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Chapter 2
Options, Arbitrage,
Martingales
The aim of this chapter is to appreciate the key connection between arbitrage
free markets and the existence of equivalent measures under which the dis
counted stock price becomes a martingale.
2.1
The ArbitrageMartingale Connection
You have already met the concept of arbitrage in PAS367/6051. We now
make this more mathematically precise.
Defnition
A selfFnancing strategy
φ
is said to be an
arbitrage oppor
tunity
if
V
φ
(0) = 0
,V
φ
(
T
)
≥
0 and
P
(
V
φ
(
T
)
>
0)
>
0
.
A market is said to be
arbitragefree
if there are no arbitrage opportunities.
So if arbitrage opportunities exist, there is a nonzero probability that
your portfolio can create wealth with no investment !
We are now going to explore the remarkable relationship between arbitrage
free markets and martingales. ±irst recall that two probability measures
P
and
P
∗
are
equivalent
if they have the same sets of measure zero (i.e.
A
∈F
and
P
(
A
) = 0 if and only if
P
∗
(
A
) = 0  see Chapter 1 of PAS401/6052 and
Problem 9).
In the case we are considering of a Fnite market model we imposed the
condition in Chapter 1 that
P
(
A
)
>
0 for all nonempty
A
. Conse
quently in this context
P
∗
is equivalent to
P
if and only if
P
∗
(
A
)
>
0 for all
nonempty
A
.
9
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View Full DocumentDefnition
A probability measure
P
∗
on (Ω
,
F
) is said to be a
martingale
measure
for (
±
S
(
n
)
,n
∈T
)i
f
1.
P
∗
is equivalent to
P
.
2. (
±
S
(
n
)
)isa
P
∗
martingale.
1
In the sequel, all expectations of random variables computed using
P
∗
will be denoted as
E
∗
.
The next result is of fundamental importance. Indeed it is usually called
the
frst Fundamental theorem oF assetpricing
. It gives a direct link between
arbitragefree markets on the one hand and existence of martingale measures
on the other.
Theorem 2.1.1
The market is arbitrageFree iF and only iF there exists at
least one martingale measure.
ProoF.
The full proof is diﬃcult and uses concepts which lie outside the
scope of this course. If you want to learn about these  go to section 4.2 in
Bingham and Kiesel. We’ll just give the easy part of the proof, i.e. we show
that if a martingale measure exists, then the market is arbitragefree.
If a martingale measure exists then the discounted wealth process (
±
V
φ
(
n
)
)
is a
P
∗
martingale (see Problem 10) and so if
φ
is a selfFnancing portfolio
and if
±
V
φ
(0) = 0, then
E
∗
(
±
V
φ
(
T
)) =
E
∗
(
±
V
φ
(0)) = 0.
Now suppose that
ψ
is an arbitrage opportunity, then by Problem 8,
V
ψ
(0) = 0
,V
ψ
(
T
)
≥
0 and
E
(
V
ψ
(
T
))
>
0
.
Let
v
1
,...,v
M
be the values of
the random variable
V
ψ
(
T
). Since
V
ψ
(
T
)
≥
0wemu
s
thave
v
i
≥
0 for all
1
≤
i
≤
M.
Let the probability distribution of
V
ψ
(
T
)be
p
i
=
P
(
V
ψ
(
T
)=
v
i
)
for 1
≤
i
≤
M
. Since
E
(
V
ψ
(
T
)) =
M
²
i
=1
v
i
p
i
>
0
,
we must have
v
i
>
0 for at least one value of
i
. Let
i
0
be any such value of
i
. Recall that each
p
i
>
0 (since it is the probability of an event in
F
and
we are assuming that these are all positive). As
P
∗
is equivalent to
P
we
have each
p
∗
i
=
P
∗
(
V
ψ
(
T
v
i
)
>
0. Hence
E
∗
(
V
ψ
(
T
))
≥
v
i
0
p
∗
i
0
>
0
.
Since
β
(
T
)
>
0 we deduce that
E
∗
(
±
V
ψ
(
T
))
≥
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 Spring '08
 mikson

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