8-4 Finite-Expiration American Put

Report
8-4 Finite-Expiration American Put
財金所 潘政宏
Consider an American put on a stock whose
price is the geometric Brownian motion:
dS(t)  rS(t)dt   S(t)dW(t)
(8.3.1)
But now the put has a finite expiration time T.
Definition 8.4.1
Let 0  t  T and x  0 be given. Assume S(x)  x. Let F , t  u  T,
(t )
u
denote the  - algebra generated by the process S( ) as  range over
[t, u], and let  denote the set of the sopping time for the filtration
t ,T
F , t  u  T taking values in [t, T] or taking the value . In other
(t )
u
words, {  u}  F for every u  [t, T]; a stopping time in  makes
(t )
u
t ,T
the decision t o stop at a time u  [t, T] based only on the path of the
stock price between ti mes t and u. The price at time t of the American
put expiring at time T is defined to be
~ 
v(t, x)  max
E[e (K - S( )) | S(t)  x] (8.4.1)

- r( - t)
t ,T
In the event that   , we interpret e  (K - S( )) to be zero.
-r
This is the case when the put expires unexercise d.
8.4.1 Analytical Characterization of
the Put Price
The finite-expiration American put price function
v(t,x) satisfies the linear complementarity
conditions :
v(t,x)  (K-x)  for all t  [0,T], x  0
(8.4.2)
1 2 2
rv(t,x)-v t (t , x)  rxvx (t , x)   x vxx (t , x)  0
2
for all t  [0,T), x  0, and (8.4.3)
for each t  [0,T) and x  0, equality holds
in either (8.4.2) or(8.4.3)
(8.4.4)
L(T-t)
Level L(T-t) depends on the time to expiration T-t
L(T) is decreases with increasing T
The set {(t , x); 0  t  T , x  0} can be divided into two
regions, the stopping set
S  {(t,x);v(t,x)  (K-x)  }
(8.4.5)
and the continuation set

C  {(t,x);v(t,x)  (K-x) }
(8.4.6)
1 2 2
rv(t , x) - vt (t , x)  rxvx (t , x)   x vxx (t , x)
2
 rK  rx  0  rx  0  rK
Because v(t,x)  K-x for 0  x  L(T-t),
we also have the left-handderivatives
v x (t , x )  1 on the curve x  L(T  t ).
The put price v(t,x) satisfies the smooth-pasting condition that
v x (t , x) is continuous, even at x  L(T-t). In other words,
v x (t , x )  v x (t , x )  1 for x  L(T  t ), 0  t  T
(8.4.7)
The smooth-pasting condition does not hold at t  T. Indeed,
L(0)  K and v(T,x)  (K-x) 
so v x (T , x )  1, whereas v x (T , x )  0. Also, v t (T , x) and
v xx (t , x) are not continuous along the curve x  L(T  t) .
(8.4.8)
Determine the function v(t,x)
1 2 2
rv(t,x)-v t (t , x)  rxvx (t , x)   x vxx (t , x)  0 ,x  L(T-t),
2
v(t,x)  K-x, 0  x  L(T-t),
v x (t , x )  v x (t , x )  1 for x  L(T  t ), 0  t  T
lim v(t , x)  0
x 
8.4.2 Probabilistic Characterization
of the Put Price
Theorem 8.4.2
Let S(u),t  u  T,be the stock price of(8.3.1) starting at S(t)  x
and with the stopping set S defined by(8.4.5). Let
 *  min{u  [t , T ];(u, S (u))  S}
(8.4.10)
where we interpret  * to be  if (u,s(u)) doesn't enter S for
any u  [t.T]. Then e-ru v(u, S (u)), t  u  T,is a supermartingale
under P, and the stopped process e -r(u  * ) v(u, S (u   * )),
t  u  T, is a martingale.
d [e- ru v(u , S (u ))]
 v(u, S (u ))de- ru  e- ru dv(u , S (u ))
 e- ru [rv(u, S (u ))du  vu (u, S (u ))du  vx (u, S (u ))dS (u )
1
 vxx (u, S (u ))dS (u )dS (u )]
2
 e- ru [rv(u, S (u ))  vu (u, S (u ))  rS (u )vx (u, S (u ))
1
  2 S 2 (u )vxx (u, S (u ))]du  e- ru S (u )vx (u, S (u ))dW (u )
2
(8.4.11)
-ru
According to Figure 8.4.1, the du term in (8.4.11) is - e rKΙ
{S(u)  L(T u)}
~
This is nonpositiv e, and so e v(u, S(u)) is a supermarti ngale under P.
- ru
In fact, starting from u  t and up until time  , we have S(u)  L(T - u),
*
so the du term is zero. Therefore, the stopped process
e  v(u   , S (u   )), t  u  T , is a martingale .
- r(u 
*
)
*
*
Corollary 8.4.3
Consider an agent with initial capital X(0)  v(0, S(0)), the initial
finite - expiration put price. Suppose this agent uses the portfolio process
(u )  v (u , S (u )) and consumes cash at rate C(u)  rKΙ
x
{S(u)  L(T u)}
per unit time.
Then X(u)  v(u, S(u)) for all times u between u  0 and the time the option
is exercised or expires. In particular , S(u)  (K - S(u)) for all times u until the

option is exercised or expires, so the agent can pay off a short option position
regardless of when the option is exercised.
Proof 8.4.3
dX  t     t  dS  t   r  X  t     t  S  t   dt  C  t  dt

dS (t )  rS (t )dt   S (t )dW (t )


d e  rt X  t   e  rt  rX  t  dt  dX  t  
 e  rt    t  dS  t   r   t  S  t  dt  C  t  dt 

 e  rt   t   S  t  dW  t   C  t  dt





d e  rt X  t   d e  rt v  t , S  t   and X  0   v  0, S  0  
we obtain X  t   v  t , S  t  
Remark 8.4.4
The proofs of Theorem 8.4.2 and Corollary 8.4.3 use the analytic
characterization of the American put price captured in Figure 8.4.1
plus the smooth-pasting condition that guarantees that v x (t , x) is
ˆ
continuous even on the curve x  L(T-t) so that the Ito-Doeblin
formula
can be applied. Here we show that the only function v(t,x) satisfying
these conditions is the function v(t,x) defined by(8.4.1)
To do this, we first fix t with 0  t  T . The supermartingale property for
e-rt v(t , S (t ))of Theorem 8.4.2 and Theorem 8.4.2(optional sampling)
implies that
e-r(t  ) v(t   , S (t   ))  E[e-r(T  ) v(T   , S (T   )) | F(t)]
e-r(t  ) v(t   , S (t   ))  E[e-r(T  ) v(T   , S (T   )) | F(t)]
For   t ,T , we have t    t , whereas T     if    and
T    T if   . Therefore, for   t ,T
e-rt v(t , S (t))  E[e-r v( , S ( )){ }  e -rT v(T , S (T )){ } | F (t )]
 E[e-r v( , S ( )) | F (t )]
(8.4.12)
where, as usual, we interpret e-r v( , S ( ))  0 if   .
Inequality (8.4.2) and the fact that (K-S(t))   K  S (t ) imply that
E[e-r v( , S ( )) | F (t )]  E[e-r ( K  S ( ))  | F (t )]
 E[e-r ( K  S ( )) | F (t )]
Putting (8.4.12) and (8.4.13) together, we have
(8.4.13)
e-rt v(t , S (t ))  E[e-r ( K  S ( )) | F (t )]
(8.4.14)
Because S(t) is a Markov process, the right-hand side of (8.4.14) is
e-rt v(t , x)  E[e-r ( K  S ( )) | S (t )  x]
since (8.4.15) holds for any   t,T , we conclude that
v(t , x)  max E[e-r( t ) ( K  S ( )) | S (t )  x]
 t ,T
(8.4.15)
(8.4.16)
For the reverse inequality, we recall form Theorem 8.4.2 that the
stopped process e-r(t  * ) v(t   * , S (t   * )) is a martingale,where  *
defined by (8.4.10) is such that v( * , S ( * ))  K  S ( * ) if  *  .
e-r(t  * ) v(t   * , S (t   * ))  E[e-r(T  * ) v(T   * , S (T   * )) | F(t)]
e-rt v(t , S (t))  E[e-r * v( * , S ( * )){ * }  e -rT v(T , S (T )){ * } | F (t )]
 E[e-r * v( * , S ( * )) | F (t )]
Finally, because v( * ,S( * ))  K-S( * ) if  *  
v(t , x)  E[e-r( * t ) ( K  S ( * )) | S (t )  x]
(8.4.17)
Equality (8.4.17) shows that equality must hold in (8.4.16),
and this is (8.4.1).

similar documents