Bass. .Financial.Mathematics.(2003).[sharethefiles.com]

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would have received if you had waited to exercise until time T . Therefore an American
call is worth no more than a European call, and hence its value must be the same as that
of a European call.
This argument does not work for puts, because selling stock gives you some money
on which you will receive interest, so it may be advantageous to exercise early. (A put is
the option to sell a stock at a price K at time T .)
Here is the more rigorous argument. Let g(x) be convex with g(0) = 0. Certainly
g(x) = (x - K)+ is such a function. We have
g(�x) = g(�x + (1 - �) � 0) d" �g(x) + (1 - �)g(0) = �g(x).
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By Jensen s inequality,
1
E [(1 + r)-(k+1)g(Sk+1) | Fk] = (1 + r)-kE g(Sk+1) | Fk
1 + r
1
e" (1 + r)-kE g Sk+1 | Fk
1 + r
1
e" (1 + r)-kg E Sk+1 | Fk
1 + r
= (1 + r)-kg(Sk).
So (1 + r)-kg(Sk) is a submartingale. By optional stopping,
E [(1 + r)-� g(S� )] d" E [(1 + r)-ng(Sn)],
so � a" n always does best.
8. Continuous random variables.
We are now going to start working towards continuous times and stocks that can
take any positive number as a value, so we need to prepare by extending some of our
definitions.
Given any random variable X, we can approximate it by r.v s Xn that are discrete.
We let
n2n
i
Xn = 1(i/2n .
2n d"X
i=-n2n
In words, if X(�) lies between -n and n, we let Xn(�) be the closest value i/2n that
is less than or equal to X(�). For � where |X(�)| > n we set Xn(�) = 0. Clearly the
Xn are discrete, and approximate X. In fact, on the set where |X| d" n, we have that
|X(�) - Xn(�)| d" 2-n.
For reasonable X we are going to define E X = lim E Xn. There are some things
one wants to prove, but all this has been worked out in measure theory and the theory of
the Lebesgue integral. Let us confine ourselves here to showing this definition is the same
as the usual one when X has a density.
Recall X has a density fX if
b
P(X " [a, b]) = fX(x)dx
a
for all a and b. In this case
"
E X = xfX(x)dx.
-"
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With our definition of Xn we have
(i+1)/2n
P(Xn = i/2n) = P(X " [i/2n, (i + 1)/2n)) = fX(x)dx.
i/2n
Then
(i+1)/2n
i i
E Xn = P(Xn = i/2n) = fX(x)dx.
2n 2n
i/2n
i i
Since x differs from i/2n by at most 1/2n when x " [i/2n, (i + 1)/2n), this will tend to
xfX(x)dx, unless the contribution to the integral for |x| e" n does not go to 0 as n �! ".
As long as |x|fX(x)dx
We also need an extension of the definition of conditional probability. A r.v. is G
measurable if (X > a) " G for every a. How do we define E [Z | G] when G is not generated
by a countable collection of disjoint sets?
Again, there is a completely worked out theory that holds in all cases. Let us give
a definition that is equivalent that works except for a very few cases. Let us suppose that
for each n the �-field Gn is finitely generated. This means that Gn is generated by finitely
many disjoint sets Bn1, . . . , Bnm . So for each n, the number of Bni is finite but arbitrary,
n
the Bni are disjoint, and their union is &!. Suppose also that G1 �" G2 �" � � �. Now *"nGn
will not in general be a �-field, but suppose G is the smallest �-field that contains all the
Gn. Finally, define P(A | G) = lim P(A | Gn).
This is a fairly general set-up. For example, let &! be the real line and let Gn be
generated by the sets (-", n), [n, ") and [i/2n, (i + 1)/2n). Then G will contain every
interval that is closed on the left and open on the right, hence G must be the �-field that
one works with when one talks about Lebesgue measure on the line.
The question that one might ask is: how does one know the limit exists? Since the
Gn increase, we know that Mn = P(A | Gn) is a martingale with respect to the Gn. It is
certainly bounded above by 1 and bounded below by 0, so by the martingale convergence
theorem, it must have a limit as n �! ".
Once one has a definition of conditional probability, one defines conditional expec-
tation by what one expects. If X is discrete, one can write X as aj1A and then one
j
j
defines
E [X | G] = ajP(Aj | Gn).
j
If the X is not discrete, one approximates as above. One has to worry about convergence,
but everything does go through.
With this extended definition of conditional expectation, do all the properties of
Section 2 hold? The answer is yes, and the proofs are by taking limits of the discrete
approximations.
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We will be talking about stochastic processes. Previously we discussed sequences
S1, S2, . . . of r.v. s. Now we want to talk about processes Yt for t e" 0. We typically let
Ft be the smallest �-field with respect to which Ys is measurable for all s d" t. As you
might imagine, there are a few technicalities one has to worry about. We will try to avoid
thinking about them as much as possible.
A continuous time martingale (or submartingale) is what one expects: Mt is in- [ Pobierz całość w formacie PDF ]

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