Equivalent Measures

24
Equivalent Measures
24.1 CHANGE OF MEASURE IN DISCRETE TIME
(i) In Section 19.2 the arbitrage theorem was applied to a simple portfolio over a single time
step. The analysis led to the concept of pseudo-probabilities, which could be calculated from
the starting value of the stock and a knowledge of the two possible values after a single time
step. It was emphasized that these pseudo-probabilities are not to be confused with actual
probabilities; they are merely computational devices, despite the fact that they display all the
mathematical properties of probabilities.
If the concept is extended from a single step model to a tree, there is a pseudo-probability
assigned to each branch of the tree. In the simplest trees, there are only two pseudo-probabilities:
up and down, which are constant throughout the tree. However, we will keep the analysis more
general and consider variable branching probabilities.
N
0
x
0
0
x
N
1
x
N
N
x
n
m
x
Figure 24.1 N -Step tree (sample
space)
Consider a large binomial tree as shown in

Figure 24.1. The underlying stochastic variable x
i
can
take the values at the nodes of the tree which are written
x
n
m
(n: time steps; m: space steps from the bottom of
the tree). The entire set of values x
0
0
,...,x
N
N
has pre-
viously been referred to as the sample space ,orthe
architecture of the tree. The set of probabilities of up or
down jumps at each of the nodes is collectively known
as the probability measure P. From a knowledge of all
the branching probabilities (it is not assumed that they
are constant throughout the tree), we can easily calcu-
late the probabilities of achieving any particular node
in the tree; these outcome probabilities can equally be
referred to as the probability measure, since branching and outcome probabilities are mechan-
ically linked to each other. We use the notation p
n
m
to denote the time 0 probability under the
probability measure P of x
i

achieving the value x
n
m
at the nth time step. Clearly, we start with
p
0
0
= 1.
We have already seen that in solving any option theory problem, we have discretion in
choosing sample space and probability measure. For example, the Cox–Ross–Rubinstein and
the Jarrow–Rudd sample spaces use different probability measures, but yield substantially the
same answers when used for computation (see Chapter 7). The purpose of this chapter is to
explore the effect of changing from one probability measure to another.
(ii) The Radon–Nikodym Derivative: Consider two alternative probability measures P and Q, i.e
two sets of outcome probabilities p
0
0
,..., p
N
N
and q
0
0
,...,q
N
N
. Either of these could be applied
to the tree (sample space ). Let us now define a quantity ξ
n
m

= q
n
m
/ p
n
m
for each node of the
tree. A new N-step tree is shown in Figure 24.2, similar to that for the x
n
m
but with nodal values
24 Equivalent Measures
equal to the quantities ξ
n
m
. This tree (), together with a probability measure, defines a new
stochastic process ξ
i
.
From these simple definitions, we may write
E
Q
[x
N
| F
0
] =
N

j=0

x
N
j
q
N
j
=
N

j=0
x
N
j
ξ
N
j
p
N
j
= E
P
[x
N
ξ
N
| F
0
] (24.1)
And similarly for any function of x
N

:
E
Q
[ f (x
N
) | F
0
] = E
P
[ f (x
N
)ξ
N
| F
0
] (24.2)
Simply by putting f (x
N
) = 1 in this last equation gives
E
P
[ξ
N
| F
0
] = 1 (24.3)
ξ
i
is a process known as the Radon–Nikodym–derivative of the measure Q with respect to
the measure P. The somewhat misleading notation dQ/dP is normally used, but the word

derivative (reinforced by the differential notation) must not be taken to denote a derivative as
in analytical differential calculus; ξ
i
is a stochastic process.
N
0
x
0
0
x
N
1
x
N
N
x
n
m
x
N
0
N
N
Figure 24.2 Sample space or Radon–Nikodym derivative
Care needs to be taken with one point: each of the ξ
n
m
is the quotient of two probabilities
and if the denominator is zero at any node, then the calculations blow up. We must have either
p

n
m
and q
n
m
both non-zero or both zero for each possible outcome. If the measures fulfill this
condition, we say that P and Q are equivalent probability measures.
(iii) The Radon–Nikodym process has an additional interesting property which follows from
equation (24.3). This equation holds true whatever the value of N, and since by definition
ξ
0
= ξ
0
0
= 1, the process ξ
i
must be a martingale.
(iv) It is a trivial generalization to extend equation (24.1) to the following which applies at step n:
E
Q
[x
N
| F
0
] = E
P
[x
N
ξ
N

| F
0
]
If we wish to express the expectation of x
N
at F
0
, subject to the condition that the process x
i
has previously hit a node with value x
n
m
(or equivalently that ξ
i
has previously achieved ξ
n
m
),
276
24.2 CHANGE OF MEASURE IN CONTINUOUS TIME
then we must introduce the probability of achieving x
n
m
on each side of the last equation; but
the probabilities must be expressed in the appropriate measure:
q
n
m
E
Q


x
N
; condition: x
n
= x
n
m


F
0

= p
n
m
E
P

x
N
ξ
N
N
; condition: x
n
= x
n
m



F
0

E
Q

x
N
; condition: x
n
= x
n
m


F
0

=
1
ξ
n
m
E
P

x
N
ξ

N
N
; condition: x
n
= x
n
m


F
0

(24.4)
(v) From the foregoing paragraphs, we can distil the following general properties for a Radon–
Nikodym derivative:
r
E
Q
[ f (x
N
) | F
0
] = E
P
[ξ
N
f (x
N
) | F
0

]
r
E
P
[ξ
j
| F
i
] = ξ
i
i < j (martingale)
r
ξ
0
= 1; ξ
i
> 0
r
ξ
i
E
Q
[ f (x
N
) | F
i
] = E
P
[ξ
N

f (x
N
) | F
i
]
24.2 CHANGE OF MEASURE IN CONTINUOUS TIME:
GIRSANOV’S THEOREM
(i) The probability measure in the tree of the last section was defined as the set of probabilities
throughout the tree. In continuous time, the probability measure is the time-dependent fre-
quency distribution for the process. For example, if W
t
is a standard Brownian motion, its
frequency distribution is a normal distribution with mean 0 and variance t. The probability of
its value lying in the interval W
t
to W
t
+ dW
t
is
dP
t
=
1
√
2πt
exp

−
1

2t
W
2
t

dW
t
A change of probability measure which is analogous to changing the set of probabilities in
discrete time simply means a change in the frequency distribution in continuous time. A
Radon–Nikodym derivative can be defined which achieves this change in probability measure:
ξ(x
t
) =
dQ
t
dP
t
=
F
Q
(x
t
)dx
Q
t
F
P
(x
t
)dx

P
t
(24.5)
where F
P
(x
t
) and F
Q
(x
t
) are the frequency distributions corresponding to the probability
measures P and Q.
As in the case of discrete distribution, the analysis breaks down if at any point we allow
dQ
t
to remain finite while dP
t
is zero. Therefore only equivalent probability measures are
considered, which ascribe zero probability to the same range of values of the variable x
t
. This
is illustrated in Figure 24.3.
Figure 24.3 Probability measures
277
24 Equivalent Measures
The first frequency distribution is normal. The middle distribution is a little unusual, but is
“equivalent” to the normal distribution. The third distribution is a very common one, but is not
equivalent to the first, since it ascribes a value 0 outside the square range.
(ii) We now investigate the process ξ

T
which is defined by
ξ
T
= e
y
T
;dy
t
=−
1
2
φ
2
t
dt − φ
t
dW
t
It is shown in Section 23.4(iv) that ξ
T
is a martingale and that ξ
0
= 1 and ξ
t
> 0. There-
fore ξ
T
displays the basic properties of a Radon–Nikodym derivative which are set out in
Section 25.1(v).

The function ξ
T
will be applied as the Radon–Nikodym derivative in transforming a drifted
Brownian motion dx
P
t
= µ
t
dt + σ
t
dW
P
t
(defined as having probability measure P) to some
other stochastic process (defined as having probability measure Q). We are interested in dis-
covering what that other stochastic process looks like, for the specific form of ξ
T
defined
above.
(iii) Before proceeding with the analysis, we recall a result described in Appendix A.2(ii). A normal
distribution N (µT ,σ
2
T ) has a moment generating function given by
M
P
() = E
P

e
x

P
T


F
0

= e
µT +
1
2
σ
2
T 
2
Furthermore, the moment generating function for the random variable x
P
T
is unique, i.e. if x
P
T
has the above moment generating function, then x
P
T
must have the distribution N (µT ,σ
2
T ).
The analogous results for a process x
P
t

with variable µ
t
and σ
t
are obtained from the
following modifications:
µT = E
P
[x
T
| F
0
] =

T
0
µ
t
dt
σ
2
T = var[x
T
] = E
Q


T
0
σ

2
t
dW
t

2




F
0

=

T
0
σ
2
t
dt
The moment generating function result above can now be written more generally as
M
P
() = E
P

e
x
P

T


F
0

= E
P

exp



x
0
+

T
0
µ
t
dt +

T
0
σ
t
dW
t






F
0

= exp



x
0
+

T
0
µ
t
dt

+
1
2

2

T
0
σ

2
t
dt

(24.6)
We will also use the following standard integral result which is a special case of the last equation
with  = 1:
E
P

exp

x
0
+

T
0
µ
t
dt +

T
0
σ
t
dW
t






F
0

= exp

x
0
+

T
0
µ
t
dt +
1
2

T
0
σ
2
t
dt

(24.7)
278
24.2 CHANGE OF MEASURE IN CONTINUOUS TIME

(iv) The effect of changing to probability measure Q by using ξ
T
as the Radon–Nikodym derivative
is as follows:
M
Q
[] = E
Q

e
x
Q
T


F
0

= E
P

ξ
T
e
x
P
T


F

0

= E
P

exp

−
1
2

T
0
φ
2
t
dt −

T
0
φ
t
dW
t

exp



x

0
+

T
0
µ
t
dt +

T
0
σ
t
dW
t





F
0

= E
P

exp

x
0

+

T
0

µ
t
−
1
2
φ
2
t

dt +

T
0
(σ
t
− φ
t
)dW
t





F

0

where we have used y
0
= 0 (since ξ
0
= 1). Use of equation (24.7) shows that the last equation
may be written
M
Q
() = exp

x
0
+

T
0

µ
t
−
1
2
φ
2
t
+
1
2


2
σ
2
t
− σ
t
φ
t
+
1
2
φ
2
t

dt

= exp



x
0
+

T
0
(µ
t

− λ
t
)dt

+
1
2

2

T
0
σ
2
t
dt

(24.8)
where we have arbitrarily defined λ
t
= σ
t
φ
t
.
Comparing equations (24.6) and (24.8) leads us to the following conclusions which constitute
Girsanov’s theorem.
The effect on a drifted Brownian motion of changing probability measure by using the
Radon–Nikodym derivative ξ
t

as defined above is as follows:
r
The measure P drifted Brownian motion is transformed into another drifted Brownian motion
(with probability measure Q).
r
The variances of the two Brownian motions are the same.
r
The only effect of the change in measure is to change the drift by an instantaneous rate λ
t
which is defined above.
Formally this may be written
W
Q
T
= W
P
T
−

T
0
λ
t
d
t
or in shorthand dW
Q
T
= dW
P

T
− λ
t
dt
The result is subject to the usual type of technical condition (the Novikov condition):
E

exp

1
2

T
0
λ
2
t
dt

< ∞
Girsanov’s theorem basically gives a prescription for changing the measure of a Brownian
motion in such a way that it remains unchanged except for the addition of a drift. So what,
you might ask? You can achieve the same effect by just adding a time-dependent term to the
underlying variable W
t
; what’s the big deal? We know already that the value of an option is the
expected value of its payoff under some pseudo-probability measure. The value of the theorem
is that it provides a recipe for applying this particular measure simply by adding a convenient
drift term in the SDE governing the process in question. This procedure is explicitly laid out
in the next section.

279
24 Equivalent Measures
(v) Girsanov’s Theorem without Stochastic Calculus: This is a theorem of great power and
usefulness in option theory, but it is worth a re-examination from the point of view of someone
without a knowledge of stochastic calculus; it leads to an intuitive understanding which does
much to demystify the theorem.
Consider a stochastic variable x
t
(x
0
= 0) distributed as N(µt,σ
2
t). The probability distri-
bution function of x
t
is
n(x
t
; µ, σ) =
1
√
2πσ
2
t
exp

−
1
2


x
t
− µt
σ
√
t

2

and the moment generating function has been shown to be
M() =

+∞
−∞
e
x
t
n(x
t
; µ, σ)dx
t
= e
µt +
1
2
σ
2
t 
2
Remember that M() uniquely defines a distribution and all its moments can be derived from

it. But by pure algebraic manipulation, the last equation could be written
M() =

+∞
−∞
e
x
t
n(x
t
; µ, σ)dx
t
≡

+∞
−∞
e
x
t

n(x
t
; µ, σ)
n(x
t
; µ

,σ)

n(x

t
; µ

,σ)dx
t
=

+∞
−∞
e
x
t

exp
−1
2σ
2
t
((x
t
− µt)
2
− (x
t
− µ

t)
2
)


n(x
t
; µ

,σ)dx
t
=

+∞
−∞
e
x
t
exp

−
1
2
φ
2
t − φ
√
t

x
t
− µ

t
σ

√
t

n(x
t
; µ

,σ)dx
t
where φ =
µ

− µ
σ
= e
(µ

−φσ)t+
1
2
σ
2
t
2
The conclusion to be drawn from this result is that any normal distribution, but always with the
same variance σ
2
t, can be used to take the expectation of a function, but the function must be
modified by multiplication by the factor exp[−
1

2
φ
2
t − φ
√
t(
x
t
−µ

t
σ
√
t
)]. Alternatively expressed,
an arbitrary choice of normal distribution (but always with the same variance) really only
affects the drift term. This may be self-evident, if we remember that a normal distribution is
entirely defined by the drift and variance, and it certainly takes some of the mystery out of
Girsanov’s theorem.
24.3 BLACK SCHOLES ANALYSIS
(i) The stochastic differential equation governing a stock price movement is assumed to be
dS
t
= µ
t
S
t
dt + σ
t
S

t
dW
RW
t
(24.9)
where µ
t
is the drift observed in the real world and the superscript RW indicates that the
Brownian motion is observed in the same real world.
Our objective now is to find the measure under which the discounted stock price S
∗
t
is a
martingale; S
∗
t
= S
t
B
−1
t
where B
t
is the zero coupon bond price. The reason we want to find
the measure is that the value of an option can be found by taking the expectation of its payoff
under this measure.
With variable (but non-stochastic) interest rates, we can define the value of the zero coupon
bond in terms of continuous, time-dependent interest rates r
t
as B

−1
t
= exp(−

1
0
r
τ
dτ ), so
280
24.3 BLACK SCHOLES ANALYSIS
that d(B
−1
t
) =−B
−1
t
r
t
dt. The process for S
∗
t
can then be written
dS
∗
t
= d

S
t

B
−1
t

= B
−1
t
dS
t
+ S
t
d

B
−1
t

= S
t
B
−1
t

(µ
t
− r
t
)dt + σ
t
dW

RW
t

(ii) Girsanov’s theorem tells us that we can change the probability measure by changing the drift
of the Brownian motion. Writing dW
RW
t
= dW
Q
t
− λ
t
dt, the last equation becomes
dS
∗
t
= S
t
B
−1
t

(µ
t
− r
t
− λ
t
σ
t

)dt + σ
t
dW
Q
t

where Q is a new measure. This is a Q-martingale if the coefficient of dt is zero, i.e. if
λ
t
=
µ
t
− r
t
σ
t
(24.10)
The term on the right-hand side of this last equation will be familiar to students of finance
theory as the Sharpe ratio. It is normally referred to in option theory as the market price of
risk.
An important and much used property of λ
t
is that it is the same for all derivatives of the
same underlying stock. Consider a stock whose process is given by dS
t
= µ
t
dt + σ
t
dW

RW
t
where µ and σ are functions of S
t
. Now consider two derivatives; Ito’s Lemma means that we
can write the processes for these as
f
(1)
t
= µ
(1)
t
dt + σ
(1)
t
dW
RW
t
; f
(2)
t
= µ
(2)
t
dt + σ
(2)
t
dW
RW
t

Let us construct a portfolio consisting of f
(2)
t
σ
(2)
t
units of the derivative f
(1)
, and − f
(1)
t
σ
(1)
t
units of f
(2)
. The portfolio value is
π
t
= f
(1)
t
f
(2)
t
σ
(2)
t
− f
(1)

t
f
(2)
t
σ
(1)
t
= f
(1)
t
f
(2)
t

σ
(2)
t
− σ
(1)
t

A change in the value of this portfolio over an infinitesimal time step dt is
dπ
t
= f
(2)
t
σ
(2)
t

d f
(1)
t
− f
(1)
t
σ
(1)
t
d f
(2)
t
= f
(1)
t
f
(2)
t

σ
(2)
t
µ
(1)
t
− σ
(1)
t
µ
(2)

t

dt
since the dW
RW
t
terms cancel. But if the return is not stochastic (i.e. is risk-free), then the return
must equal the interest rate:
σ
(2)
t
µ
(1)
t
− σ
(1)
t
µ
(2)
t
σ
(2)
t
σ
(1)
t
= r
t
or
µ

(1)
t
− r
t
σ
(1)
t
=
µ
(2)
t
− r
t
σ
(2)
t
(= λ
t
)
(iii) Let us now return to equation (24.9) and rewrite this in terms of the measure Q, using the above
value for λ
t
. Simple substitution gives us
dS
t
= r
t
S
t
dt + σ

t
S
t
dW
Q
t
(24.11)
In a nutshell, we have changed the real-world SDE by changing to the alternative measure Q
which turns the discounted stock price into a martingale; the effect of this switch is merely to
replace the real-world stock drift by the risk-free interest rate. The measure is therefore usually
referred to as the risk-neutral measure. This analysis is simply a sophisticated re-statement
of the principle of risk neutrality on which we based the first three parts of this book.
(iv) Continuous Dividends: The effect of a continuous dividend rate q is easy to include in the
above framework. We use constant rates for simplicity. The effect of a dividend is that the
holder of the shares receives a cash throw-off. It was shown in Chapter 1 that this can be
281
24 Equivalent Measures
incorporated into the calculations by writing the stock price as S
t
e
+qt
. The discounted share
value is therefore
S
∗
t
= S
t
e
qt

e
−rt
so that
dS
∗
t
= S
∗
t

(µ − (r − q)) dt + σ dW
RW
t

or from the previous analysis
dS
t
= (r − q)S
t
dt + σ S
t
dW
Q
t
(24.12)
(v) Forward Price: This can be written F
t
= S
t
e

(r−q)(T −t)
. We saw in the last subsection that
S
t
e
−(r−q)t
is a Q-martingale, i.e. dS
∗
t
= S
∗
t
σ dW
Q
t
. Multiply both sides by e
(r−q)T
to give
dF
t
= σ F
t
dW
Q
t
(24.13)
Using the results of Section 23.4(iv) gives
F
t
= e

−
1
2
σ
2
t+σ dW
Q
t
(24.14)
282
25
Axiomatic Option Theory
25.1 CLASSICAL VS. AXIOMATIC OPTION THEORY
(i) In the first three parts of this book, option theory was developed from a very few key concepts:
(A) Perfect Hedge: An option may be perfectly hedged. Previously, it was just assumed that
this is possible, and it was shown in Section 4.4 that if such a hedge exists, then it
must be a self-financing portfolio. Now, using arbitrage arguments we have shown that
a discounted derivatives price is a martingale. It was also shown that the discounted
value of a self-financing portfolio consisting of stock plus cash is also a martingale; the
martingale representation theorem therefore proves that an option can be perfectly hedged.
The reader who has the inclination to play with these two set of arguments as explicitly
laid out in Sections 4.3 and 21.5 will quickly realize how closely the two analyses are
related. Stochastic theory has just added a lot of fancy words.
(B) Risk Neutrality: For discrete models, the derivation of risk neutrality is very closely related
in classical and axiomatic option theory. In both cases, we start by examining a single step:
in the classical case, we get the result in Section 4.1 by saying that a portfolio consisting
of an option plus a hedge must be risk-free and therefore have a return equal to the interest
rate; in the axiomatic case, we use the arbitrage theorem to prove the same result.
For continuous models, the arguments appear to diverge rather more. We inferred risk
neutrality in the classical case from the fact that the real-world drift does not appear in the

Black Scholes equation. In axiomatic theory, risk neutrality falls out of the application of
Girsanov’s theorem and a consideration of the properties of martingales.
(C) The Black Scholes Equation: This was derived in Section 4.2 by constructing a continuous
time portfolio of derivative plus hedge and requiring its rate of return to equal the interest
rate. In Section 23.5 it appears as a consequence of the fact that the discounted option
price is a P-martingale, which in turn is a consequence of the arbitrage theorem. Both
derivations are critically dependent on Ito’s lemma, which was introduced with a lot of
hand waving in Section 3.4 and which is of course a central pillar of stochastic calculus.
It is not possible to derive an options theory without some recourse to stochastic calculus,
albeit the very rough and ready description of Ito’s lemma given in Chapter 3.
(D) Risk-neutral Expectations: Use of these to price options was introduced with a minimum
of fuss (or rigor) in Section 4.1. Using the axiomatic approach, it was shown in Section
22.3(iv) to result from the fact that the discounted option price is a P-martingale (and
hence from the arbitrage theorem).
(ii) At this point the reader faces the awkward question “was it all worth it?” Despite our rather
robust approach, stochastic calculus has been seen to be a tool of great subtlety; but we don’t
seem to have any additional specific results to what we had before.
Without beating about the bush, our view is that if someone is interested in equity-type
options (including FX and commodities), he is likely to find most results he needs through the
classical statistical approach to option theory. His main problem will be reading the technical