From our formulas for stochastic integrals, this means

t

|Hs ’ Hs |2 ds ’ 0.

n m

E

0

This says that Hs is a Cauchy sequence in the space L2 (with respect to the norm ·

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 ’ Hs |2 ds ’ 0.

n

E

0

n

In particular Hs ’ Hs , and this implies Hs is adapted. Another consequence, due to Fatou™s

t 2

lemma, is that E 0 Hs ds.

t

Let Ut = 0 Hs dWs . Then as above,

t

n 2

(Hs ’ Hs )2 ds ’ 0.

n

E |(V ’ cn ) ’ Ut | = E

0

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

81

Note 3. Here is the proof of Proposition 18.3. By Ito™s formula with Xs = ’iuWs + u2 s/2

and f (x) = ex ,

t t

Xt Xs

eXs (u2 /2)ds

e =1+ e (’iu)dWs +

0 0

t

eXs (’iu)2 ds

1

+ 2

0

t

eXs dWs .

= 1 ’ iu

0

2

If we multiply both sides by e’u t/2

, which is a constant and hence adapted, we obtain

t

’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 ∞ 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 ’∞ to

∞, 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 ’ Wt ).

From Proposition 18.3 we now have that

t u

g(Wu ’ Wt ) = d +

f (Wt ) = c + Hs dWs , Ks dWs .

0 t

Set H r = Hr if 0 ¤ s < t and 0 otherwise. Set K r = Kr if s ¤ 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

82

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.

83

19. Completeness.

Now let Pt be a geometric Brownian motion. As we mentioned in Section 16, if

Pt = P0 exp(σWt + (µ ’ r ’ σ 2 /2)t), then given Pt we can determine Wt and vice versa,

so the σ ¬elds generated by Pt and Wt are the same. Recall Pt satis¬es

dPt = σPt dWt + (µ ’ r)Pt dt.

De¬ne a new probability P by

dP

= Mt = exp(aWt ’ a2 t/2).

dP

By the Girsanov theorem,

Wt = Wt ’ at

is a Brownian motion under P. So

dPt = σPt dWt + σPt adt + (µ ’ r)Pt dt.

If we choose a = ’(µ ’ r)/σ, we then have

dPt = σPt 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 = σ ’1 Pt’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 < ∞ and

t

V =c+ Hs d W s .

0

But then using (19.2) we have

t

Hs σ ’1 Ps dPs .

’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 ≥ 0 be a ¬xed

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 “no arbitrage” 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 ’ 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 pro¬t of erT (W0 ’ 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 ’Ks shares of stock at time s. By the same argument, since

we can™t get a riskless pro¬t, we must have W0 ≥ 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 = e’rt (St ’ K)+ = (e’rt St ’ e’rt K)+ = (Pt ’ e’rt K)+ .

Recall that under P the stock price satis¬es

dPt = σPt dWt ,

where Wt is a Brownian motion under P. So then

2

Pt = P0 eσWt ’σ t/2

.

85

Hence

E V = E [(PT ’ e’rT K)+ ] (20.2)

2

= E [(P0 eσWT ’(σ ’ e’rT K)+ ].

/2)T

2

We know the density of WT is just (2πT )’1/2 e’y /(2T ) , so we can do some calculations

(see Note 1) and end up with the famous Black-Scholes formula:

W0 = x¦(g(x, T )) ’ Ke’rT ¦(h(x, T )),

z 2

e’y /2

√1

where ¦(z) = dy, x = P0 = S0 ,

’∞

2π

log(x/K) + (r + σ 2 /2)T

√

g(x, T ) = ,

σT

√

h(x, T ) = g(x, T ) ’ σ T .

It is of considerable interest that the ¬nal formula depends on σ but is completely

independent of µ. The reason for that can be explained as follows. Under P the process Pt

satis¬es dPt = σPt dWt , where Wt is a Brownian motion. Therefore, similarly to formulas

we have already done,

2

Pt = P0 eσWt ’σ t/2 ,

and there is no µ present here. (We used the Girsanov formula to get rid of the µ.) The

price of the option V is

E [PT ’ e’rT K]+ , (20.3)

which is independent of µ since Pt is.

Note 1. We want to calculate

2

E (xeσWT ’σ ’ e’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

√

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 √

σ T Z’σ 2 T /2

> e’rT K

xe

if and only if

√

log x + σ T Z ’ σ 2 T /2 > ’r + log K,

86

or if

Z > (σ 2 T /2) ’ r + log K ’ log x.