The Mathematics of Arbitrage

30 2 Models of Financial Markets on Finite Probability Spaces

The converse inclusion follows by symmetry. In the financial market mod-
elled byXwe can chooseWt=V^1 tas num ́eraire. The passage fromXtoSext
is then done by usingWas a new num ́eraire and the inclusion we just proved
then yields

K(S)⊂

1

WT

K(X)=VTK(X).

This shows thatK(S)=VTK(X) as required.

Theorem 2.5.4 (change of num ́eraire). Let S satisfy the no-arbitrage
condition, let V =1+H^0 ·S be such that Vt > 0 for all t,andlet

X=

(

S^1

V,...,

Sd

V,

1

V

)

.ThenXsatisfies the no-arbitrage condition too and

Qbelongs toMe(S)if and only if the measureQ′defined bydQ′=VTdQ
belongs toMe(X).

Proof.SinceK(X)=V^1 TK(S)wehavethatXsatisfies the no-arbitrage prop-
erty by directly verifying Definition 2.2.3. By Proposition 2.2.10 an equivalent
probability measureQis inMe(S) if and only if, for allf∈K(S), we have
EQ[f] = 0. But this is the same as

EQ

[

VT

f

VT

]

= 0, for allf∈K(S),

which is equivalent toEQ[VTg] = 0 for allg∈K(X). This happens if and
only if the probability measureQ′, defined asdQ′=VTdQ,isinMe(X).
(Note that by the martingale property we haveEQ[VT]=V 0 =1.)

Remark 2.5.5.The process (Vt)Tt=0is aQ-martingale for everyQ∈Me(S).
Now ifdQ′ =VTdQ, then we have the following so-called Bayes’ rule for
f∈L∞(Ω,F,P):

EQ′[f|Ft]=

EQ[fVT|Ft]

EQ[VT|Ft]

=

EQ[fVT|Ft]

Vt

=EQ

[

f

VT

Vt

∣

∣

∣

∣Ft

]

.

The previous equality can also be written as

VtEQ′[f|Ft]=EQ[fVT|Ft].

From this it follows that (Zt)Tt=0is aQ′-martingale if and only if (ZtVt)Tt=0is a
Q-martingale. This statement can also be seen as the martingale formulation
of Theorem 2.5.4 above.