The Mathematics of Arbitrage

2.5 Change of Num ́eraire 27

Proof.As observed in the proof of Theorem 2.3.2 and Proposition 2.3.1, every
Q∈Me(S) can be written as ddQP =f 0 dQ

0

dP whereQ

(^0) |F
0 =P|F 0 ,Q
(^0) ∈
Me(S,F 0 )andwheref 0 isF 0 -measurable, strictly positive andEP[f 0 ]=1.
But otherwisef 0 is arbitrary. Now forQ∈Me(S)wehaveddQP=f 0 dQ
0
dP and
hence
EQ[f]=EQ

[

EQ 0 [f|F 0 ]

]

≤EQ[a 1 ]=EP[a 1 f 0 ].

Thus supQ∈Me(S)EQ[f]≤‖a 1 ‖∞.
To prove the converse inequality we need some more approximations. First
for givenε>0, we choosef 0 ,F 0 -measurable,f 0 >0,EP[f 0 ]=1andso
thatEP[f 0 a 1 ]≥‖a 1 ‖∞−ε.Givenf 0 we may takeQ^1 ∈Me(S,F 0 )sothat
E[f 0 (a 1 −EQ 1 [f|F 0 ])]≤ε. This is possible since the family{EQ[f|F 0 ]|Q∈
Me(S,F 0 )}is a lattice and since all these functions are in theL∞-ball with

radius‖f‖∞.NowtakeQ^0 defined bydQ

0

dP =f^0

dQ^1

dP. ClearlyQ

(^0) ∈Me(S)
and we have
EQ 0 [f]=EP

[

f 0

dQ^1

dP

f

]

=EP

[

f 0 EQ 1 [f|F 0 ]

]

sinceQ^1 |F 0 =P

≥EP[f 0 a 1 ]−εby the choice ofQ^1

≥‖a 1 ‖∞− 2 εby the choice off 0. 

2.5 Change of Num ́eraire

In the previous sections we have developed the basic tools for the pricing and
hedging of derivative securities. Recall that we did our analysis in adiscounted
modelwhere we did choose one of the traded assets as num ́eraire.
How do these things change, when we pass to a new num ́eraire, i.e., a new
unit in which we denote the values of the stocks? Of course, the arbitrage
free prices should remain unchanged (after denominating things in the new
num ́eraire), as the notion of arbitrage should not depend on whether we do
the book-keeping ineor in $. On the other hand, we shall see that the risk-
neutral measuresQdo depend on the choice of num ́eraire. We will also show
how, conversely, a change of risk neutral measures corresponds to a change of
num ́eraire.

Let us analyse the situation in the appropriate degree of generality: the
model of a financial marketŜ=(Ŝ^0 t,Ŝt^1 ,...,Ŝdt)Tt=0is defined as in 2.1 above.
Recall that we assumed that the traded assetŜ^0 serves as num ́eraire, i.e., we

have passed from the valueŜtjof thej’th asset at timetto its valueStj=
Ŝjt
Ŝ^0 t,
expressed in units ofŜt^0. This led us in (2.3) to the introduction of the process