
:
Lemma 2.20 If
fY
k
g
n
k=0
is another process satisfying
Y
k
G
k
;k =0;1;::: ;n;
a.s.,
and
f1 + r
,k
Y
k
g
n
k=0
is a supermartingale, then
Y
k
V
k
;k =0;1;::: ;n;
a.s.
CHAPTER 6. Properties of American Derivative Securities
87
Proof: The optional sampling theorem for the supermartingale
f1 + r
,k
Y
k
g
n
k=0
implies
f
IE 1 + r
,
Y
jF
k
1 + r
,k
Y
k
; 8 2 T
k
:
Therefore,
V
k
= 1 + r
k
max
2T
k
f
IE 1 + r
,
G
jF
k
1 + r
k
max
2T
k
f
IE 1 + r
,
Y
jF
k
1 + r
,k
1 + r
k
Y
k
= Y
k
:
Lemma 2.21 Define
C
k
= V
k
,
1
1+r
f
IEV
k+1
jF
k
= 1 + r
k
n
1 + r
,k
V
k
,
f
IE 1 + r
,k+1
V
k+1
jF
k
o
:
Since
f1 + r
,k
V
k
g
n
k=0
is a supermartingale,
C
k
must be non-negative almost surely. Define
k
!
1
;::: ;!
k
=
V
k+1
!
1
;::: ;!
k
;H, V
k+1
!
1
;::: ;!
k
;T
S
k+1
!
1
;::: ;!
k
;H, S
k+1
!
1
;::: ;!
k
;T
:
Set
X
0
= V
0
and define recursively
X
k+1
=
k
S
k+1
+1+rX
k
, C
k
,
k
S
k
:
Then
X
k
= V
k
8k:
Proof: We proceed by induction on
k
. The induction hypothesis is that
X
k
= V
k
for some
k 2f0;1;::: ;n , 1g
, i.e., for each fixed
!
1
;::: ;!
k
we have
X
k
!
1
;::: ;!
k
=V
k
!
1
;::: ;!
k
:
We need to show that
X
k+1
!
1
;::: ;!
k
;H= V
k+1
!
1
;::: ;!
k
;H;
X
k+1
!
1
;::: ;!
k
;T=V
k+1
!
1
;::: ;!
k
;T:
We prove the first equality; the proof of the second is similar. Note first that
V
k
!
1
;::: ;!
k
, C
k
!
1
;::: ;!
k
=
1
1+ r
f
IEV
k+1
jF
k
!
1
;::: ;!
k
=
1
1+ r
~pV
k+1
!
1
;::: ;!
k
;H+ ~qV
k+1
!
1
;::: ;!
k
;T :
88
Since
!
1
;::: ;!
k
will be fixed for the rest of the proof, we will suppress these symbols. For
example, the last equation can be written simply as
V
k
, C
k
=
1
1+r
~pV
k+1
H + ~qV
k+1
T :
We compute
X
k+1
H =
k
S
k+1
H + 1 + rX
k
, C
k
,
k
S
k
=
V
k+1
H , V
k+1
T
S
k+1
H , S
k+1
T
S
k+1
H , 1 + rS
k
+1 + rV
k
, C
k
=
V
k+1
H , V
k+1
T
u , dS
k
uS
k
, 1 + rS
k
+~pV
k+1
H + ~qV
k+1
T
= V
k+1
H , V
k+1
T ~q +~pV
k+1
H + ~qV
k+1
T
= V
k+1
H :
6.3 Compound European Derivative Securities
In order to derive the optimal stopping time for an American derivative security, it will be useful to
study compound European derivative securities, which are also interesting in their own right.
A compound European derivative security consists of
n +1
different simple European derivative
securities (with the same underlying stock) expiring at times
0; 1;::: ;n
; the security that expires
at time
j
has payoff
C
j
. Thus a compound European derivative security is specified by the process
fC
j
g
n
j =0
, where each
C
j
is
F
j
-measurable, i.e., the process
fC
j
g
n
j =0
is adapted to the filtration
fF
k
g
n
k=0
.
Hedging a short position (one payment). Here is how we can hedge a short position in the
j
’th
European derivative security. The value of European derivative security
j
at time
k
is given by
V
j
k
=1+r
k
f
IE1 + r
,j
C
j
jF
k
;k=0;::: ;j;
and the hedging portfolio for that security is given by
j
k
!
1
;::: ;!
k
=
V
j
k+1
!
1
;::: ;!
k
;H, V
j
k+1
!
1
;::: ;!
k
;T
S
j
k+1
!
1
;::: ;!
k
;H, S
j
k+1
!
1
;::: ;!
k
;T
;k =0;::: ;j , 1:
Thus, starting with wealth
V
j
0
, and using the portfolio
j
0
;::: ;
j
j,1
, we can ensure that at
time
j
we have wealth
C
j
.
Hedging a short position (all payments). Superpose the hedges for the individual payments. In
other words, start with wealth
V
0
=
P
n
j =0
V
j
0
. At each time
k 2f0;1;::: ;n , 1g
,firstmakethe
payment
C
k
and then use the portfolio
k
=
k
k+1
+
k
k+2
+ :::+
k
n
CHAPTER 6. Properties of American Derivative Securities
89
corresponding to all future payments. At the final time
n
, after making the final payment
C
n
,we
will have exactly zero wealth.
Suppose you own a compound European derivative security
fC
j
g
n
j =0
. Compute
V
0
=
n
X
j =0
V
j
0
=
f
IE
2
4
n
X
j =0
1 + r
,j
C
j
3
5
and the hedging portfolio is
f
k
g
n,1
k=0
. You can borrow
V
0
and consume it immediately. This leaves
you with wealth
X
0
= ,V
0
. In each period
k
, receive the payment
C
k
and then use the portfolio
,
k
. At the final time
n
, after receiving the last payment
C
n
, your wealth will reach zero, i.e., you
will no longer have a debt.
6.4 Optimal Exercise of American Derivative Security
In this section we derive the optimal exercise time for the owner of an American derivative security.
Let
fG
k
g
n
k=0
be an American derivative security. Let
be the stopping time the owner plans to
use. (We assume that each
G
k
is non-negative, so we may assume without loss of generality that the
owner stops at expiration – time
n
– if not before). Using the stopping time
,inperiod
j
the owner
will receive the payment
C
j
= I
f =j g
G
j
:
In other words, once he chooses a stopping time, the owner has effectively converted the American
derivative security into a compound European derivative security, whose value is
V
0
=
f
IE
2
4
n
X
j =0
1 + r
,j
C
j
3
5
=
f
IE
2
4
n
X
j =0
1 + r
,j
I
f =j g
G
j
3
5
=
f
IE 1 + r
,
G
:
The owner of the American derivative security can borrow this amount of money immediately, if
he chooses, and invest in the market so as to exaclty pay off his debt as the payments
fC
j
g
n
j =0
are
received. Thus, his optimal behavior is to use a stopping time
which maximizes
V
0
.
Lemma 4.22
V
0
is maximized by the stopping time
= minfk ; V
k
= G
k
g:
Proof: Recall the definition
V
0
4
= max
2T
0
f
IE 1 + r
,
G
= max
2T
0
V
0
