Change of num´eraire
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Chapter 33
Change of num
´
eraire
Consider a Brownian motion driven market model with time horizon
T
. For now, we will have
one asset, which we call a “stock” even though in applications it will usually be an interest rate
dependent claim. The price of the stock is modeled by
dS t= rt Stdt + tS t dW t;
(0.1)
where the interest rate process
rt
and the volatility process
t
are adapted to some filtration
fF t; 0 t T
g
.
W
is a Brownian motion relative to this filtration, but
fF t; 0 t T
g
may be larger than the filtration generated by
W
.
This is not a geometric Brownian motion model. We are particularly interested in the case that the
interest rate is stochastic, given by a term structure model we have not yet specified.
We shall work only under the risk-neutral measure, which is reflected by the fact that the mean rate
of return for the stock is
rt
.
We define the accumulation factor
t = exp
Z
t
0
ru du
;
so that the discounted stock price
S t
t
is a martingale. Indeed,
d
S t
t
=
S t
t
t dW t:
The zero-coupon bond prices are given by
B t; T =IE
"
exp
,
Z
T
t
ru du
F t
= IE
t
T
F t
;
325
326
so
B t; T
t
= IE
1
T
F t
is also a martingale (tower property).
The
T
-forward price
F t; T
of the stock is the price set at time
t
for delivery of one share of stock
at time
T
with payment at time
T
. The value of the forward contract at time
t
is zero, so
0=IE
t
T
ST,Ft; T
F t
= tIE
S T
T
Ft
, Ft; T IE
t
T
F t
= t
S t
t
, F t; T B t; T
= S t , F t; T B t; T
Therefore,
F t; T =
St
Bt; T
:
Definition 33.1 (Num
´
eraire) Any asset in the model whose price is always strictly positive can be
taken as the num´eraire. We then denominate all other assets in units of this num´eraire.
Example 33.1 (Money market as num´eraire) The money market could be the num´eraire. At time
t
,the
stock is worth
S t
t
units of money market and the
T
-maturity bond is worth
Bt;T
t
units of money market.
Example 33.2 (Bond as num´eraire) The
T
-maturity bond could be the num´eraire. At time
t T
, the stock
is worth
F t; T
units of
T
-maturity bond and the
T
-maturity bond is worth 1 unit.
We will say that a probability measure
IP
N
is risk-neutral for the num´eraire
N
if every asset price,
divided by
N
, is a martingale under
IP
N
. The original probability measure
IP
is risk-neutral for the
num´eraire
(Example 33.1).
Theorem 0.71 Let
N
be a num´eraire, i.e., the price process for some asset whose price is always
strictly positive. Then
IP
N
defined by
IP
N
A=
1
N0
Z
A
N T
T
dIP; 8A 2FT
;
is risk-neutral for
N
.
CHAPTER 33. Change of num´eraire
327
Note:
IP
and
IP
N
are equivalent, i.e., have the same probability zero sets, and
IP A=N0
Z
A
T
N T
dIP
N
; 8A 2FT
:
Proof: Because
N
is the price process for some asset,
N=
is a martingale under
IP
. Therefore,
IP
N
=
1
N 0
Z
N T
T
dIP
=
1
N 0
:IE
N T
T
=
1
N 0
N 0
0
=1;
and we see that
IP
N
is a probability measure.
Let
Y
be an asset price. Under
IP
,
Y=
is a martingale. We must show that under
IP
N
,
Y=N
is
a martingale. For this, we need to recall how to combine conditional expectations with change of
measure (Lemma 1.54). If
0 t T T
and
X
is
F T
-measurable, then
IE
N
X
F t
=
N 0 t
N t
IE
N T
N 0 T
X
F t
=
t
N t
IE
N T
T
X
F t
:
Therefore,
IE
N
Y T
N T
F t
=
t
N t
IE
N T
T
Y T
N T
F t
=
t
N t
Y t
t
=
Y t
N t
;
which is the martingale property for
Y=N
under
IP
N
.
33.1 Bond price as num
´
eraire
Fix
T 2 0;T
and let
B t; T
be the num´eraire. The risk-neutral measure for this num´eraire is
IP
T
A=
1
B0;T
Z
A
BT; T
T
dIP
=
1
B 0;T
Z
A
1
T
dIP 8A 2FT:
328
Because this bond is not defined after time
T
, we change the measure only “up to time
T
”, i.e.,
using
1
B 0;T
B T;T
T
and only for
A 2FT
.
IP
T
is called the
T
-forward measure. Denominated in units of
T
-maturity bond, the value of the
stock is
F t; T =
St
Bt; T
; 0 t T:
This is a martingale under
IP
T
, and so has a differential of the form
dF t; T =
F
t; T F t; T dW
T
t; 0 t T;
(1.1)
i.e., a differential without a
dt
term. The process
fW
T
;0tTg
is a Brownian motion under
IP
T
. We may assume without loss of generality that
F
t; T 0
.
We write
F t
rather than
F t; T
from now on.
33.2 Stock price as num
´
eraire
Let
S t
be the num´eraire. In terms of this num´eraire, the stock price is identically 1. The risk-
neutral measure under this num´eraire is
IP
S
A=
1
S0
Z
A
S T
T
dIP; 8A 2FT
:
Denominated in shares of stock, the value of the
T
-maturity bond is
B t; T
S t
=
1
F t
:
This is a martingale under
IP
S
, and so has a differential of the form
d
1
F t
= t; T
1
F t
dW
S
t;
(2.1)
where
fW
S
t; 0 t T
g
is a Brownian motion under
IP
S
. We may assume without loss of
generality that
t; T 0
.
Theorem 2.72 The volatility
t; T
in (2.1) is equal to the volatility
F
t; T
in (1.1). In other
words, (2.1) can be rewritten as
d
1
F t
=
F
t; T
1
F t
dW
S
t;
(2.1’)
CHAPTER 33. Change of num´eraire
329
Proof: Let
g x=1=x
,so
g
0
x=,1=x
2
;g
00
x=2=x
3
.Then
d
1
F t
= dg F t
= g
0
F t dF t+
1
2
g
00
Ft dF t dF t
= ,
1
F
2
t
F
t; T F t; T dW
T
t+
1
F
3
t
2
F
t; T F
2
t; T dt
=
1
F t
h
,
F
t; T dW
T
t+
2
F
t; T dt
i
=
F
t; T
1
F t
,dW
T
t+
F
t; T dt:
Under
IP
T
; ,W
T
is a Brownian motion. Under this measure,
1
F t
has volatility
F
t; T
and mean
rate of return
2
F
t; T
. The change of measure from
IP
T
to
IP
S
makes
1
F t
a martingale, i.e., it
changes the mean return to zero, but the change of measure does not affect the volatility. Therefore,
t; T
in (2.1) must be
F
t; T
and
W
S
must be
W
S
t=,W
T
t+
Z
t
0
F
u; T du:
33.3 Merton option pricing formula
The price at time zero of a European call is
V 0 = IE
1
T
S T , K
+
= IE
S T
T
1
fS T K g
, KIE
1
T
1
fST K g
= S 0
Z
fS T K g
S T
S 0 T
dIP , KB0;T
Z
fSTK g
1
B 0;TT
dIP
= S 0IP
S
fS T Kg,KB0;TIP
T
fST Kg
=S0IP
S
fF T Kg,KB0;TIP
T
fFT Kg
=S0IP
S
1
F T
1
K
, KB0;TIP
T
fFT Kg:
