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Mathematical Finance, Vol. 14, No. 2 (April 2004), 295–315
A NOTE ON COMPLETENESS IN LARGE FINANCIAL MARKETS
MARZIA DE DONNO
Dipartimento di Matematica, Università di Pisa
We study completeness in large financial markets, namely markets containing countably many assets. We investigate the relationship between asymptotic completeness in
the global market and completeness in the finite submarkets, under a no-arbitrage
assumption. We also suggest a way to approximate a replicating strategy in the large
market by finite-dimensional portfolios. Furthermore, we find necessary and sufficient
conditions for completeness to hold in a factor model.
KEY WORDS: large financial market, self-financing portfolio, completeness, factor models, cylindrical
stochastic integration
1. INTRODUCTION
The present paper deals with the question of completeness in a large financial market.
The notion of large financial market was introduced by Kabanov and Kramkov (1994)
as a sequence of “finite dimensional markets,” also called “small markets” by Klein
and Schachermayer (1996a, 1996b) and Klein (2000). In this framework, Kabanov and
Kramkov (1994, 1998) provided an extension of the Fundamental Theorem of Asset
Pricing, relating the notion of asymptotic arbitrage to contiguity properties of sequences of probability measures. Some further results on this topic are due to Klein and
Schachermayer (1996a, 1996b). A general version of the Fundamental Theorem of Asset
Pricing can be found in Klein (2000); she introduced the notion of asymptotic free lunch
as an extension to the large market of the classical free lunch condition for the finitedimensional case (see, e.g., Kreps 1981), and gave a very general result on the connection
between this condition and contiguity of equivalent martingale measures.
The main result on completeness in a finite-dimensional market is the Second
Fundamental Theorem of Asset Pricing, which states the equivalence between market
completeness and uniqueness of the equivalent martingale measure (see, e.g., Harrison
and Pliska 1981).
In the context of a large financial market, Bättig (1999) and Bättig and Jarrow (1999)
proposed a definition of completeness that is independent of either the notion of no
arbitrage or equivalent martingale measures. This allows us to understand a counterexample due to Artzner and Heath (1995). Bättig also provided an example where the
existence of an equivalent martingale measure can exclude the possibility of replicating a
claim, to show how the two notions of no arbitrage and completeness are independent in
The author gratefully acknowledges financial support from the CNR Strategic Project “Modellizzazione
matematica di fenomeni economici.”
Manuscript received January 2002; final revision received November 2002.
Address correspondence to the author at Dipartimento di Matematica, Università di Pisa, via Buonarroti
2, 56127 Pisa, Italy; e-mail: [email protected].
C 2004 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148, USA, and 9600 Garsington Road, Oxford
OX4 2DQ, UK.
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M. DE DONNO
practice. However, he also showed that under a no-arbitrage assumption his definition of
completeness is equivalent to the traditional one.
As a special case of a large financial market, one can define a market as a sequence
of processes on one fixed filtered probability space; one assumes, in fact, that the market
contains an infinite, countable, number of assets. This is the model studied by Björk and
Näslund (1998), who obtained an extension of some classical results of arbitrage pricing
theory to a continuous time setting, in the case of factor models. For other results of the
same type, one can see the references in the paper by Björk and Näslund.
As for completeness, the main difference between the finite and the infinite-dimensional
case is that when an infinite number of assets is available, diversification of some kind of
risk can be obtained by making use of “well-diversified” portfolios, that is, portfolios that
contain a small quantity of a large number of assets. Björk and Näslund (1998) analyzed
this question in a very simple financial market, where asset prices satisfy a factor structure.
They showed that the market can be completed when it is possible to form well-diversified
portfolios, as limit of finite-dimensional portfolios. Furthermore, they pointed out the
differences between the case when the systematic risk and the idiosyncratic risk are of the
same type and the case when they are not.
The purpose of this paper is to give a precise mathematical formulation to diversification in a general setting. Under a no-arbitrage assumption, we introduce the notion of
“generalized” strategies as strategies that may involve an infinite number of assets; then
we formulate completeness as the possibility of replicating all claims or (in mathematical terms) as the property of representing all sufficiently integrable random variables as
“stochastic integrals” with respect to the asset prices. To this aim, we need an integral
with respect to infinite-dimensional processes; the appropriate tool seems to be a theory
of cylindrical integration recently developed by Mikulevicius and Rozovskii (1998, 1999).
We recall their main results, adapting to our particular case, in Section 2.
In Section 3, we can rigorously define the notion of strategy and completeness in the
large market. Since the stochastic integral is defined only for cylindrical martingales,
we assume the existence of an equivalent measure under which the discounted price
processes are martingales. Delbaen and Schachermayer (1999) proved that in the finitedimensional case, this is a stronger condition than no arbitrage and it is in fact equivalent
to no free lunch with vanishing risk. Kabanov and Kramkov (1994) showed that, under
this assumption, in a large financial market there are no asymptotic arbitrage opportunities of first and second kind. Finally, Klein (2000) proved that there is no asymptotic
free lunch and, conversely, if one assumes no asymptotic free lunch in our setting, an
equivalent martingale measure for all assets does exist. So we can work in the “martingale modeling” setup. We take as class of contingent claims the set of square integrable
(under the equivalent martingale measure) random variables: then a self-financing strategy will be represented as an integrable (with respect to the infinite-dimensional price
process) process and the value of a self-financing portfolio as the stochastic integral of
the strategy with respect to the assets. We distinguish between “naive” strategies, which
are instantaneously based on a finite number of assets, and generalized strategies, which
are limits of “naive” strategies but may involve an infinite number of assets. The market
is said to be complete if each claim can be replicated by either a “naive” or generalized
strategy—namely, if it can be represented as a stochastic integral. We also investigate
the relationship between completeness in the large financial market and completeness in
the finite-dimensional submarkets. We find that when all the finite markets are complete
the global market is complete. The converse is not true: if the large market is complete,
we can ideally replicate a claim investing in all the assets, but a finite number of assets is
COMPLETENESS IN LARGE FINANCIAL MARKETS
297
generically not sufficient to obtain replication, as shown by Björk and Näslund (1998).
Note that this negative result may hold even if the claim is defined in terms of only a finite
number of the underlying assets. In other words, the possibility of forming well-diversified
portfolios may complete an otherwise incomplete market.
In Section 4, we study the simple factor model introduced by Björk and Näslund
(1998) in more generality; we characterize completeness, giving necessary and sufficient
conditions which include the results that can be found therein.
Finally, in Section 5 we deal with the problem of approximating strategies: in a large
financial market a hedgeable claim can be a priori replicated by a strategy possibly based
on all the assets. We prove that, in fact, the claim can be approximately hegded by a
portfolio containing a finite, sufficiently large, number of assets and depending only on
the information generated by these assets.
2. STOCHASTIC INTEGRATION WITH RESPECT TO A SEQUENCE
OF MARTINGALES
For all definitions and notions on the theory of stochastic processes, we mainly refer to
Jacod and Shiryaev (1987). Let be given a stochastic basis (, F, IF = (Ft )t∈[0,T ] , P), fulfilling the usual assumptions (in the sense of Defs. I.1.2 and I.1.3 of Jacod and Shiryaev).
2
(P) will be defined
We denote by M2 (P) the set of square integrable martingales ( Mloc
2
in the obvious way) and by L (FT ) the set of square integrable random variables. We will
often make use of the standard identification between a square integrable martingale and
its terminal value, which is an element of L2 (FT ).
Let M = (Mi )i≥1 be a sequence of locally square
integrable martingales; our aim is
to give a mathematical sense to the expression F dM as an element of M2 (P). The
problem is twofold: we first need to define a suitable space of integrands, then we have
to define an integral, the cylindrical integral, for this class of functions. The solutions to
both are contained in the theory of cylindrical integration developed by Mikulevicius
and Rozovskii (1998, 1999), which we will briefly recall, adapting to our particular case.
We denote by E the set of all real-valued sequences (i.e., IRIN ) endowed with the topology of pointwise convergence, and by E its topological dual, that is, the set of linear
combination of Dirac measures; each of these measures can be identified with an element
of E, a sequence with all but finitely many components equal to zero. We denote by {ei }i≥1
the canonical basis in E; , E ,E will denote the duality operator. We recall the following
definition.
DEFINITION 2.1. A locally square integrable cylindrical martingale on E is a linear
2
(P).
mapping M, defined on E and with values in Mloc
It is then obvious that a sequence of locally square integrable martingales M = (Mi )i≥1
is a cylindrical martingale on E. Given the cylindrical martingale M, it is possible to
construct an increasing predictable (real-valued) process A and a process Q, defined on
× [0, T ], with values in the set of linear mappings from E to E such that, setting
ij
Qt,ω = ei , Qt,ω e j E ,E , the following properties hold:
(i) IE [AT ] < ∞.
(ii) Qij is a predictable process, for all i, j.
ij
ji
(iii) Q is symmetric; that is, Qt,ω = Q t,ω , for fixed (ω, t) ∈ × [0, T ], for all i, j ∈ IN.
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M. DE DONNO
ij
(iv) Q is nonnegative definite; that is, i , j ≤n ci Qt,ω c j ≥ 0, for fixed (ω, t) ∈ ×
[0, T ], for all n ∈IN, c1 , . . . , cn ∈ IR.
t ij
(v) Mi , Mj t (ω) = 0 Qt,ω dAt (ω), for all (ω, t) ∈ × [0, T ], for all i, j ∈ IN.
The proof of this fact is a simple extension of a well-known property of locally square integrable, finite-dimensional, martingales (see, e.g., Jacod and Shiryaev 1987,
Thm. II.2.9).
Let F be a predictable function, with values in E ofthe form F = i ≤k Fi ei for
some k ∈ IN. Then F is integrable with respect to M and F dM is a square integrable
martingale if and only if F̄ k = (F1 , . . . , Fk ) is integrable with respect to the IRk -valued
martingale M̄ k = (M1 , . . . Mk ) and F̄ k d M̄ k is a square integrable martingale, namely,
if and only if
T ij
Fi (s)Qs F j (s) dAs < ∞
IE
0
i , j ≤k
(see, e.g., Jacod and Shiryaev 1987, Sec. III. 4). We denote by L2 ( M̄ k ) the set of such
functions; then the integral is well-defined on ∪k≥1 L2 ( M̄ k ).
We define L2 (M, E ) as the set of E -valued processes F, which are predictable in the
sense that each component is predictable and which satisfy the following integrability
condition:
T ij
Fi (s)Qs F j (s) dAs < ∞.
IE
0
i , j ≥1
As intuition may suggest, the definition of the stochastic integral can be extended to this
class of functions. This has been proved by Mikulevicius and Rozovskii (1999, Prop. 2.4).
PROPOSITION 2.2. The map I : F → F dM defined on ∪k≥1 L2 ( M̄ k ) has an extension
to L2 (M, E ), still denoted by F → I(F), such that
(i) I(F) ∈ M2 (P) and
I(F), I(F)
t =
(iii )
0 i , j ≥1
(ii) I(F) is linear.
|I(F)|2M2 (P)
t
Fi (s)Qijs F j (s) dAs .
T
= IE [I(F), I(F)
T ] = IE
0
=
Fi (s)Qijs F j (s) dAs
i , j ≥1
|F|2L2 (M,E ) .
This means that we can extend the integral to all the functions F which pointwise have
finitely many components different from zero, though there may not exist any i such
that Fi ≡ 0. The problem of this extension is that the image of the map I is not closed
in M2 (P), or equivalently in L2 (FT ), as we can see in the following example.
EXAMPLE
2.3. Consider the cylindrical martingale (Mi )i≥1 defined by
Mi = W + Ni ,
where W is a Wiener process, and Ni is a sequence of independent compensated Poisson
processes all having
the same intensity λ = 1 such that Ni is independent of W for all i.
Let F n = n −1 i ≤n ei . The functions (F n ) are in L2 (M, E ) and
COMPLETENESS IN LARGE FINANCIAL MARKETS
T
F n (s) dM(s) =
0
299
M1 (T) + · · · + Mn (T)
n
2
converges to W (T) in L (FT ). On the other hand, F n converges pointwise (and in l 2 ) to 0.
So it is intuitively obvious that there exists no “naive” integrand F such that I(F) = W. In
fact, it can be shown that there exists no F in L2 (M, E ) which satisfies this requirement.
Thus, we see that there is a need to enlarge the space of integrands.
Our next step will be to look for a completion of the set of integrands. Precisely
because it seems that E is too small as a reasonable value space of integrands, we try
to extend it. For simplicity, we assume that Q is positive definite (see Remark 2.5). We
fix (ω, t) ∈ × [0, T ]. The Ito isometry given by Proposition 2.2(iii) makes it natural to
define on E a norm by setting:
|x|2Et = x, Qt,ω x
E ,E =
(2.1)
∞
ij
xi Qt,ω x j ,
i , j ≥1
where the sum contains a finite number of terms. The norm is induced by an obvious
scalar product, which induces on E a pre-Hilbert structure. This norm depends on (t, ω);
for the sake of simplicity we omit ω, but we keep t to remind us of this dependence. We
denote by Et the space E with the norm induced by Qt ; it is not difficult to see that Et is
not complete. We can take its completion, which we denote by Ht and which is a Hilbert
space. The set Ht is generically not even included in E, hence the canonical injection from
E to E cannot be extended to an injection from Ht to E.
Let F be a process such that F(t) takes values in Ht , the function F is predictable in
the sense that (F(t), ei ) Ht is predictable for all i, and the following integrability condition
holds:
T
|F(t)|2Ht dAt < ∞.
IE
0
The class of such functions is denoted by L2 (M, H ). The Ito isometry given by Proposition 2.2(iii) makes it natural to extend the integral to this set.
PROPOSITION 2.4. The map I : F → F dM defined on the set L2 (M, E ) can be
extended to L2 (M, H ) in such a way that
(i) I(F) ∈ M2 (P) and
t
I(F), I(F)
t =
(ii) I(F) is linear.
(iii )
|I(F)|2M2 (P)
0
|F(s)|2Hs dAs .
T
= IE [I(F), I(F)
T ] = IE
0
|F(t)|2Ht
dAt = |F|2L2 (M,H ) .
(iv) I(L2 (M, H )) is closed in L2 (FT ) and coincides with the stable subspace generated
by M in M2 (P), that is, the smallest closed subspace of M2 (P), which contains
all Mi , possibly stopped, and is stable for stochastic integration.
The results contained in (i), (ii), and (iii) are due to Mikulevicius and Rozovskii (1999,
Prop. 2.5); (iv) is a straightforward extension of a well-known result (see, e.g., Protter
1990, Thm. IV.35).
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M. DE DONNO
REMARK 2.5. We assumed, for sake of simplicity, that Q is positive definite. When Q
is only nonnegative definite, (2.1) defines a seminorm on E . The construction of H and
of the integral can still be carried on, replacing E with the quotient space E /ker Q.
2.6. In order to be able to approximate processes in L2 (M, H ), it can be
shown that, starting from the canonical basis in E, and by a standard orthogonalization
procedure, it is possible to construct an orthonormal basis in H , which we denote by
{hi }i≥1 , such that hi ∈ span(e1 , . . . , ei ). It follows that every F, such that F(s) ∈ Hs , can be
written in the form
(F(s), h i (s)) Hs h i (s) =
λi (s)h i (s),
F(s) =
REMARK
i ≥1
i ≥1
where λi (s) = (F(s), h (s)) Hs and
= i (λi (s))2 < ∞. Then the function F can
be approximated by the sequence F (s) = i ≤n λi (s)h i (s) (Mikulevicius and Rozovskii
n
1999, Cors. 2.2 and 2.3). Notice that F is IRn -valued for all n.
|F(s)|2Hs
n
i
In light of these results, we go back to the analysis of Example 2.3. In this case, it
makes sense to set At = t, Qij = 1 + δij . With this choice, Q, hence H , does not depend
on (s, ω). The space E will have a pre-Hilbert structure induced by the norm
2
2
|x| E =
xi +
(xi )2 .
i ≥1
i ≥1
Thus, a sequence (x ) in E is a Cauchy sequence if and only if ( i xin ) is a Cauchy
sequence in IR and ((x ni )i≥1 ) is a Cauchy sequence in l 2 . The limit
of such a sequence
is the
can be represented as a point (x0 , x) ∈ IR ⊕ l 2 where x0 = limn i xin , whereas x
limit of (x n ) in the l 2 -norm; this limit point is an element of E if and only if x0 = i xi .
It follows that H is isomorphic to a subset of IR ⊕ l 2 ; it can proved in fact that H is
isomorphic to IR ⊕ l 2 (see, e.g., Lemma 4.2 and Theorem 4.3 below).
A function F ∈ L2 (M, H ) has the form (F0 , (Fi )i≥1 ), where Fi is predictable for all i ≥ 0
and
T
2
(Fi (s)) ds < ∞.
IE
n
i ≥0
0
LEMMA 2.7. In the hypotheses and setting of the previous example, let F ∈ L2 (M, H ).
Then
Fi dN i .
(2.2)
I(F) =
F dM =
F0 dW +
i ≥1
Proof . By Remark 2.6, the function F is the limit in L2 (M, H ) of a sequence (F n )
such that Fin ≡ 0 for i > n. This means that

 

2
T 

2
Fin (s) − F0 (s) +
Fin (s) − Fi (s)
IE 
(2.3)
ds 

0  i ≤n
i ≥1
tends to 0 as n goes to ∞. Furthermore,
n
n
n
Fin d Ni ;
F dM =
Fi dM i =
Fin dW +
I(F ) =
i ≤n
i ≤n
i ≤n
COMPLETENESS IN LARGE FINANCIAL MARKETS
301
denoting by J (F) the right-hand side of (2.2), we have that |I(F n ) − J (F)|2M2 (P) is equal
to (2.3), hence it goes to 0 as n → ∞. Thus, necessarily, I(F) = J (F).
3. GENERALIZED PORTFOLIOS
We take as given a large financial market on a finite fixed-time interval [0, T ]. The notion
of “large financial market” was introduced by Kabanov and Kramkov (1994) as a sequence of small markets. We consider a special case of this definition: we study a financial
market containing countably many assets, following the approach by Björk and Näslund
(1998). The prices of the assets are modeled by a sequence of stochastic processes defined
on a given filtered probability space (, F, IF = (Ft )t≤T , P) fulfilling the usual assumptions; for simplicity, we assume that F0 is the trivial σ -algebra and that F = FT . The
price processes ((Si (t))t≤T )i≥1 are supposed to be adapted, càdlàg, and locally bounded.
We also assume the existence of a process S0 , which represents the riskless bond and
which we can take, without loss of generality, identically 1, as the price processes had
already been discounted.
We suppose that IF is the filtration generated by (Si )i≥1 . For all n, we consider the nth
small market ((, F n , IFn , Pn ), (S0 , S̄ n )), based on the first n assets: S̄ n is the asset prices
vector (Si )i≤n ; by IFn = (Ftn )t≤T we denote the (completed) filtration generated by S̄ n ,
with F n = FTn ; finally, we set Pn = P|F n . We denote by P, P n the predictable σ -fields
relative respectively to the filtrations IF, IFn : the σ -field P coincides with ∨n P n ; that is,
the minimal σ -field which contains all P n .
ASSUMPTION 3.1. There exists an equivalent martingale measure Q, namely a probability measure Q equivalent to P, such that all Si are locally square integrable martingales
with respect to Q.
Under this assumption, Qn = Q|F n is a martingale measure for S̄ n and it is equivalent
to Pn . This ensures that there are no arbitrage opportunities in the small markets (see,
e.g., Delbaen and Schachermayer 1999). Kabanov and Kramkov (1994, 1998) proved
that on the large market there are no asymptotic arbitrage opportunities of the first and
second kind. Klein (2000) showed that there is even no asymptotic free lunch; conversely,
if one assumes no asymptotic free lunch in the present setting, one can find an equivalent
martingale measure for all Si .
We denote by S the global, arbitrage-free market ((, F, IF, Q), (Si )i ≥0 ), and by S n
the small market ((, F n , IFn , Qn ), (S0 , S̄ n )).
For the sake of notations, we will write M2 instead of M2 (Q) and analogously
define M2n as the class of real-valued square integrable martingales on (, F n , IFn , Qn ).
The set L2 (F) will denote the set of Q-square integrable random variables, and L2 (F n )
will be the subset of L2 (F) of random variables which are F n -measurable.
The process S = (Si )i≥1 is a cylindrical martingale under Q; so, we can define Q, A,
and H as in Section 2.
Our aim is to analyze the question of completeness in this framework: we define as
market observable contingent claim on S every random variable X ∈ L2 (F). We call a
market observable finite contingent claim a claim on the market S which, in fact, belongs
to a small market S n for some n, namely a random variable X ∈ L2 (F n ). Björk and
Näslund (1998) distinguished between (finite) market observable contingent claim and
(finite) contingent claim (see Def. 3.2 in Björk and Näslund). However, in the present
setting, we will not consider nonobservable claims. For this reason, and for simplicity,
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M. DE DONNO
we will use “contingent claim” and “market observable contingent claim” to denote the
same object.
The main difficulty in studying completeness in a large financial market comes from
the fact that an investor can build a portfolio using all the assets in the market. For this
reason, we will distinguish between strategies based on either a finite or an infinite number
of assets.
DEFINITION 3.2. We call a “naive” strategy a pair π = (ξ, F), where ξ is a real-valued
predictable process and F is an element of L2 (S, E ).
The process ξ represents the quantity invested in the riskless bond, and F is the investment
in the risky assets; a “naive” strategy consists in investing instantaneously in a finite
number of stocks, even if it may involve all of them.
DEFINITION 3.3. For a naive strategy π = (ξ, F), the value of the corresponding portfolio is given by the process
Fi (t)Si (t)
V π (t) = ξ (t)S0 (t) + F(t) · S(t) = ξ (t) +
i ≥1
(where F(t) · S(t) = F(t), S(t)
E ,E and the sum is, in fact, a finite sum for all (t, ω)); we
will say that the portfolio, or, equivalently, the strategy, is self-financing if it satisfies the
following condition:
(3.1)
dV π (t) = ξ (t) dS0 (t) + F(t) dS(t) = F(t) dS(t).
The self-financing condition can be written also in integral form:
t
F(s) dS(s).
V π (t) = V(0) +
0
We recall that if a portfolio is self-financing then the investing strategy and the initial
value of the portfolio uniquely determine the amount invested in the riskless bond. So,
in fact, a self-financing portfolio is specified by the pair (V (0), F).
We saw in Section 2 that the set L2 (S, E ) is not complete (with respect to the norm
induced by the stochastic integral) and that its completion is L2 (S, H ). Then, it becomes
natural to give the following definition.
DEFINITION 3.4. We call generalized strategy a pair π = (ξ, F), where ξ is a real-valued
predictable process and F is an element of L2 (S, H ).
A generalized strategy is the limit of a sequence of naive portfolios and may contain
all stocks in the market. Yet, it is not clear what the value is of a generalized portfolio:
the main problem is to make sense of the mathematical expression F(t) · S(t), when F is
H -valued. The following result is helpful (Mikulevicius and Rozovskii 1999, Prop. 2.2,
Cor. 2.2).
PROPOSITION 3.5. For fixed (ω, t), a norm can be defined on QE by setting, for x ∈ E ,
|Qx| QE = |x| E .
QE has a pre-Hilbert structure; its completion H is a Hilbert space and it is such that
COMPLETENESS IN LARGE FINANCIAL MARKETS
303
(i) H can be continuously embedded in E.
(ii) H is the topological dual of H.
(iii) The mapping Q : E → E can be extended to a mapping from H to H.
When S(t) is Ht -valued for all t, then it is obvious to define F(t) · S(t) by exploiting the
duality between Ht and Ht , namely
F(t) · S(t) = F(t), S(t)
H ,H .
(3.2)
Unfortunately, in most cases, this does not happen. Take, for instance, the cylindrical
martingale W = (Wi )i≥1 defined by a sequence of independent Wiener processes; then Q
is the identity function and H = H = l 2 but W(t) is not in l 2 for t > 0.
However, given F ∈ L2 (S, H ), there exists a sequence F n ∈ L2 (S, E ) such that F n
converges to F in L2 (S, H ). Given an initial value V (0), the pair (V (0), F n ) uniquely
determines a self-financing portfolio, with a certain money-holding
ξ n ; if we denote by
t
n
n
2
V (t) its value at time t, then V − V (0) converges in M to 0 F(s) dS(s). We then give
the following definition.
DEFINITION 3.6.
(i) Given V (0) ∈ IR, F ∈ L2 (S, H ), F n ∈ L2 (S, E ), such that F n converges to F in
L2 (S, H ), we define the sequence of the induced self-financing portfolio value
processes by setting
t
V n (t) = V(0) +
F n (s) dS(s);
0
the induced money holdings will be given by:
ξ n (t) = V n (t) − V(0) − F n (t) · S(t).
(ii) We define the generalized induced self-financing portfolio value process by the
formula
t
V F (t) = V(0) +
F(s) dS(s).
0
It must be pointed out that such a process is obtained as a limit portfolio: it exists as a
value of a self-financing portfolio, but it may be not possible to specify either the part
invested in the riskless bond or the risky investment.
DEFINITION 3.7. We say that a claim X is attainable if there exist an initial value V (0)
and a self-financing naive strategy F(F ∈ L2 (S, E )) such that V F (T) = X; we also say
that the strategy F replicates the claim X.
We say that a claim X is asymptotically attainable if there exist an initial value V (0)
and a generalized strategy F (F ∈ L2 (S, H )) such that V F (T) = X.
In other words, a claim X is asymptotically attainable if there exists a sequence of selffinancing naive porfolios whose final values tend to X. We also recall that, when a claim
is attainable, V(0) = IE [X].
DEFINITION 3.8. The market is said to be complete if each claim is either attainable
or asymptotically attainable.
304
M. DE DONNO
Bättig (1999), making use of functional analytic methods, gave a very general definition of completeness in a large financial market, which is invariant with respect to a
change in probability and independent of no arbitrage. However, he also proved (in his
Thm. 3) that, under a no-arbitrage assumption, his notion of completeness is equivalent to
ours.
Completeness can also be stated in terms of equivalent martingale measures. More
precisely, a market is complete under an equivalent martingale measure Q if it is an
extreme point in the set of all martingale measures. Artzner and Heath (1995) constructed
an example of a financial market containing countably many assets such that there exist
two equivalent martingale measures under which the market is complete. The following
proposition summarizes results obtained by Jacod (1979) for conditions (i) and (ii) and
by Pratelli (1996) for condition (iii).
PROPOSITION 3.9. Completeness of the market is equivalent to uniqueness of the equivalent martingale measure when one of the following conditions is fulfilled:
(i) The market contains a finite number of assets.
(ii) Every asset price process has continuous trajectories.
(iii) The filtration IF (generated by the asset price processes) is strictly left continuous;
that is, for all stopping times τ , we have Fτ = Fτ − .
We now want to adapt to the case of large markets a mathematical formulation of
completeness that is widely used in the finite-dimensional markets because it allows us
to exploit properties of square integrable martingales.
PROPOSITION 3.10. The market is complete if and only if I(L2 (S, H )) = L2 (F); that
is, by Proposition 2.4, the stable subspace generated by M coincides with L2 (F).
Proof . I(L2 (S, H ))= L2 (F) if and only if each X ∈ M2 admits a representation of
the form X = IE [X] + F dS for some F ∈ L2 (S, H ). This is equivalent to saying that
for each claim X there exists a self-financing strategy which replicates it.
In fact, the claims that are available in the market are finite contingent claims. Hence
it makes sense to give the following definitions.
DEFINITION 3.11. The market S is said to be complete on the set of finite contingent
claims if each finite contingent claim is either attainable or asymptotically attainable.
DEFINITION 3.12. The market S is said to be finitely complete if for every finite contingent claim there exists a small market S n where the claim is attainable.
Completeness on the set of finite contingent claims means that each finite contingent
claim can be hedged by a portfolio either naive or generalized. Furthermore, if finite
completeness holds (Definition 3.12) then the replicating portfolio will be based on a
finite number of assets. Hence, if the market is finitely complete, it is trivially complete
on the set of finite contingent claims. The converse does not hold, as will be clear from
examples to be discussed below.
The next lemma gives a rather technical characterization of completeness on the set of
finite contingent claims (closely related to Proposition 3.10).
COMPLETENESS IN LARGE FINANCIAL MARKETS
305
LEMMA 3.13. The market is complete on the set of finite contingent claims if and only
if for every M ∈ M2 , which is orthogonal to Si for all i and such that M(0) = 0, the random
variable M(T) is orthogonal to all finite contingent claims X such that IE [X] = 0.
Proof . Assume that the market is complete on the set of finite contingent claims
= 0 and M strongly orthogonal to all Si ; then
and let M be in M2 such that M(0)
M is orthogonal to all integrals F dS. Let now X be a finite contingent claim such
that IE [X] = 0. By hypothesis,
Tit is attainable. Hence, there exists a generalized strategy
F ∈ L2 (S, H ) such that X = 0 F(s) dS(s). It follows that M(T) is orthogonal to X.
Conversely, let X be a finite contingent claim and project it on the stable subspace
generated by S:
T
X = IE [X] +
F(s) dS(s) + M(T),
0
where M ∈ M2 is strongly orthogonal to Si and such that M(0) = 0. Then, by hypothesis,
M(T) is orthogonal to X −
IE [X], which is a finite contingent claim with zero mean. This
condition entails that IE M(T)2 = 0, hence M ≡ 0. The self-financing portfolio with
initial value IE [X] and strategy F replicates the claim.
This characterization allows us to show that if all finite contingent claims are attainable
then the market is complete.
PROPOSITION 3.14. The market S is complete if and only if it is complete on the set of
finite market observable contingent claims.
Proof . Necessity is trivial. In order to prove sufficiency, by Proposition 3.10, we need
to show that the stable subspace generated by S coincides with L2 (F). This is equivalent
to proving that every U ∈ M2 , orthogonal to all Si and such that U0 = 0, is identically
zero (see Protter 1990, Cors. 1 and 3 to Thm. IV.36). Let U be such a martingale, and set,
for all t,
Utn = IE Ut | Ftn ;
then Utn = IE[UTn | Ftn ]; namely, for all n, (Utn )t ∈ M2n .
The random variable UTn is a finite contingent claim. Since we are assuming that the
market is complete on the set of finite contingent claim, by Lemma 3.13, UTn is orthogonal
to UT . It follows that IE[(UTn )2 ] = IE[UTn UT ] = 0, hence Utn ≡ 0. But for fixed t, (Utn )n∈IN
is a martingale with respect to the filtration (Ftn )n∈IN and admits Ut as terminal variable.
Thus, by the martingale convergence theorem, (Utn ) tends almost surely to Ut , as n → ∞,
and this implies Ut ≡ 0.
The following proposition rigorously states an intuitive result: when each finite submarket is complete, then global completeness holds. This is an easy consequence of
Proposition 3.14.
PROPOSITION 3.15. Assume that the market S n is complete for all n. Then also the
market S is complete.
306
M. DE DONNO
EXAMPLE
3.16. Assume that Si follows the dynamics
dSi (t)
= βi dY(t) + σi d Zi (t),
Si (t−)
where Y , (Zi )i≥1 are independent locally square integrable martingales, βi , σi ∈ IR+ , and
there exist c, C ∈ IR+ such that 0 < c ≤ infi (βi ∧ σi ) ≤ supi (βi ∨ σi ) ≤ C < ∞. We take,
as in the diffusion model of Björk and Näslund (1998), Y = W0 , (Zi )i≥1 = (Wi )i≥1 to be
independent Wiener processes.
Following Proposition 3.3 of Björk and Näslund (1998), one can build n independent
Wiener processes W̄1 , . . . , W̄n such that
n
dSi (t) Dij d W̄i (t),
=
Si (t)
j =1
and the matrix (Dij )1≤i,j≤n is nonsingular; then σ (S1 , . . . , Sn ) = σ (W̄1 , . . . , W̄n ). It is well
known that, given an n-dimensional Brownian motion, every square integrable martingale
with respect to its completed natural filtration can be represented as a stochastic integral
with respect to the Brownian motion (Protter 1990, Thm. IV. 42). Henceforth the finite
market S n is complete and, by Proposition 3.15, the global market also is complete.
Moreover condition (ii) of Proposition 3.9 holds and this implies uniqueness of the
equivalent martingale measure.
The converse of Proposition 3.15 does not hold. We can consider the previous example,
but now we take as Y a Wiener process W and as (Zi ) a sequence of independent compensated Poisson processes, independent also of W : in this case none of the small markets is
complete. Nevertheless, we will show in Section 4 that, under some assumptions on the
coefficients and on the intensities of Zi , we do in fact have global completeness. In other
words, diversification can complete an otherwise incomplete market.
4. COMPLETENESS IN FACTOR MODELS
In some cases, the identification of the space H , hence of the generalized strategies,
becomes easier if more information on the model is available; a particular structure can
help in characterizing generalized strategies. This is the case of factor models; we study
two simple examples, one- and two-factor models, where the random sources are Wiener
or Poisson processes and the coefficients are constant. The results can be extended to
more general factor models adding some proper hypotheses.
4.1. One-Factor Models
Consider the simple model studied by Björk and Näslund (1998): the asset prices follow
the dynamics
dSi (t)
= βi dW (t) + σi dN i (t),
Si (t−)
where βi , σi are strictly positive constants, W is a Wiener process, (Ni )i≥1 is a sequence of
independent, compensated Poisson processes, such that every N i is independent of W . For
simplicity, we assume all Ni to have the same intensity, λi = 1. When this is not the case,
it will be sufficient to replace σi2 by σi2 λi in the discussion that follows. In financial terms,
COMPLETENESS IN LARGE FINANCIAL MARKETS
307
W represents the systematic risk in the market, and Ni is the idiosyncratic component of
risk for asset i.
Such a market is not finitely complete in the sense of Definition 3.12 (see also Björk
and Näslund 1998, Prop. 4.2). We will prove that under some further assumption on
coefficients, global completeness—that is, completeness of the infinite market S—holds.
Denote by D = (Di )i≥1 the square integrable
t cylindrical martingale defined by Di (t) =
βi W (t) + σi Ni (t), or, equivalently, Di (t) = 0 (Si (s−))−1 dSi (s). Then we have the following
lemma.
LEMMA 4.1. The market is complete if and only if the stable subspace generated by D
coincides with L2 (F).
Proof . By Proposition 3.10, the market is complete if and only if the stable subspace
generated by S coincides with L2 (F) and this is equivalent to saying that if M ∈ M2
is such that M0 = 0 and is strongly orthogonal to all Si , then M ≡ 0. It is not difficult to see that a square integrable martingale is strongly orthogonal to all Si if and
only if it is strongly orthogonal to all Di . As a consequence, the claimed equivalence is
obtained.
So we can concentrate on D instead of S. From an economic point of view, this means
that we consider relative portfolios; that is, in the case of naive strategies we specify the
relative proportion of the total portfolio that is invested in an asset, instead of the exact
amount of money. We set At = t so that for all (t, ω), the covariance operator for the
cylindrical martingale D is defined as
Qiit,ω = βi2 + σi2 ,
Qijs,ω = βi β j
for i = j .
Notice that Q does not depend either on ω or t, hence H will not depend either.
Furthermore,
dSi , Sj t
= dDi , D j t = Qij dt.
Si (t−)Sj (t−)
The norm induced by Q for x ∈ E is given by
2
2
|x| E =
βi xi +
σi2 (xi )2 ;
i ≥1
i ≥1
n
a sequence (x n ) in E is a Cauchy sequence if and only if
i βi xi is a Cauchy sequence
l 2 . Thus the limit of (x n ) in H can be represented
in IR and (σi x ni ) is a Cauchy sequence in
as a pair x̃ = (x0 , x), where x0 = limn i βi xin ∈ IR, and x = (xi )i≥1 is the limit in l 2 of
(σi x ni )i≥1 ; the norm of this element is given by |x̃|2H = x02 + |x|l22 . So, H is isomorphic to
a closed subset of IR ⊕ l 2 .
An element of the set L2 (D, H ) is a predictable process with values in H , which can
be represented as a pair
F̃ = (F0 , F), where F0 takes values in IR, F is l 2 -valued, and the
T
2
2
random variable ( 0
i ≥0 (Fi (s)) ds) is in L (FT ). Furthermore, it can be shown (as in
Lemma 2.7) that
T
T
T
F̃(s) dD(s) =
F0 (s) dW (s) +
Fi (s) dN i (s).
0
0
i ≥1
0
308
M. DE DONNO
Note, however, that H may be isomorphic to a proper subspace of IR ⊕ l 2 . In fact, we will
show that the equality holds if and only if the market S is complete. It is well known that
the set of martingales {W , (Ni )i≥1 } has the predictable representation property on M2
(in the sense of Protter 1990, Sec. IV.3); that is, every M ∈ M2 , with M(0) = 0, can be
written in the form
T
T
(4.1)
F0 (s) dW (s) +
Fi (s) dN i (s),
M(T) =
0
i ≥1
0
where (Fi )i≥0 are predictable processes such that
T
T
2
2
(4.2)
(F0 (s)) ds +
(Fi (s)) ds < ∞,
IE
0
i ≥1
0
and this representation is unique. As a consequence, we deduce the above-mentioned
result.
PROPOSITION 4.2. The market S is complete if and only if H is isomorphic to IR ⊕ l 2 .
Proof . By Lemma 4.1, completeness of S means that for all X ∈ L2 (FT ) there exists
a function F̃ = (F0 , (Fi )i ≥1 ) ∈ L2 (D, H ) such that
T
T
T
F̃(s) dD(s) = IE [X] +
(4.3) X = IE [X] +
F0 (s) dW (s) +
Fi (s) dN i (s).
0
0
i ≥1
0
Fix α ∈ IR, y ∈ l 2 . The random variable X = αW(T) + i ≥1 yi Ni (T) is in L2 (FT ); thus,
it admits a representation as in (4.3). By uniqueness of representation, F0 = α, Fi = yi ,
hence (α, y) must be an element of H .
Conversely, let X be a contingent claim. Since (W , (Ni )i≥1 ) have the predictable representation property, there exist predictable functions F0 with values in IR and F with values
Morein l 2 such that representation (4.1) holds. By hypothesis, F̃ = (F0 , F) is H -valued.
T
over the integrability condition holds, hence we can write X = IE [X] + 0 F̃(s) dD(s),
which means completeness.
By using Proposition 4.2, it is possible to characterize completeness of the factor models
in terms of its coefficients βi , σi .
THEOREM 4.3. The market S is complete if and only if
(4.4)
β2
i
= ∞.
2
σ
i ≥1 i
Proof . We start proving necessity by contradiction. By Proposition 4.2, we can equivalently assume that H is isomorphic to IR ⊕ l 2 . Then, for all α ∈ IR there existsa Cauchy
n
2
sequence x n in E such that (σi x ni )
i≥1 goes to zero in l (as n goes to ∞) and ( i ≥1 βi xi )
2
2
converges to α. Assume now that i ≥1 βi /σi < ∞. Then
2 β2
2
i
n
2
n
βi xi
≤
σi xi
,
2
i ≥1
i ≥1 σi
i ≥1
but this forces the left-hand side to go to zero, which gives a contradiction.
COMPLETENESS IN LARGE FINANCIAL MARKETS
309
To show sufficiency, we use Lemma 4.1. Let M ∈ M2 be such that M(0) = 0 and M is
orthogonal to Di for all i. By the predictable representation property of {W , (Ni )i≥1 }, there
exists F̃ = (F0 , (Fi )i ≥1 ) which satisfies condition (4.2) and is such that representation (4.1)
holds. By orthogonality, necessarily, for all i:
dM, Di t
Q-a.s.
= βi F0 (t) + σi Fi (t) = 0
dt
hence Fi (t) = − βσii F0 (t); condition (4.2) entails i ≥0 (Fi (t))2 < ∞ a.s. Since i ≥1 βi2 /σi2 =
+∞, it must be F0 (t) = 0 and then Fi (t) = 0 for all i, which implies M = 0.
REMARK 4.4. Björk and Näslund (1998) proved completeness of the market in a
heuristic way. They assumed that the coefficients βi and σi satisfy the following condition
(which is a special case of condition (4.4)): There exist β, σ > 0 such that limi βi = β
and limi σi = σ , where the limit is taken possibly passing to a subsequence. Thanks to
this hypothesis, risk can be diversified: a portfolio, called “asymptotic asset,” is obtained
as limit of well-diversified portfolios, so that it has no idiosyncratic component of risk.
Therefore, it is possible to replicate all finite contingent claims (which implies global
completeness by Proposition 3.14) with strategies based on finitely many assets; this is
what Björk and Näslund called completeness in the asymptotic market. This surprising
result can be explained by our approach: the asymptotic portfolio is an element of the
stable subspace generated by (Si )i≥1 and it is, in our language, a generalized portfolio.
REMARK 4.5. It is clear that an analogous result holds if we take as systematic risk a
Poisson process and as idiosyncratic risks a sequence of independent Wiener processes.
Such a choice could find an economic reason in the fact that there may occur some shocks
in the market which affect all the securities.
4.2. Two-Factor Models
Let us consider a slightly more complicated model, which has also been studied by
Björk and Näslund (1998):
dSi (t)
= βi dW (t) + γi dN(t) + σi dN i (t),
Si (t−)
where now in the systematic risk component there is one more random source represented
by an independent (of W and Ni ) compensated Poisson process with intensity 1. The
set {W , N, (Ni )i } has the predictable representation property. We assume, for simplicity,
that mini (βi ∧ γi ) ≥ ε > 0. Björk and Näslund proved that, in this case, completeness in
the asymptotic model can be achieved by adding two “independent” asymptotic assets
under the following assumption: there exist two subsequences I and J such that limi∈I βi =
βI , limi∈J βi = βJ , limi∈I γi = γI , limi∈J γi = γJ and βI /γI = βJ /γJ (Prop. 5.3 in Björk and
Näslund). The intuitive idea behind this result is that if the idiosyncratic risk can be
eliminated by diversification then two more assets are needed to hedge the systematic
risk, which derives from two independent random sources.
This fact can be formulated in a more general way, as a consequence of our results. We
put again At = t, so that Q does not depend on t and ω and it is given by
Qii = βi2 + γi2 + σi2 ,
Qij = βi β j + γi γ j
In this case, the norm induced by Q on E will be
fori = j .
310
M. DE DONNO
|x|2E
=
2
βi xi
+
i ≥1
2
γi xi
+
i ≥1
σi2 xi2 ;
i ≥1
n
n
the sequence (x n ) is a Cauchy sequence in H if and only if both
i βi xi and
i γi xi
are Cauchy sequences in IR, and (σi x ni ) is a Cauchy sequence in l 2 . Analogously to
Section 4.1, one can prove the following result.
PROPOSITION 4.6. The market S is complete if and only if H is isomorphic to IR2 ⊕ l 2 .
Once again, completeness can be characterized by some conditions on the coefficients of
the dynamics of the asset prices.
THEOREM 4.7. The market S is complete if and only if the following conditions hold:
(i)
βi2
i ≥1 σi2
γi2
i ≥1 σi2
= ∞.
(ii)
= ∞.
(iii) The sequence (βi /γi ) has no limit (as i → ∞).
Proof . If the market is complete, for the same reason as in Theorem 4.3 conditions (i)
and (ii) must hold. Furthermore, assume that condition (iii) does not hold and that the
sequence (βi /γi ) converges to some real number. Then it is bounded by some constant K
and
2
2
2
βi xi ≤ K
γi xi ,
i ≥1
i ≥1
which is in contradiction with the isomorphism stated in Proposition 4.6. If (βi /γi ) converges to ∞, then (γi /βi ) goes to zero and again we obtain a contradiction.
Conversely, take, as in the proof of Theorem 4.3, a martingale M ∈ M2 such that
M0 = 0 and M is strongly orthogonal to Si for all i. By the predictable representation
property of the set {W , N, (Ni )i },
t
M(t) =
0
t
G 0 (s) dW (s) +
0
F0 (s) dN(s) +
i ≥1
t
Fi (s) dN i (s)
0
and the orthogonality condition entails: βi G0 (t) + γi F0 (t)
+ σi Fi (t) = 0 a.s. It follows that
Fi (t) = −(βi G0 (t) + γi F0 (t))/σi and from the condition i ≥1 (Fi )2 < ∞ Q-a.s., we can say
that
βi
lim γi
G 0 + F0 = 0.
i →∞
γi
By (iii), we necessarily have G0 = F0 = Fi = 0 or, equivalently, M ≡ 0.
REMARK 4.8. Intuitively, conditions (i) and (ii) guarantee the existence of two asymptotic assets (in the language of Björk and Näslund 1998). Condition (iii) ensures that
these assets are nonredundant.
COMPLETENESS IN LARGE FINANCIAL MARKETS
311
5. APPROXIMATING STRATEGIES
We have seen that, given an attainable claim, the replicating strategy can be a generalized
strategy, which cannot in fact be achieved in a real market. We have also seen (while
completing the set of integrands) that such a strategy can be approximated by naive
strategies. The problem is that these strategies, even if they involve a finite number of
assets, depend on the global market in the sense that we need the information generated
by all the assets to determine the approximating strategies; so, in fact, we need to know all
the market to obtain in the limit a replicating portfolio. Our next aim is to find a sequence
of strategies such that, at the nth step of approximation, the portfolio really belongs to
the small market S n ; namely, it is based on S0 , S1 . . . , Sn and uniquely on the information
generated by these assets.
THEOREM 5.1. Let H ∈ L2 (F), IE [H] = 0, and assume that H is in the stable subspace
generated by S.Then there exists a sequence (ξ̄ n ) such that ξ̄ n is in L2 ( S̄ n ) (as defined in
Section 2) and ξ̄ n d S̄ n converges to H in L2 (F) as n → ∞.
In financial terms, if H is an attainable claim, the replicating portfolio can be approximated by a sequence of portfolios based on the strategies ξ̄ n in such way that ξ̄ n belongs to
the small market S n for all n.
The idea of the proof consists of taking an approximating strategy and projecting
it on the space of random variables that are measurable (predictable) with respect to
S0 , . . . , Sn .
Proof . The proof is based on three steps:
By hypothesis, there exists ξ ∈ L2 (S, H ) such that H = ξ dS.
Fix ε > 0; there exists a sequence ξ m of the form ξ m = i ≤m ξim ei and mε such that ξ m
satisfies the inequality
Step 1.
ξ − ξ m L2 (S,H ) < δ,
where δ = 6ε and m ≥ mε (this also implies that ξ m ∈ L2 (S, H ); see Remark 2.6).
Step 2. Fix m̄ ≥ mε . Denote by ξ m̄,n the predictable projection of ξ m̄ on F n
for n ≥ m̄. Let Am̄ be a P m̄ -measurable increasing process such that dSi , Sj = Qim̄j dAm̄ ,
for i , j ≤ m̄, for some Qm̄ predictable, with values in the set of symmetric nonnegative
definite m̄ × m̄ matrices.
can
The process Am̄ is P n -measurable for all n ≥ m̄; then, for j ≤ m̄, the process ξ m̄,n
j
n
,
with
respect
to
the
σ
-field
P
and
to
be seen as the conditional expectation of ξ m̄
j
m̄,n
m̄
the measure dQ ⊗ dA (see Protter 1990, Thm. III.25); trivially, ξ j = 0 for j > m̄.
m̄
n
So, (ξ m̄,n
j )n is a dQ ⊗ dA -martingale with respect to the filtration (P )n≥1 (which is
such that ∨n P n = P), and has ξ m̄
j as terminal variable. By the martingale convergence
m̄,n
m̄
theorem, ξ j tends to ξ j , dQ ⊗ dAm̄ -a.s., as n → ∞.
m̄ m̄ m̄
m̄ m̄
m̄
Since ξ m̄
j = 0 for j > m, we have (Q ξ , ξ ) =
i , j ≤m̄ Qi j ξi ξ j , and a similar expression holds for ξ m̄,n . Applying Jensen’s inequality for conditional expectation to the convex
function x → (Qm̄ x, x), we obtain
m̄ m̄,n m̄,n (5.1)
≤ IEdQ⊗dAm̄ (Qm̄ ξ m̄ , ξ m̄ ) | P n ,
Q ξ ,ξ
and integrating in dQ ⊗ dAm̄ we have
312
M. DE DONNO
(5.2)
T
IE
0
m̄,n
Qm̄
, ξtm̄,n
t ξt
dAm̄
t
m̄
≤ IE
0
The left-hand side of (5.2) is equal to IE [
L2 (S, H ). Furthermore,
m̄
m̄
Qm̄
t ξt , ξt
dAm̄
t
= ξ m̄ 2L2 (S,H ) .
T
m̄,n
, ξtm̄,n ) dAt ]; then we can say that ξ m̄,n ∈
0 (Qt ξt
(Qm̄ (ξ m̄,n − ξ m̄ ), ξ m̄,n − ξ m̄ ) → 0
dQ ⊗ dAm̄ -a.s.
as n → ∞
and it is a sequence of uniformly integrable random variables; therefore, it also converges
in L1 (dQ ⊗ dAm̄ ). This is equivalent to
lim ξ m̄ − ξ m̄,n 2L2 (S,H ) = 0,
n→∞
which implies that there exists n ε ≥ m̄ such that, for all n ≥ nε ,
ε
ξ m̄ − ξ m̄,n L2 (S,H ) < .
2
Step 3. We can use the same trick as in (5.1) and (5.2) to obtain, for all n ≥ n ε ≥ m̄,
the inequality
ξ m̄,n − ξ n,n L2 (S,H ) ≤ ξ m̄ − ξ n L2 (S,H ) ≤ 2δ.
n
2
n
Finally, we set ξ̄ nj = ξ n,n
j , for j ≤ n, mε ≤ m̄ ≤ n ε ≤ n. Then, ξ̄ ∈ L ( S̄ ) by (5.2) and
IE
T
H−
2 ξ̄ d S̄
n
n
0
= h − ξ n,n 2L2 (S,H )
2
≤ ξ − ξ m̄ L2 (S,H ) + ξ m̄ − ξ m̄,n L2 (S,H ) + ξ m̄,n − ξ n,n L2 (S,H )
≤ ε2 .
5.1. An Example: Asymptotic Model
The construction carried on in Theorem 5.1 is rather technical and not so easy to realize.
However, the structure of the model may help us find a proper approximating sequence
in an easier way. Consider the jump model as in Section 4.1 or in Björk and Näslund
(1998, Sec 4); it consists of a sequence of assets with the following price processes:
dSi (t) = Si (t−)[βi dW (t) + σi d Ni (t)],
where W is a Wiener process, Ni (t) = N̂i (t) − λi t is a compensated Poisson process (or
equivalently, N̂i is a Poisson process with intensity λi ), W , N1 , N2 , . . . are independent,
and βi , σi , λi are positive numbers that satisfy the following conditions:
lim βi = β
i →∞
sup σi2 λi ≤ M < ∞.
i ≥1
The market (Si )i≥1 is complete by Theorem 4.3, but it is not finitely complete in the sense
specified in Section 4.1. Björk and Näslund added the asymptotic asset
dS∞ (t) = S∞ (t)βdW (t),
COMPLETENESS IN LARGE FINANCIAL MARKETS
313
which is obtained as limit of well-diversified portfolios, and proved (Prop. 4.3 in Björk and
Näslund 1998) that, given a finite contingent claim H, which is measurable with respect
to F K = σ (S1 , . . . , SK ) = σ (W, N1 , . . . , NK ), it can be replicated by investing in a ideal
portfolio containing S∞ , S1 , . . . , SK ; namely, there exist ζ∞ , ζ1 , . . . , ζK , predictable with
respect to IFK (which coincides with the natural filtration associated to W , N1 , . . . , NK )
such that
T
T
ζ∞ (t) dS∞ (t) +
ζi (t) dSi (t)
H = IE [H] +
0
i ≤K
T
= IE [H] +
0
0
ξ∞ (t)
dS∞ (t) +
S∞ (t)
i ≤K
T
0
ξi (t)
dSi (t),
Si (t−)
where ξi (t) = ζi (t)Si (t−). Though the asymptotic asset S∞ does not exist in the real market, we can use asymptotic completeness to build a sequence of strategies that approximately replicate H.
For n > K, consider the strategy ψ n defined by

ξ∞

ξi + n if 1 ≤ i ≤ K
(5.3)
ψin = ξn∞
if K + 1 ≤ i ≤ n


0
if i > n.
to IFn , for n ≥ K, and it is in L2 (S, H ).
It is predictable with respect to IFK , hence
Tto nIF and
dS1 (t) dS2 (t)
We want to estimate IE[(H − IE [H] − 0 ψ (t)( S1 (t)− , S2 (t−) , . . . , ))2 ].
Denote by ψ̃ n the process with values in IRn+1 defined by (ξ∞ , − ξn∞ , . . . , − ξn∞ ). Then
T
dS1 (t) dS2 (t)
dS∞ (t)
dSn (t)
ψ n (t)
,
, . . . = ψ̃ n (t)
,...,
H − IE [H] −
S1 (t−) S2 (t−)
S∞ (t−)
Sn (t−)
0
and
(5.4)
2 dS∞ (t)
dSn (t)
ψ̃ (t)
,...,
IE
S∞ (t)
Sn (t−)
0
T
ξ 2 (t) ∞
2
2
2
2
βi + σi λi dt
ξ∞ (t)β +
= IE
n2
0
i ≤n
T
ξ 2 (t)
∞
βi dt
− IE
2β
n
0
i ≤n
T
2
(t) ξ∞
+ IE
βi β j dt
n 2 i , j ≤n
0
T
n
T
= C n IE
0
(ξ∞ (t))2 dt
= ε,
where C n = (β − n1 j ≤n β j )2 + n12 i ≤n σi2 λi tends to 0 as n → ∞. By the integrability
T
condition, IE [ 0 (ξ∞ (t))2 dt] is bounded; then we can make ε arbitrarily small (hence, we
can replicate H arbitrarily well), investing in the strategy ψ n , which is based on the first
n assets, with n sufficiently large.
314
M. DE DONNO
Let us show it in a concrete example. Take as H the claim which pays 1 if the first asset
has not any jump, namely H = 1{ N̂1 (T)=0} . Björk and Näslund (1998) proved that H can
be replicated by investing in S1 , in the asymptotic asset S∞ , and in the riskless bond S0 ,
respectively:
ζ1 (t) =
M(t−)
σ1 S1 (t−)
ζ∞ (t) =
M(t−)β1
,
σ1 β S∞ (t−)
ζ0 (t) = M(t) − ζ∞ (t)S∞ (t) − ζi (t)Si (t),
where M(t) = IE [H | Ft ] = IE [H | Ft1 ] = e−λ1 (T−t) 1{ N̂1 (t)=0} .
Set
ξ1 (t) = ζ1 (t)S1 (t−) =
M(t−)
,
σ1
ξ∞ (t) = ζ∞ (t)S∞ (t−) =
M(t−)β1
σ1 β
and define ψ n as in (5.3). We can explicitely compute (5.4). Indeed,
T
T
T
β12
β12
2
2
IE
(ξ∞ (t)) dt = 2 2 IE
M (t−) dt = 2 2
IE M2 (t−) dt.
σ
β
σ
β
0
0
0
1
1
Since IE [M2 (t−)] = e−λ1 (T−t) IE [M(t−)] = e−λ1 (T−t) e−λ1 T , we obtain
T
β 2 (1 − e−λ1 T )e−λ1 T
2
IE
(ξ∞ (t)) dt = 1
.
σ12 β 2 λ1
0
Thus we are able to estimate the approximation error. To get some numerical results, let
i +1
us take, for example, β = 0.15, βi = β + (−1)i 0.05, σi = 0.1, λi = 0.1, and T = 1 year.
Then, n = 17 assets are sufficient to replicate H with an error of ε ≤ 0.01, n = 32 assets
give ε ≤ 0.005.
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