8.3.4 Probabilistic Characterization of the Put Price Presenter: Chih-tai,Shen Jan,05 2012 Stat,NCU.
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Transcript of 8.3.4 Probabilistic Characterization of the Put Price Presenter: Chih-tai,Shen Jan,05 2012 Stat,NCU.
Theorem 8.3.5
Let S(t) be the stock price given by (8.3.1) and let be given by (8.3.9) with . Then (S(t)) is a supermartingale under , and the stopped process is a martingale .
(8.3.1) (8.3.9) Supermartingale:
Theorem 8.3.5
PROOF:use Ito-Doeblin formula
d[]=(S(t))d(S(t))+(S(t))d(S(t)d(S(t)]
=[ (S(t))+ (S(t))]dt
+ (S(t))d
because of (8.3.16) and (8.3.17)the dt term is either 0 or depend on whether
Theorem 8.3.5
When it has the downward tendency. If the initial stock price above ,then prior to the time when the stock price first reaches , the dt term in is zero and hence ) is a martingale.
Corollary 8.3.6
Recall that the set of all stopping times ,not just those of the (8.3.9) . we have
where is the initial stock price . In other words ,s the perpetual American put price of Definition 8.3.1
Corollary 8.3.6
PROOF: (i)because is a supermartingale under , we have from Theorem 8.2.4 (optional sampling) for every stopping time
Supermartingale:
*( ) max [ ( ( ))]r
Lv x E e K S
T
*
* *
( ) ( )
Dominated Convergence Theorem (Theorem 1.4.9):
( ( )) is bounded
lim { ( ( ))} {lim ( ( ))}
L
r t r tL L
t t
v S t
E e v S t E e v S t
* *( ) [ ( ( ))] [ ( ( ))] r r
L Lv x E e v S E e K S T
(8.3.18): ( ) ( ) for all 0 v x K x x
* ( ) max [ ( ( ))]rLv x E e K S
T
(ii)
*
*
*
*
* * *
* * * *
* * * *
*
( )
Dominated Convergence Theorem
* *
replace by
( ( )) : martingale under P
( ) [ ( ( ))]
( ( )) ( ) ( ) ( ( ))
if (and i
L
L
L L L L
L
r t
L
r
L L L
r r r r
L L L L
L
e v S t
v x E e v S
e v S e v L e K L e K S
*
*
* *
*
* *
s interpreted to ve zero if ), we see that
( ) [ ( ( ))] (8.3.23)
It follow that
( ) max [ ( ( ))]
L
L
L
r
L L
r
L L
v x E e K S
v x E e K S
T
*( ) max [ ( ( ))]r
Lv x E e K S
T
Discounted European option prices are martingales under the risk-neutral probability measure. Discounted American option prices are martingales up to the time they should be exercised. If they are not exercised when they should be, they tend downward.
Since a martingale is a special case of a supermartingale, and processes that tend downward are supermartingales, discounted American option prices are supermartingales.