The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.

1. Pricing Exotics using Change of Numeraire
Swati Mital
March 28, 2016
1 Preliminaries
The intention of this essay is to show how change of numeraire technique is
used in pricing derivatives with complex payoffs. In the first instance, we
apply the technique to pricing European Call Options and then use the same
method to price an exotic Power Option.
Fundamental Theorem of Asset Pricing tells us that under no arbitrage
conditions there exists at least one risk neutral measure that is equivalent to
the original probability measure. Moreover, any discounted traded asset is a
martingale under this measure.
Hence, we can derive time-t value of a derivative P(t) = P(t, S(t)) on an
underlying asset S(t) with payoff P(T) at time T by choosing an Equivalent
Martingale Measure Q and a numeraire N(t) such that that discounted value
of the derivative is a martingale in it’s own filtration. This is expressed as,
P(t)
N(t)
= EQ P(T)
N(T)
Ft (1)
Often in derivative pricing, we change the numeraire and, hence, select a
new measure such that Equation (1) is simplified. This is most commonly
done in pricing of interest rate derivatives but has applications in equities
and other asset classes.
1

2. 2 European Call Options
We start off with the simplest case of a European Call Option and derive the
Black Scholes formula using Change of Numeraire technique. We know that
the payoff of a European call option at time T is, C(T) = (ST − K)+
. We
select the cash numeraire to price this option. A cash numeraire is simply
the discounted value at time 0 of $1 paid at time t using interest rate r(u)
B(t) = e− t
0 rudu
(2)
Using Equation (1), we can now write the price of the European Call
Option at time t, C(t), as,
C(t) = B(t)EQ (S(T) − K)+
B(T)
Ft
= B(t)EQ S(T)
B(T)
1S(T)≥K Ft − B(t)EQ K
B(T)
1S(T)≥K Ft
(3)
We change the measure in first expression of Equation (3) from Q where
B(t) is the numeraire to measure QS
where the stock price S(t) is the nu-
meraire.
In the second expression we change the numeraire to the price of a zero
coupon bond, P(t, T) = e− T
t rudu
= B(t)
B(T)
, in the forward T-measure, QT
.
The change of numeraire is a trick to simplify Equation (3) into something
we can easily solve. This gives us,
C(t) = S(t)EQS
1S(T)≥K Ft − KP(t, T)EQT 1
P(T, T)
1S(T)≥K Ft
= S(t)EQS
1S(T)≥K Ft − KP(t, T)EQT
1S(T)≥K Ft
= S(t)QS
(S(T) ≥ K) − KP(t, T)QT
(S(T) ≥ K)
(4)
So far you may have noticed that we haven’t relied on the particular
2

3. dynamics of the underlying asset price process to get to Equation (4). One
of the assumptions of the Black Scholes formula is that the underlying stock
price follows a Geometric Brownian Motion. That means under Q measure
S(t) = S(0) exp (r − σ2
2
)t + σWQ
t where WQ
is a Q-Brownian Motion.
Hence we can derive Equation (5) below where Φ is the CDF of a standard
normal distribution.
Q (S(t) ≥ K) = Q S(0) exp (r −
σ2
2
)t + σWQ
t ≥ K
= Q

WQ
t ≤
log S(0)
K
+ (r − σ2
2
)t
σ


= Φ


log S(0)
K
+ (r − σ2
2
)t
σ
√
t


(5)
We have derived the above assuming that the Brownian Motion was under
the original Q measure. However, in Equation (4), we changed the measure
to QS
and QT
. So, we use Girsanov’s Theorem in Brownian setting to change
the drift of the underlying asset process.
First to note that since under the Black-Scholes assumption, the interest
rates are constant, the price of a zero coupon bond reduces to P(t, T) =
e−r(T−t)
and, hence, the forward-T measure, QT
is the same as the Q measure.
Hence, that leaves us to deal with QS
Brownian Motion.
dQS
dQ
=
B(T)/B(t)
S(t)/S(T)
=
e−r(T−t)
S(t) exp((r − σ2
/2)(T − t) + σ(WT − Wt))
S(t)
= exp σ(WT − Wt) −
σ2
2
(T − t)
(6)
Hence, by, Girsanov’s Theorem, we can express QS
-Brownian Motion in
terms of Q-Brownian Motion as WQS
t = WQ
t − σt. We use this to transform
Equation (5) as shown below,
3

4. QS
(S(t) ≥ K) = QS

WQS
t + σ2
t ≤
log S(0)
K
+ (r − σ2
2
)t
σ


= Φ


log S(0)
K
+ (r + σ2
2
)t
σ
√
t


(7)
We can now substitute Equation (5) and (7) in Equation (4) to give us
the Black-Scholes formula for European Call Option.
C(t) = S(t)Φ


log S(t)
K
+ (r + σ2
2
)(T − t)
σ
√
T − t

−Ke−r(T−t)
Φ


log S(t)
K
+ (r − σ2
2
)(T − t)
σ
√
T − t


(8)
Although you may never have to derive the Black Scholes formula using
the Change of Numeraire technique, I find the above illustration very useful
when thinking about complex option payoffs and changing to Equivalent
Martingale Measure to price them.
3 Exotic Power Options
Let’s consider a European option with payoff on power of the stock price and
power of the strike. At expiry, it’s payoff is given by,
V (ST , T) =
1
K2
(K3
− S(T)3
)+
(9)
We assume that the underlying price process, S(t) follows the Geometric
Brownian Motion with risk free rate r and no dividend. We apply Ito’s
Lemma and derive the Stochastic Differential Equation for S3
as,
dS3
S3
= (3r + 3σ2
)dt + 3σdWQ
t (10)
4

5. Now, to price this derivative we follow the same steps as we used to price
the European Call Option. We start by writing the risk neutral expectation
under the cash numeraire.
V (S(t), t) = P(t, T)KEQ
1K3≥S(T)3 Ft − B(t)EQ S(T)3
B(T)K2
1K3≥S(T)3 Ft
= P(t, T)KQ K3
≥ S(T)3
−
S(t)3
K2
QS
K3
≥ S(T)3
(11)
Using Equation (10), we can compute the Radon-Nikodyn density as,
dQS
dQ
=
B(T)/B(t)
S(t)3/S(T)3
= exp (3r −
3
2
σ2
)(T − t) + 3σWQ
T−t
= exp 3σWQ
T−t −
9
2
σ2
(T − t) exp (3r + 3σ2
)(T − t)
(12)
Using Girsanov’s Theorem we can express WQS
t = WQ
t − 3σt and the
constant exp{(3r + 3σ2
)(T − t)} is taken care of in the end after we have
found a suitable CDF function.
Under the Q measure, we can express the indicator function as,
5

6. Q K3
≥ S(T)2
= Q K3
≥ S(0)3
exp (3r −
3
2
σ2
)T + 3σWQ
T
= Q

WQ
T ≤
log K
S(0)
− (r − 1
2
σ2
)T
σ


= Φ


log K
S(0)
− (r − 1
2
σ2
)T
σ
√
T


(13)
Using Brownian Motion transformation, we can now solve for QS
K3
≥ S(T)3
as,
QS
K3
≥ S(T)2
= QS
K3
≥ S(0)3
exp (3r −
15
2
σ2
)T + 3σWQS
T
= QS

WQS
T ≤
log K
S(0)
− (r − 5
2
σ2
)T
σ


= Φ


log K
S(0)
− (r − 5
2
σ2
)T
σ
√
T


(14)
Hence, using the simple trick of change of numeraire we can price this
seemingly complex option payoff.
4 Final Remarks
Even though a lot of the material covered in this essay can be found in
textbooks, I find that most standard text books favor probabilistic approach
to pricing over pricing by change of numeraire. I would like to thank Dr.
Johannes Ruf, University of Oxford, for introducing me to this way of pricing,
which has saved me several lines of unnecessary calculations in the last year.
6