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From our formulas for stochastic integrals, this means
t
|Hs в€’ Hs |2 ds в†’ 0.
n m
E
0

This says that Hs is a Cauchy sequence in the space L2 (with respect to the norm В·
n
given
2
1/2
t
by Y 2 = E 0 Ys2 ds ). Measure theory tells us that L2 is a complete metric space, so
there exists Hs such that
t
|Hs в€’ Hs |2 ds в†’ 0.
n
E
0
n
In particular Hs в†’ Hs , and this implies Hs is adapted. Another consequence, due to FatouвЂ™s
t 2
lemma, is that E 0 Hs ds.
t
Let Ut = 0 Hs dWs . Then as above,
t
n 2
(Hs в€’ Hs )2 ds в†’ 0.
n
E |(V в€’ cn ) в€’ Ut | = E
0

Therefore Ut = V в€’ c, and U has the desired form.

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Note 3. Here is the proof of Proposition 18.3. By ItoвЂ™s formula with Xs = в€’iuWs + u2 s/2
and f (x) = ex ,
t t
Xt Xs
eXs (u2 /2)ds
e =1+ e (в€’iu)dWs +
0 0
t
eXs (в€’iu)2 ds
1
+ 2
0
t
eXs dWs .
= 1 в€’ iu
0
2
If we multiply both sides by eв€’u t/2
, which is a constant and hence adapted, we obtain
t
в€’iuWt u
e = cu + Hs dWs (18.3)
0

for an appropriate constant cu and integrand H u .
If f is a smooth function (e.g., C в€ћ with compact support), then its Fourier transform
f will also be very nice. So if we multiply (18.3) by f (u) and integrate over u from в€’в€ћ to
в€ћ, we obtain
t
f (Wt ) = c + Hs dWs
0
for some constant c and some adapted integrand H. (We implicitly used Proposition 18.2,
because we approximate our integral by Riemann sums, and then take a limit.) Now using
Proposition 18.2 we take limits and obtain the proposition.

Note 4. The argument is by induction; let us do the case n = 2 for clarity. So we suppose

V = f (Wt )g(Wu в€’ Wt ).

From Proposition 18.3 we now have that
t u
g(Wu в€’ Wt ) = d +
f (Wt ) = c + Hs dWs , Ks dWs .
0 t

Set H r = Hr if 0 в‰¤ s < t and 0 otherwise. Set K r = Kr if s в‰¤ r < t and 0 otherwise. Let
s s
Xs = c + 0 H r dWr and Ys = d + 0 K r dWr . Then
s
X, Y = H r K r dr = 0.
s
0

Then by the Ito product formula,
s s
Xs Ys = X0 Y0 + Xr dYr + Yr dXr
0 0
+ X, Y s
s
= cd + [Xr K r + Yr H r ]dWr .
0

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If we now take s = u, that is exactly what we wanted. Note that Xr K r + Yr H r is 0 if r > u;
this is needed to do the general induction step.

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19. Completeness.
Now let Pt be a geometric Brownian motion. As we mentioned in Section 16, if
Pt = P0 exp(ПѓWt + (Вµ в€’ r в€’ Пѓ 2 /2)t), then given Pt we can determine Wt and vice versa,
so the Пѓ п¬Ѓelds generated by Pt and Wt are the same. Recall Pt satisп¬Ѓes

dPt = ПѓPt dWt + (Вµ в€’ r)Pt dt.

Deп¬Ѓne a new probability P by
dP
= Mt = exp(aWt в€’ a2 t/2).
dP
By the Girsanov theorem,
Wt = Wt в€’ at
is a Brownian motion under P. So

dPt = ПѓPt dWt + ПѓPt adt + (Вµ в€’ r)Pt dt.

If we choose a = в€’(Вµ в€’ r)/Пѓ, we then have

dPt = ПѓPt dWt . (19.1)

Since Wt is a Brownian motion under P, then Pt must be a martingale, since it is a
stochastic integral of a Brownian motion. We can rewrite (19.1) as

dWt = Пѓ в€’1 Ptв€’1 dPt . (19.2)

Given a Ft measurable variable V , we know by Theorem 18.1 that there exist a
t 2
constant and an adapted process Hs such that E 0 Hs ds < в€ћ and
t
V =c+ Hs d W s .
0

But then using (19.2) we have
t
Hs Пѓ в€’1 Ps dPs .
в€’1
V =c+
0

We have therefore proved
Theorem 19.1. If Pt is a geometric Brownian motion and V is Ft measurable and square
integrable, then there exist a constant c and an adapted process Ks such that
t
V =c+ Ks dPs .
0

Moreover, there is a probability P under which Pt is a martingale.
The probability P is called the risk-neutral measure. Under P the present day value
of the stock price is a martingale.

84
20. Black-Scholes formula, I.
We can now derive the formula for the price of any option. Let T в‰Ґ 0 be a п¬Ѓxed
real. If V is FT measurable, we have by Theorem 19.1 that

T
V =c+ Ks dPs , (20.1)
0

and under P, the process Ps is a martingale.

Theorem 20.1. The price of V must be E V .

Proof. This is the вЂњno arbitrageвЂќ principle again. Suppose the price of the option V at
time 0 is W . Starting with 0 dollars, we can sell the option V for W dollars, and use the
W dollars to buy and trade shares of the stock. In fact, if we use c of those dollars, and
invest according to the strategy of holding Ks shares at time s, then at time T we will
have
erT (W0 в€’ c) + V

dollars. At time T the buyer of our option exercises it and we use V dollars to meet that
obligation. That leaves us a proп¬Ѓt of erT (W0 в€’ c) if W0 > c, without any risk. Therefore
W0 must be less than or equal to c. If W0 < c, we just reverse things: we buy the option
instead of sell it, and hold в€’Ks shares of stock at time s. By the same argument, since
we canвЂ™t get a riskless proп¬Ѓt, we must have W0 в‰Ґ c, or W0 = c.
Finally, under P the process Pt is a martingale. So taking expectations in (20.1),
we obtain
E V = c.

The formula in the statement of Theorem 20.1. is amenable to calculation. Suppose
we have the standard European option, where

V = eв€’rt (St в€’ K)+ = (eв€’rt St в€’ eв€’rt K)+ = (Pt в€’ eв€’rt K)+ .

Recall that under P the stock price satisп¬Ѓes

dPt = ПѓPt dWt ,

where Wt is a Brownian motion under P. So then
2
Pt = P0 eПѓWt в€’Пѓ t/2
.

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Hence

E V = E [(PT в€’ eв€’rT K)+ ] (20.2)
2
= E [(P0 eПѓWT в€’(Пѓ в€’ eв€’rT K)+ ].
/2)T

2
We know the density of WT is just (2ПЂT )в€’1/2 eв€’y /(2T ) , so we can do some calculations
(see Note 1) and end up with the famous Black-Scholes formula:

W0 = xО¦(g(x, T )) в€’ Keв€’rT О¦(h(x, T )),

z 2
eв€’y /2
в€љ1
where О¦(z) = dy, x = P0 = S0 ,
в€’в€ћ
2ПЂ

log(x/K) + (r + Пѓ 2 /2)T
в€љ
g(x, T ) = ,
ПѓT
в€љ
h(x, T ) = g(x, T ) в€’ Пѓ T .

It is of considerable interest that the п¬Ѓnal formula depends on Пѓ but is completely
independent of Вµ. The reason for that can be explained as follows. Under P the process Pt
satisп¬Ѓes dPt = ПѓPt dWt , where Wt is a Brownian motion. Therefore, similarly to formulas
we have already done,
2
Pt = P0 eПѓWt в€’Пѓ t/2 ,

and there is no Вµ present here. (We used the Girsanov formula to get rid of the Вµ.) The
price of the option V is
E [PT в€’ eв€’rT K]+ , (20.3)

which is independent of Вµ since Pt is.

Note 1. We want to calculate
2
E (xeПѓWT в€’Пѓ в€’ eв€’rT K)+ ,
T /2
(20.4)

where Wt is a Brownian motion under P and we write x for P0 = S0 . Since WT is a normal
в€љ
random vairable with mean 0 and variance T , we can write it as T Z, where Z is a standard
mean 0 variance 1 normal random variable.
Now в€љ
Пѓ T Zв€’Пѓ 2 T /2
> eв€’rT K
xe

if and only if
в€љ
log x + Пѓ T Z в€’ Пѓ 2 T /2 > в€’r + log K,

86
or if
Z > (Пѓ 2 T /2) в€’ r + log K в€’ log x.
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