3.2 Finite Market Models: General Setting
We start by proving Theorem 3.34
Proof. By the FFTAP, there exists at least an EMM, say Q. Given a hedging
strategy ↵, by definition VT↵ = X and by Lemma 3.28 the discounted portfolio
value of ↵ is a Q-martingale, hence the equation (3.39) holds, where the
right-hand member does not depend on ↵, and so niether do V ⇤ and V ↵ .
Moreover, the left-hand member in (3.39) does not depend on the particular
EMM chosen, therefore the claim holds for all EMMs.
To be more precise, equalities between stochastic processes are usually of
two types, as in the following definition.
Definition 3.35. Let X = (Xt )t2T , Y = (Yt )t2T be two stochastic processes
on a probability space (⌦, F, P). Then, X and Y are modifications of one
another if
P(Xt = Yt ) = 1
8t 2 T.
Instead, X and Y are indistinguishable if
P(Xt = Yt 8t 2 T) = 1.
Note that in Definition 3.35 there is no condition on the set T of time
indexes, which can be either discrete (discrete-time stochastic processes) or
dense (continuous-time stochastic processes), as well as either finite (finitehorizon stochastic processes) or not (infinite-horizon stochastic processes).
In general, it is clear that two indistinguishable stochastic processes are also
modifications of one another, but the converse is not true. However, in continuous time, we can easily prove that almost-surely right- or left-continuity
is enough to make the converse hold true.
In a finite market model, we deal with finite-horizon discrete-time stochastic
processes, where T = {0, 1, . . . , T }. In this framework, the two notions of
modifications and indistinguishable processes are equivalent. Indeed: assume
X and Y are modifications of one another and denote
Nt := {! 2 ⌦ : Xt (!) 6= Yt (!)},
t = 0, 1, . . . , T ;
77
78
3. Finite Market Models
then, by Definition 3.35, all Nt , t = 0, . . . , T , are null sets, and so is their
union, i.e.
P(N ) = 0,
N :=
[
Nt ;
t=0,1,...,T
but the null set N is exactly the complement of
{! 2 ⌦ : Xt (!) = Yt (!) 8t = 0, 1, . . . , T } =
\
Ntc ,
t=0,1,...,T
which has thus probability one under P. Therefore, X and Y are modifications one of another if and only if they are indistinguishable. Then, if we
consider another probability measure Q ⇠ P, it is clear that indistinguishable processes under P are also indistinguishable under Q and viceversa.
Moreover, in a finite market model, since P({!}) > 0 for all ! 2 ⌦, then
all almost-sure equalities are in fact sure equalities, which means that X
and Y are indistinguishable processes on (⌦, F, P), if and only if they are
indistinguishable on (⌦, F, Q), if and only if
Xt (!) = Yt (!) 8t = 0, 1, . . . , T, 8! 2 ⌦.
Now, we improve Theorem 3.34 by showing that not only the portfolio
value process of any hedging strategy for X is unique, but it also is the unique
arbitrage-free price process for the ECC X. This result generalizes Theorem
3.8, already proven for the Binomial model, and Corollary 3.10. In order to
do that, we have to introduce the notion of trading strategies in the extended
market where, besides the stock and the bond, the (attainable) ECC X can
also be traded at any time t = 0, 1, . . . , T with price process C = (Ct )t=0,...,T ,
that is a real-valued F-adapted stochastic process such that CT = X.
Definition 3.36. Given an attainable ECC X with price process C =
(Ct )t=0,...,T , a trading strategy in the secondary market (S, X) is a collection (V0 , ↵, ), where V0 2 R denotes the value of the initial investment, ↵ =
(↵0 , ↵1 , . . . , ↵d ) = {(↵t1 , ↵t2 )}t=1,...,T is a Rd+1 -valued F-predictable stochastic process representing the holdings in the bond and stock as usual (see
3.2 Finite Market Models: General Setting
Definition 3.21), and
tic process where
t
79
= ( t )t=1,...,T is a real-valued F-predictable stochas-
represents the number of contingent claims held in the
portfolio over the time period (t
1, t], t = 1, . . . , T . The portfolio value of
(V0 , ↵, ) at any time t 2 {1, . . . , T } is given by
Vt↵, = ↵t · St +
t Ct .
(3.40)
The trading strategy (V0 , ↵, ) in the secondary market (X, S) is self-financing
if it satisfies
(
V 0 = ↵ 1 · S0 +
↵ t · St +
t Ct
1 C0
=
Pd
i=0
= ↵t+1 · St +
↵1i S0i +
t+1 Ct
1 C0 ,
t = 1, . . . , T.
(3.41)
A self-financing strategy4 (↵, ) in the secondary market (X, S) is an arbitrage
if
(i) V0↵, = 0,
(ii) VT↵,
0,
(iii) P(VT↵, > 0) > 0, or equivalently EP [VT↵, ] > 0.
Note that, if we consider the ECC X as a primary security and we add it
to the other risky securities S 1 , . . . , S d , we get a finite market model based
on d + 2 securities, with price porcess S = (S 0 , S 1 , . . . , S d , C), and Definition
3.36 gets covered by the definitions given in Section 3.2.3.
Now, using the above definitions and Theorem 3.34, we are able to prove
the following theorem, which generalizes the results obtained in the binomial
model.
Theorem 3.37. Assume that the NA condition holds true and that X is an
attainable ECC. Then, the stochastic process
St0 EQ [X ⇤ |Ft ],
t = 0, . . . , T
(3.42)
is the unique arbitrage-free price process for X, where Q is an equivalent
martingale measure for S ⇤ .
4
As usual, we identify a self-financing trading strategy by its holding processes only,
since the initial investment is univocally determined by (3.41).
80
3. Finite Market Models
Proof. First, by Theorem 3.34, the process in (3.42) coincides with the portfolio value of any hedging strategy ↵ for X, that is
Vt↵ = St0 EQ [X ⇤ |Ft ],
t = 0, . . . , T.
Now, let C = (Ct )t=0,...,T be the price process for X, where CT = X, we
have to prove two things: that if C is di↵erent from (3.42) then the extended
market allows for arbitrage opportunities, and that if C coincides with (3.42),
no arbitrage opportunities can be realized.
(Uniqueness). Suppose that, for some s 2 {0, . . . , T
1}, P(Cs > Vs↵ ) > 0,
and denote A := {! 2 ⌦ : Cs (!) > Vs↵ (!)} 2 Fs . Consider the following
trading strategy: invest nothing until at time s, then, if in a scenario ! 2 A,
at time s sell one unit of the ECC, invest an amount Vs↵ (!) of the proceeds in
the primary market according to ↵s+1 and invest the remainder Cs (!) Vs↵ (!)
in the bond (to be held fixed over (s, T ]), then continue to trade according
to ↵ over the following time periods. Formally, this strategy is defined by:
V0 = 0 and
8
<0,
(˜
↵t , ˜t )(!) = ⇣
: ↵0 (!) +
t
Cs (!) Vs↵ (!)
Ss0
⌘
t  s;
, ↵t1 (!), . . . , ↵td (!), 1 1A (!), t > s.
The strategy (V0 , ↵
˜ , ˜ ) is self-financing: until time t = s
1 the conditions in
(3.41) are trivially satisfied, as they are from time t = s+1 to maturity, since
the portfolio evolves as the self-financing portfolio ↵, being the additional
holdings in the ECC and bond fixed over the time period (s, T ]; at time
˜
t = s we have Vs↵,˜
= 0 and
✓
Cs
↵
˜ s+1 · Ss Cs = ↵s+1 · Ss +
Vs↵
Ss0
Moreover, the final portfolio value is
✓
Cs Vs↵ 0
↵,˜
˜
V T = ↵ T · ST +
ST
Ss0
✓
◆
0
↵ ST
= (Cs Vs ) 0 1A
Ss
0,
◆
Ss0
Cs 1A = 0.
CT
◆
1A
3.2 Finite Market Models: General Setting
˜
and P(VT↵,˜
81
0) = P(A) > 0. Thus, (˜
↵, ˜ ) is an arbitrage.
Analogously, supposed that, for some s 2 {0, . . . , T
1}, P(Cs < Vs↵ ) > 0,
and denoted B := {! 2 ⌦ : Cs (!) < Vs↵ (!)} 2 Fs , we could prove that the
strategy (ˆ
↵, ˆ ), defined by
8
<0,
(ˆ
↵t , ˆt )(!) = ⇣
: ↵0 (!) +
t
Vs↵ (!) Cs (!)
,
Ss0
⌘
t  s;
↵t1 (!), . . . , ↵td (!), 1 1B (!), t > s.
is an arbitrage.
Therefore, if P(9s 2 {0, . . . , T
1} : Cs 6= Vs↵ ) > 0, then there are
arbitrage opportunities.
(Arbitrage-free). The second part of the proof is about showing that
if the price process for X is defined as in (3.42), then it does not allow
for arbitrage opportunities. Following a reductio ad absurdum, suppose that
C = V ↵ and that (¯
↵, ¯ ) is an arbitrage. Denote by V ⇤ and V̄ ⇤ the discounted
¯
value process of ↵ and (¯
↵, ¯ ) respectively, Vt⇤ := Vt↵ /St0 and V̄t⇤ := Vt↵,¯
/St0
for all t = 0, . . . , T , and compute
EQ [V̄t⇤ |Ft 1 ] = EQ [¯
↵t · St⇤ + ¯t Vt⇤ |Ft 1 ]
=↵
¯ t · EQ [St⇤ |Ft 1 ] + ¯t EQ [Vt⇤ |Ft 1 ]
=↵
¯ t St⇤
1
+ ¯t Vt⇤ 1
= V̄t⇤ 1
by linearity of the conditional expectation, by Lemma 3.28 and by selffinancing of (¯
↵, ¯ ). Hence, V̄ ⇤ is also a (Q, F)-martingale, and its expectation
under Q is constant:
EQ [V̄T⇤ ] = EQ [V̄0⇤ ] = 0,
since an arbitrage has zero initial value. On the other hand, an arbitrage
has non-negative final value V̄T⇤
0. A non-negative random value with null
expectation is necessarily almost-surely null, i.e. Q(V̄T⇤ = 0) = 1, but Q ⇠ P
and so it would follow that P(V̄T⇤ = 0) = 1, a contradiction.
82
3. Finite Market Models
3.2.5
The Second Fundamental Theorem of Asset Pricing
Once solved the hedging and pricing problems for attainable ECCs, it
remains to identify the attainable claims. A more convenient situation would
be a model where all ECCs can be replicated, that is the subject of the present
section.
Definition 3.38. A financial market model is said complete if all European
contingent claims are attainable.
In a complete finite market model, Theorem 3.37 solves the arbitragefree pricing problem for all ECCs. A necessary and sufficient condition for
completeness of the model is given by the following fundamental theorem.
Theorem 3.39 (Second Fundamental Theorem of Asset Pricing (SFTAP)).
A finite market model satisfying NA is complete if and only if there is a
unique equivalent martingale measure for S ⇤ .
Proof.
()) Assume that the market is arbitrage-free, i.e. satisfies NA, and complete.
By contradiction, suppose that there are two di↵erent equivalent martingale
measures for S ⇤ , Q1 , Q2 . Now, consider any event A 2 F and take the
European contingent claim which takes value 1 on A and 0 otherwise, X =
1A . Since the market is complete, X is attainable, thus by Theorem 3.37 its
arbitrage-free price at time 0 is unique and equal to


Q 1 1A
Q 2 1A
E
=E
.
ST0
ST0
Since A was arbitrarily chosen in F, the equation above translates into
Q1 (A) = Q2 (A)
8A 2 F,
therefore Q1 = Q2 , that is there cannot be two distinct EMMs for S ⇤ .
(() Assume that the equivalent martingale measure for S ⇤ , Q, is unique.
By contradiction, suppose that the (arbitrage-free) market is not complete.
3.2 Finite Market Models: General Setting
83
As in the proof of the FFTAP, we identify random variables with vectors in
RN . We denote by
V := {V ↵,⇤ | ↵ self-financing strategy}
the set of all discounted portfolio values of self-financing strategies, which is
a vector space, and remark that the market is not complete if and only if
V ( RN .
Let us define a scalar product h·,·iQ in RN as follows:
Q
hX, Y iQ = E [XY ] =
N
X
i=1
xi yi Q({!i }),
X = (x1 , . . . , xN ), Y = (y1 , . . . , yN ),
for all X, Y, 2 RN ,
xi = X(!i ), yi = Y (!i ), i = 1, . . . , N.
Then, there exists a vector ⇠ 2 RN \ {0} orthogonal to V, i.e. such that
h⇠, XiQ = EQ [⇠X] = 0 8X 2 V.
In particular, as every constant belongs to V (it can be reached by taking
both the initial investment and ↵0 equal to the final constant discounted
value and ↵1 = . . . = ↵d = 0), by taking X = 1 we get EQ [⇠] = 0.
Now, chosen arbitrarily
as
Q ({!i }) =
✓
> 1, we define another probability measure Q
⇠i
1+
k⇠k1
◆
Q({!i }),
i = 1, . . . , N,
where k⇠k1 := max |⇠i |. If we can prove that Q is another EMM for S ⇤ ,
1iN
then we have a contradiction, that ends of the proof.
First, Q is a probability measure, since
Q (⌦) =
=
N
X
i=1
N
X
i=1
=1+
Q ({!i })
N
1 X
Q({!i }) +
⇠i Q({!i })
k⇠k1 i=1
1
EQ [⇠] = 1.
k⇠k1
84
3. Finite Market Models
Moreover, Q ⇠ P because, by definition, it assigns strictly positive probability to each elementary event. It remains to be proved that S ⇤ is a Q martingale. Again, by Proposition A.37, this is equivalent to prove that, for
all bounded R-valued predictable processes ,
EQ [GT ( , S ⇤ k )] = 0 8k = 1, . . . , d.
For all k = 1, . . . , d, we have
Q
⇤k
E [GT ( , S )] =
N
X
i=1
N
1 X
GT ( , S )(!i )Q({!i }) +
⇠i GT ( , S ⇤ k )(!i )Q({!i })
k⇠k1 i=1
⇤k
= EQ [GT ( , S ⇤ k )] +
= 0,
1
EQ [⇠GT ( , S ⇤ k )]
k⇠k1
since GT ( , S ⇤ k ) 2 V and Q is an EMM for S ⇤ .
Another equivalent characterization of completeness is provided by the
following version of the martingale representation theorem.
Theorem 3.40. Assume that the finite market model satisfies NA and let Q
be an equivalent martingale measure. Then, the model is complete if and only
if every real-valued (Q, F)-martingale M = (Mt )t=0,...,T has a representation
of the form
Mt = M 0 +
t
X
s=1
↵
ˆ s · (Ŝs⇤
Ŝs⇤ 1 ),
t = 0, . . . , T,
(3.43)
for some Rd -valued F-predictable stochastic process ↵
ˆ = (↵1 , . . . , ↵d ), where
we denoted Ŝ ⇤ = (S ⇤,1 , . . . , S ⇤,d ).
Proof.
()). Suppose the model is complete and take an arbitrary (Q, F)-
martingale M = (Mt )t=0,...,T . Then, X = ST0 MT is an ECC and, by completeness, there exists a hedging strategy ↵ for X. Denoted by V ⇤ the discounted
portfolio value process of ↵, it holds VT⇤ = X ⇤ = MT , and
Vt⇤ = EQ [X ⇤ |Ft ] = EQ [MT |Ft ],
t = 0, . . . , T,
3.2 Finite Market Models: General Setting
85
by Theorem 3.34. But also, since M is a martingale,
Mt = EQ [MT |Ft ],
t = 0, . . . , T.
Thus, by (3.31), we obtain
Mt = Vt⇤ = V0 + G⇤t
=
V0⇤
+
= M0 +
t
X
s=1
t
X
s=1
↵s · (Ss⇤
Ss⇤ 1 )
↵
ˆ s · (Ŝs⇤
Ŝs⇤ 1 ),
where ↵
ˆ = (↵1 , . . . , ↵d ) and we denoted by G⇤ the discounted gain process
of ↵. This gives a representation in the form of (3.43).
((). Suppose the martingale representation holds and consider any ECC X.
We can define a (Q, F)-martingale M = (Mt )t=0,...,T by
Mt = EQ [X ⇤ |Ft ],
t = 0, . . . , T.
By Lemma 3.24, we know that, given ↵
ˆ as in the statement of the theorem,
there exists a unique real-valued predictable process ↵0 such that
↵ = (↵0 , ↵
ˆ ) = (↵0 , ↵1 , . . . , ↵d )
is a self-financing trading strategy with initial value M0 . The discounted
value V ⇤ = V ↵ /S 0 of such strategy at any time t 2 {0, . . . , T } is
Vt⇤ = V0⇤ + G⇤t
= M0 +
= M0 +
t
X
s=1
t
X
s=1
= Mt ,
↵s · (Ss⇤
Ss⇤ 1 )
↵
ˆ s · (Ŝs⇤
Ŝs⇤ 1 )
from which VT↵ = MT ST0 = X, hence ↵ is a hedging strategy for X and X is
attainable.