Mean-variance hedging in large financial markets∗ Luciano CAMPI

†‡

Abstract We consider a mean-variance hedging (MVH) problem for an arbitrage-free large financial market, i.e. a financial market with countably many risky assets modelled by a sequence of continuous semimartingales. By using the stochastic integration theory for a sequence of semimartingales developed in De Donno and Pratelli [7], we extend the results about change of num´eraire and MVH of Gourieroux, Laurent and Pham [14] to this setting. We also consider, for all n ≥ 1, the market formed by the first n risky assets and study the solutions to the corresponding n-dimensional MVH problem and their behaviour when n tends to infinity. Key words: hedging, large financial market, stochastic integral for a sequence of semimartingales, num´eraire, artificial extension method. MS Classification (2000): 91B28, 60H05, 60G48

1

Introduction

In this paper we study the hedging problem for a future stochastic cash flow F , delivered at time T , in a large financial market. Let us consider first a market consisting of n + 1 primitive assets X = {S 0 , S}: one R t bond with price process St0 = exp( 0 rs ds) and n risky assets whose price process is a continuous n-dimensional semimartingale S = (S 1 , · · · , S n ). A criterion for determining a “good” hedging strategy is to solve the mean-variance hedging (MVH) problem introduced by F¨ollmer and Sondermann [13]: h i2 min E F − VTx,ϑ , (1) ϑ∈Θ

where VTx,ϑ

=

ST0

µ Z x+ 0

T

¡ ¢ ϑt d S/S 0 t

¶ (2)

∗ This research has been partially supported by the EU HCM-project Dynstoch (IMP, Fifth Framework Programme). † CEREMADE, Universit´e Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775 Paris Cedex 16. E-mail: [email protected]. ‡ I would like to thank Huyˆen Pham for fruitful suggestions and Marc Yor for his remarks. I am also grateful to Marzia De Donno and Maurizio Pratelli for their interest in this work.

1

is the terminal value of a self-financed portfolio in the primitive assets, with initial investment x and quantities ϑ invested in the risky assets. This problem has been solved by F¨ollmer and Sondermann [13] and Bouleau and Lamberton [3] in the martingale case: S 0 deterministic and S/S 0 is a martingale under the objective probability P thanks to direct application of the Galtchouk-Kunita-Watanabe (abbr. GKW) projection theorem. Extensions to more general cases were later studied by Gourieroux, Laurent and Pham [14] (abbr. GLP), Rheinl¨ander and Schweizer [28] and Pham et al. [26], in a continuous-time framework under more or less restrictive conditions. In particular GLP, by adding a num´eraire as an asset to trade in, show how self-financed portfolios may be expressed with respect to this extended assets family, without changing the set of attainable contingent claims. They introduce the hedging num´eraire V (e a), relate e it to the variance-optimal martingale measure P and, using this num´e raire both as a deflator and to extend the primitive assets family {S 0 , S}, transform the original MVH problem (1) into an equivalent and simpler one; this transformed quadratic optimization problem is solved by the GKW projection theorem under a martingale measure for the hedging num´eraire extended assets family {V (a), S 0 , S}. This procedure gives an explicit description of the optimal trading strategy for the original MVH-problem. Here, we seek to extend this approach R t to a large financial market, i.e. a market with one bond with price process St0 = exp( 0 rs ds) and countably many risky assets whose price process is a sequence of continuous semimartingales S = (S i )i≥1 . Large financial markets have been studied for the first time in a rigorous way by Kabanov and Kramkov [17, 18], who analyzed, under the assumption of market completeness, the relation between no-arbitrage in each finite submarket and the notions of asymptotic noarbitrage of the first and second type in the whole market. Klein and Schachermayer [19] studied the same problem in the incomplete case. More recently, Bjork and N¨aslund [2] and De Donno [4] have investigated the completeness of some special models, where the assets prices dynamics are driven by one Brownian motion (common to all assets) and countably many independent Poisson processes (one for each asset). Finally, we remark that also the Heath, Jarrow and Morton [15] interest rate model with a countable maturities set fits into this framework. In order to extend the GLP artificial extension method to an infinite-dimensional setting, a one has to use a stochastic integration (SI) theory with respect to a sequence of semimartingales, i.e. with respect to a semimartingale taking values in the space RN of all real sequences, which is much more delicate to use than the vectorial one. Mikulevicius and Rozovskii [24, 25] developed a SI theory for cylindrical martingales, i.e. martingales taking values in a topological vector space (see also De Donno [4], De Donno and Pratelli [6] and De Donno et al. [5] for financial applications). More recently, De Donno and Pratelli [7] have proposed a stochastic integral for a sequence of semimartingales, generalizing the SI theory by Mikulevicius and Rozovskii in this particular case. We will use their construction for making our MVH problem meaningful. Finally, we stress the fact that the techniques we use in our proofs are very different from the finite case. The main reason is that here we don’t have the Itˆo (and then integration by parts) formula at our disposal, which represents the main tool in proving results in GLP. 2

This paper is organized as follows. In Section 2, we recall some basic facts on stochastic integration with respect to a sequence of semimartingales S = (S i )i≥1 and obtain an infinitedimensional version of the GKW-decomposition theorem. In Section 3, we describe the model and define the set Θ of trading strategies, by using the SI theory of the previous section, and we show that the set of all attainable contingent claims is closed in L2 (P ). In Section 4 we generalize the artificial extension method to our setting, by showing invariance properties of state-price densities and especially of self-financed portfolios: the infinite-dimensional Memin’s theorem turns out to be crucial to show this property. We recall here that the original finite version of Memin’s theorem (see Memin [22])Rstates that, if S is an Rd -valued semimartingale, then the set of all stochastic integrals ξdS, for ξ predictable and S-integrable, is closed with respect to the Emery topology. In Section 5, we show how to use the artificial extension method for solving the MVH problem in this framework. Unfortunately we don’t have here an explicit expression of the correspondences relating the solutions of the initial and the transformed MVH-problem. For this reason, in Section 6 we define the finite-dimensional MVH problems, corresponding n to consider only the market formed by the bond S 0 and the first n risky assets S := (S 1 , . . . , S n ), n ≥ 1; we show that the sequence of solutions of these finite-dimensional problems converges to the solution of the original one (1).

2

Some preliminaries on stochastic integration with respect to a sequence of semimartingales

In this section we will follow very closely the treatment of the stochastic integration for countable many semimartingales, that the reader can find in De Donno and Pratelli [7]. We remark that there is a huge literature on stochastic integration for martingales and semimartingales taking values in infinite-dimensional spaces. Here we quote only the pioneering work by Kunita [21] and the book by M´etivier [23]. Let (Ω, F, P ) be a probability space with a filtration F ={Ft , t ∈ [0, T ]} satisfying the usual conditions of right-continuity and P -completeness, where T > 0 is a fixed finite horizon. Letting p ≥ 1, we will denote by Hp (P ) the set of all real-valued martingales M on the given filtered probability space, such that M ∗ = supt∈[0,T ] |Mt | is in Lp (P ) (see Jacod’s book [16] for more details). Moreover, we denote by S(P ) the space of real semimartingales, equipped with Emery’s topology (see Emery [12]). S(P ) is a complete metric space. Let S = (S i )i≥1 be a sequence of semimartingales. We denote by E the set of all real-valued sequences (i.e. RN ), endowed with the topology of pointwise convergence and by E 0 its topological dual, namely the space of all signed measures on N which have a finite support; each of them can be identified with a sequence with all but finitely many components equal to 0. We will denote by ei the element, both of E 0 and E such that eij = δi,j (where δi,j is the Dirac delta); h·, ·iE 0 ,E will denote the duality between E 0 and E. We denote by U the set of not necessarily bounded operators on E and, for all h ∈ U, we denote by D(h) the domain of h (D(h) ⊂ E). We say that a sequence (hn ) ⊂ E 0 converges 3

to h ∈ U if limn hn (x) = h(x), for every x ∈ D(h). We say that a process ξ taking values in U is predictable if there exists a sequence (ξ n ) of E 0 -valued predictable processes, such that for all (ω, t), and for all x ∈ D(ξt (ω)), one has ξt (ω) = limn ξtn (ω). 0 Finally, predictable process ξ is called a simple integrand if it has the Pa E -valued form ξ = i6n ξ i ei = (ξ 1 , ξ 2 , · · · , ξ n , 0, · · · ), where ξ i are real-valued predictable bounded processes. We note that one can define the stochastic integral of a simple integrand ξ with respect to S in the following obvious way: Z Z X ξdS = ξ i dS i .

6

i n

We are now able to give the following definition: Definition 1 (De Donno and Pratelli [7]) Let ξ be a predictable U-valued process. We say that ξ is integrable with respect to S if there exists a sequence (ξ n ) of simple integrands such that: 1. ξ n converges to ξ pointwise; R 2. ξ n dS converges to a semimartingale Y in S(P ). R We call ξ a generalized integrand and define ξdS := Y . Moreover, we denote by L(S, U) the set of generalized integrands. This notion of stochastic integral is well-defined, as shown by De Donno and Pratelli [7] (Proposition 5.1), and moreover there is an infinite-dimensional extension of Memin’s theorem (Th´eor`eme V.4 [23]), which states that the set of stochastic integrals with respect to a semimartingale is closed in S(P ): Theorem 2 (De Donno and Pratelli [7], Theorem 5.2) Let be given a sequence of R semimartingales S = (S i )i>1 and a sequence (ξ n ) of generalized integrands: assume that ( ξ n dS) is R an Cauchy R sequence in S(P ). Then there exists a generalized integrand ξ such that ξ dS → ξdS in S(P ). In the sequel, we will need an infinite-dimensional version of the GKW-decomposition for a sequence of continuous local martingales. For this reason, we briefly recall the Mikulevicius and Rozovskii [24] theory of SI for a sequence of locally square integrable martingales and show how to extend it to a sequence of continuous local martingales. We assume that S i = M i ∈ H2 (P ) for all i > 1. It is easy to see that there exist: 1. an increasing predictable real-valued process (At ) with E[AT ] < ∞, 2. a family C = (C ij )i,j >1 of predictable real-valued process, P such that C is symmetric ij ji and non-negative definite, in the sense that C = C and i,j 6l xi C ij xj > 0, for all l ∈ N, for all x ∈ Rl , dP dA a.s., 4

such that

­ i ® M , M j t (ω) =

For a simple integrand ξ = "µZ E 0

¶2 #

T

ξs dMs

P

6

i nξ

 Z  =E

T

0

Z 0

t

ij Cs,ω dAs (ω).

(3)

i ei ,

the Itˆo isometry holds:   Z X ­ i ® i j j   ξs ξs d M , M s = E

6

 T

0

i,j n

X

6

ξsi ξsj Csij dAs  .

(4)

i,j n

Consider C for fixed (ω, t) and assume for simplicity that C is positive definite. The above Itˆo isometry makes it natural to define on E 0 a norm by setting: |x|2E 0

t,ω

= hx, Ct,ω xiE 0 ,E =

∞ X

ij xi Ct,ω xj ,

(5)

i,j=1

where the sum contains a finite number of terms. The norm is induced by an obvious scalar product, which makes E 0 a pre-Hilbert space. This norm depends on (ω, t): for simplicity, we omit ω, but we keep t to remind us about this dependence and denote by Et0 the space E 0 with norm induced by Ct . It is not difficult to see that Et0 is not complete, but we can take its completion which we denote by Ht0 and which is a Hilbert space. Ht0 is generically not included in E, hence the canonical injection from E 0 to E cannot be extended to an injection from Ht0 to E. Moreover, Ht0 can be characterized as the topological dual of the completion Ht of the pre-Hilbert space Ct E 0 with respect to the norm induced by the scalar product (Ct x, Ct y)Ct E 0 = hx, Ct yiE 0 ,E = hy, Ct xiE 0 ,E . The following theorem is essentially due to Mikulevicius and Rozovskii [24] (see their Proposition 11, p. 145). Nonetheless we prefer to follow the formulation of this result given by De Donno and Pratelli [7], since it fits better into their more general theory of SI for a sequence of semimartingales, as introduced before. Theorem 3 (De Donno and Pratelli [7], Theorem 3.1) Let ξ be a U-valued process such that: 1. D(ξω,t ) ⊃ Hω,t for all (ω, t); 0 ; 2. ξω,t |Hω,t ∈ Hω,t

3. ξt (Ct en ) is predictable for all n; RT 4. E[ 0 |ξt |2H 0 dAt ] < ∞. t

n converges to ξ Then, there exists a sequence ξ n of simple integrands, such that ξω,t ω,t in R n 2 0 is a Cauchy sequence in H (P ). As a consequence, we can Hω,t for all (ω, t) and ξ dM R R n define the stochastic integral ξdM as the limit of the sequence ξ dM .

5

Remark 4 (Mikulevicius and Rozovskii [24], De Donno [4]) When Ct is only non negativedefinite, (5) defines a seminorm on E 0 . The construction of Ht0 and of the stochastic integral can be carried on replacing E 0 with the quotient space E 0 / ker Ct . R Remark 5 The set of all stochastic integrals ξdM , with ξ fulfilling the four conditions of Theorem 3, is a closed set in H2 (P ) and coincides with the stable subspace generated by M in H2 (P ). It is an immediate extension of the analogous result in the finite-dimensional case. When M = (M i )i>1 is a sequence of continuous local martingales, it is quite easy to extend the previous construction. Indeed, from Dellacherie [11] we know that there exists a uniform localization for the sequence (M i ), i.e. a sequence (τn ) of stopping times such that i τn → T and M·∧τ is a bounded martingale for all i > 1. To see this, it suffices to apply n Th´eor`eme 2 and Th´eor`eme 3 in [11], p. 743, to the sequence M . This property allows us to define by localization, in the usual way, a stochastic integral with respect to M and for all U-valued processes ξ such that: 1. D(ξω,t ) ⊃ Hω,t for all (ω, t); 0 ; 2. ξω,t |Hω,t ∈ Hω,t

3. ξt (Ct en ) is predictable for all n; RT 4. 0 |ξt |2H 0 dAt < ∞ P -a.s.. t

Finally, again by using the uniform localization, it is easy to prove a GKW-decomposition theorem in our infinite-dimensional setting: Proposition 6 Let M = (M i )i>1 be a sequence of continuous local martingales and N be a real-valued local martingale. Then, there exist an integrand ξ satisfying conditions 1.-4. above, and a real-valued local martingale L vanishing at zero and orthogonal to each M i , such that Z N = N0 + ξdM + L. (6) Proof. Apply the Dellacherie uniform localization and Remark 5 and proceed exactly as, for instance, in Jacod [16], Th´eor`eme (4.27) if N is locally square-integrable. Otherwise, use the following argument by Ansel and Stricker [1] (D.K.W. cas 3): write N as the sum N = N c + N d , where N c and N d are its continuous and purely discontinuous parts. N d is orthogonal to all continuous local martingales and N c is locally bounded and then it can R c be written as N = ξdM + U with U orthogonal to M and ξ satisfying conditions 1. to 4.. To conclude, it suffices to set L = U + N d . ¤

6

3

The market model

Let (Ω, F, P ) be a probability space with a filtration F ={Ft , t ∈ [0, T ]} satisfying the usual conditions of right-continuity and P -completeness, where T > 0 is a fixed finite horizon. We also assume that F0 is trivial and FT = F. There exist countably many primitive assets i modelled by a sequence R t of real valued processes X = (S )i≥0 : one bond whose price process 0 is given by St = exp 0 rs ds, with r a progressively measurable process with respect to F, interpreted as the instantaneous interest rate, and countably many risky assets, whose price processes S i are continuous semimartingales. In the rest of the paper, we shall make the standing assumption on the bond process S 0 : ST0 ,

1 ∈ L∞ (P ) , ST0

(7)

RT which is equivalent to assuming that | 0 rs ds| ≤ c, P a.s. for some positive constant c. We let ½ ¾ 1 dQ 2 i 0 ∈ L (P ) , every S /S is a Q-local martingale M2 = Q ¿ P : 0 ST dP denote the set of P -absolutely continuous probability measures Q on F with square integrable state price density (1/ST0 )dQ/dP and such that each component of the sequence S/S 0 is a local martingale under Q. By Me2 = {Q ∈ M2 : Q ∼ P } we denote the subset of M2 formed by the probability measures that are equivalent to P . Throughout the paper, we make the natural standing assumption: Me2 6= ∅.

(8)

This assumption is related to some kind of no-arbitrage condition and we refer to Delbaen [8] and to the seminal paper by Kreps [20] for a general version of this fundamental theorem of asset pricing for a potentially infinite family of price processes. For the more specific framework of large financial markets, we refer to Kabanov and Kramkov [17, 18] and Klein and Schachermayer [19]. We denote by Θ the space of all generalized integrands ϑ ∈ L(S/S 0 , U) such that R RT 0 2 e ϑd(S/S 0 ) is a Q-martingale. 0 ϑt d(S/S )t is in L (P ) and for all Q ∈ M2 Notice (see the discussion in De Donno [4], pp. 8-11) that in general one cannot define the value process of a trading strategy ϑ in the usual way: the expression ϑt · (S/S 0 )t is not always well-defined. This is because ϑt takes values in the space U which, in most cases, is strictly bigger than E 0 , and so we cannot use duality to define a product between the strategy and the price process. For this reason we give the following:

7

Definition 7 For a trading strategy ϑ ∈ Θ the value process of the corresponding selffinanced portfolio with respect to the primitive asset family {S 0 , S} and with initial value x ∈ R is given by µ ¶ Z t ¡ ¢ x,ϑ 0 0 Vt = Vt = St x + ϑs d S/S s t ∈ [0, T ] (9) 0

Finally we denote by GT (x, Θ) the set of investment opportunities (or attainable claims) with initial value x ∈ R: ¶ ¾ ½ µ Z T ¡ ¢ 0 0 GT (x, Θ) := ST x + ϑs d S/S s : ϑ ∈ Θ ⊆ L2 (P ) . 0

Proposition 8 The set GT (x, Θ) is closed in L2 (P ). RT Proof. Let ϑn be a sequence in Θ such that ST0 (x + 0 ϑns d(S/S 0 )s ) converges in L2 (P ) Rt to a random variable V . Take some Q ∈ Me2 and set Ytn := 0 ϑns d(S/S 0 )s , t ∈ [0, T ]. By Remark 2.2. in Delbaen and Schachermayer [10] the sequence of Q-martingales (Ytn )t∈[0,T ] converges in H1 (Q). Now, since convergence in H1 (Q) implies that in S(Q) (see Theorem 14 in Protter [27], p. 208) and Emery’s topology is invariant under a change of an equivalent probability measure (see Th´eor`eme II.5 in Memin [22], p. 20), the sequence (Ytn )t∈[0,T ] converges in S(P ) too. Now, thanks to Theorem 2 we can exhibit a generalized integrand ϑ ∈ L(S/S 0 , U) such that V = ST0

µ Z x+

T

0

¶ ¡ ¢ ϑs d S/S 0 s .

The other two properties, characterizing the set Θ, are obviously satisfied by the process ϑ. ¤ This proposition makes our MVH-problem meaningful, ensuring the existence and uniqueness of its solution.

4

Extending the GLP artificial extension method

Definition 9 A num´eraire is defined as a self-financed portfolio with respect to the primitive assets family X = {S 0 , S}, characterized by a trading strategy a ∈ Θ and a value process V (a) = V 1,a as in (9) with unit initial value and intermediate values assumed to be almost surely strictly positive. Remark 10 To avoid misunderstandings of our notation, observe that we have denoted by “a” the strategy used as num´eraire, instead of the integrand in its exponential representation as in GLP.

8

To such a num´eraire a we can associate a new countable family of assets consisting of this num´eraire and the primitive assets. This assets family is called, as in GLP, a-extended assets family and its price process is given by {V (a), X}. Its price process renormalized in the new num ´eraire is {1, X (a)} := {1, X/V (a)} . Notice that when a = 0, V (a) is the initial bond price process S 0 and X(a) = (1, S/S 0 ). Given a num´eraire a ∈ Θ, we define ½ ¾ 1 dQ (a) i 2 M2 (a) = Q (a) ¿ P : ∈ L (P ) , every X (a) is a Q (a) -local martingale VT (a) dP and Me2 (a) = {Q (a) ∈ M2 (a) : Q (a) ∼ P } . As in GLP Proposition 3.1, and with the same proof, we have the following characterization of the set Me2 (a) of equivalent a-martingale measures in terms of the set Me2 of equivalent martingale measures: Proposition 11 Let a ∈ Θ be a num´eraire and V (a) its value process. There is a one-toone correspondence between M2 (a) (resp. Me2 (a)) and M2 (resp. Me2 ): Q(a) ∈ M2 (a) (resp. Me2 (a)) if and only if there exists Q ∈ M2 (resp. Me2 ) such that VT (a) dQ dQ (a) = . dP ST0 dP

(10)

We denote by Φ(a) the space of trading strategies with respect to the a-extended assets RT family {V (a), X}, i.e. the set of all φ(a) ∈ L(X(a), U) such that VT (a) 0 φt (a)dXt (a) ∈ R L2 (P ) and V (a) φ(a)dX(a) is a local Q(a)-martingale for all Q(a) ∈ Me2 (a). For a trading strategy φ(a) ∈ Φ(a) the value process of the corresponding self-financed portfolio with respect to the a-extended assets family {V (a), X} and with initial value x ∈ R is given by µ ¶ Z x,φ(a)

Vt = Vt

t

= Vt (a) x +

0

φs (a) dXs (a)

t ∈ [0, T ]

(11)

Let us denote by GT (x, Φ(a)) the set of terminal values of self-financed portfolios with respect to the a-extended assets family {V (a), X}, and with initial value x: GT (x, Φ (a)) :=

½ µ Z VT (a) x +

T

0

¶ ¾ φs (a) dXs (a) : φ (a) ∈ Φ (a) .

In the finite assets case (GLP, Proposition 3.2, p. 186-188) the artificial extension leaves invariant the investment opportunity set, and there are explicit expressions for the correspondences linking the investment opportunities sets. We briefly recall this result: let an n be a n-dimensional num´eraire and V (an ) = V 1,a its value process,

9

n

• to a self-financed portfolio (V n , ϑn ) with respect to {S 0 , S }, corresponds the selffinanced portfolio (V n , φn (an )) = (V n , (η n (an ), ϑn (an ))) w.r.t. {V (an ), S 0 , S} given by n V n − ϑnt S t ηtn (an ) = t and ϑnt (an ) = ϑnt , t ∈ [0, T ]; (12) St0 • to a self-financed portfolio (V n , φn (an )) = (V n , (η n (an ), ϑn (an ))) w.r.t. {V (an ), S 0 , S} n corresponds the self-financed portfolio (V n , ϑn ) w.r.t. {S 0 , S } given by ϑnt

=

ϑnt (an )

+

Vn ant t

n

− φnt (an )X t , Vt (an )

t ∈ [0, T ].

(13)

Note that the above expressions involve some scalar products between strategies and price processes, which in our infinite-dimensional setting are not well-defined. This makes very difficult to find the infinite-dimensional analogues of the GLP-correspondences above. Nonetheless, the following proposition states their existence for our large financial market and, as a straightforward consequence, the invariance of the investment opportunities set under a change of num´eraire. Proposition 12 Let a ∈ Θ be a num´eraire and V (a) its value process. 1. Let ϑ ∈ Θ be a trading strategy with respect to the primitive assets family {S 0 , S} and let V denote the value process of the corresponding self-financed portfolio. Then there exists a trading strategy φ(a) ∈ Φ(a) with respect to the a-extended assets family {V (a), X} with the same value process V . 2. Let φ(a) = (η(a), ϑ(a)) ∈ Φ(a) be a trading strategy with respect to the a-extended assets family {V (a), X} and let V denote the value process of the corresponding selffinanced portfolio. Then there exists a trading strategy ϑ ∈ Θ with respect to the primitive assets family {S 0 , S} with the same value process V . 3. We have then in particular that GT (x, Θ) = GT (x, Φ (a)) .

(14)

Proof. 1. Let ϑ ∈ Θ be a trading strategy with respect to the primitive assets family R 0 )). By Definition 1 there exists a sequence {S 0 , S} with value process V = S 0 (V0 + ϑd(S/S R R ϑn of simple integrands w.r.t. S/S 0 such that ϑn d(S/S 0 ) converges in S(P ) to ϑd(S/S 0 ). We associate to each approximating strategy ϑn a self-financed portfolio with respect to the primitive assets family, whose value process is given by ¶ µ Z t ¡ n 0¢ n 0 n Vt = St V0 + ϑs d S /S s , t ∈ [0, T ]. 0

10

By GLP, Proposition 3.2 (i), there exists a trading strategy φn (a) = (η n (a), ϑn (a)) given by (12) with V n (a) instead of V n (an ) and with the same value process V n , i.e. µ ¶ µ ¶ Z t Z t ¡ ¢ 0 n 0 n St V0 + ϑs d S/S s = Vt (a) V0 + φs (a)dXs (a) , t ∈ [0, T ]. 0

0

By the multidimensional version of Proposition 4 in Emery (1979), we have that µ ¶ µ ¶ Z Z ¡ ¢ ¡ ¢ S0 S0 n 0 0 V0 + ϑ d S/S → V0 + ϑd S/S V (a) V (a) R in S(P ), as n → ∞, and so the sequence φn (a)dX(a) is convergent in S(P ). Now, by the infinite-dimensional version of Memin’s theorem (Theorem 2) there exists a generalized R n R integrand φ(a) ∈ L(X(a), U) such that V0 + φ (a)dX(a) → V0 + φ(a)dX(a) in S(P ), as n → ∞, and obviously for all t ∈ [0, T ] µ ¶ µ ¶ Z t Z t ¡ ¢ 0 0 St V0 + ϑs d S/S s = Vt (a) V0 + φs (a) dXs (a) . 0

0

R Finally, by Proposition 11 and since ϑ ∈ Θ, the process φ(a)dX(a) is a local Q(a)RT martingale for all Q(a) ∈ Me2 (a), and also VT (a) 0 φs (a)dXs (a) ∈ L2 (P ), i.e. φ(a) ∈ Φ(a). 2. Let φ(a) ∈ Φ(a) be a trading strategy with respect to the a-extended assets family {V (a), X} with value process of the corresponding self-financed portfolio given by V (a)(V0 + R φ(a)dX(a)). By definition of Φ(a), there exists a sequence of simple integrands φn (a) = (η n (a), ϑn (a)), with η n (a) real-valued, converging pointwise to φ(a) and such that Z Z n φ (a)dX(a) → φ(a)dX(a) in S(P ) as n → ∞. Denote by V n the value process of R the approximating self-financed portfolio corresponding to φn (a), i.e. V n = V (a)(V0 + φns (a)dX(a)), and consider the following sequence of strategies ϑn with respect to {S 0 , S} defined by the GLP correspondence (13): ϑnt = ϑnt (a) + at ψtn (a), where

t ∈ [0, T ], n

ψn =

V n − φn (a)X . V (a)

We remark that the process ϑn takes values in U. Now, if we proceed as in the second part of the proof of Proposition 3.2 in GLP (observe that, by definition of generalized integrand, φn (a) is bounded, which implies ψ n (a) locally bounded), we obtain d(V n /S 0 )t = ψtn d(V (a)/S 0 )t + ϑnt (a)d(S/S 0 )t . 11

Being d(V (a)/S 0 )t = at d(S/S 0 )t with a ∈ L(S/S 0 , U), if we approximate a by Ra sequence R k k 0 a of simple integrands converging to a and such that a d(S/S ) → ad(S/S 0 ) R n pointwise k 0 in S(P ), also the sequence ψ (a)a d(S/S ) converges in S(P ) with n fixed and k tending to infinity and then, by Theorem 2, there exists a generalized integrand ζ n such that µ ¶ µ ¶ V (a) S n n ψt (a)d = ζt d , S0 t S0 t and moreover, since ψ n (a)ak converges pointwise to ψ n (a)a, ζ n = ψ n (a)a. Furthermore ¶ µ µ ¶ Z t Z t ϑns d(S/S 0 )s = Vt (a) V0 + φns (a)dXs (a) , t ∈ [0, T ]. St0 V0 + 0

0

Finally, by letting n tend to infinity and by using the same argument (infinite-dimensional version of Memin’s theorem) as in the previous part of the proof (after having inverted the rˆoles of ϑn and φn (a)), one can easily show that there exists a strategy ϑ ∈ Θ, whose value process equals V . The proof of the point 2 is now complete. 3. This statement follows trivially from the first two items of this proposition. ¤ We just mentioned that, since it is not possible in this setting to define a product between strategies and price processes, we are not able to find an explicit expression for the infinite-dimensional GLP correspondences. We only know that the two strategies sets are related by the equality of their value processes, i.e. given a strategy ϑ (resp. φ(a)) its corresponding strategy φ(a) (resp. ϑ) satisfies the following equation: µ ¶ µ ¶ Z t Z t 0 0 St V0 + ϑs d(S/S )s = Vt (a) V0 + φs (a)dXs (a) , t ∈ [0, T ]. 0

0

The previous proposition ensures the existence of a unique solution to this equation when ϑ (resp. φ(a)) is fixed. Nonetheless, we observe that if, given a trading strategy ϑ w.r.t. n {S 0 , S}, its approximating sequence ϑn is such that ϑn S converges pointwise to some process U , then the corresponding trading strategy φ(a) = (η(a), ϑ(a)) is given by ηt (a) =

V t − Ut St0

ϑt (a) = ϑt ,

t ∈ [0, T ].

Analogously, if, given a trading strategy φ(a) w.r.t. {V (a), S 0 , S}, its approximating sen quence φn (a) is such that φn (a)X converges pointwise to a process W , then the corresponding trading strategy ϑ is given by ϑt = ϑt (a) + at

Vt − Wt , Vt (a)

t ∈ [0, T ].

Remark 13 In Section 6, we will see that there exists a sequence of predictable trading strategies ϑn,∗ , that both solve the MVH-problem arisen by considering only the first n risky assets, and its value processes converge to the value process of ϑ∗ , solution to problem (1), in L2 (P ) as n tends to infinity. 12

5

The MVH problem

We would like to apply the artificial extension method introduced by GLP to the following “large” mean-variance hedging optimization problem: · µ ¶¸2 Z T ¡ ¢ J(x, F ) := min E F − ST0 x + ϑt d S/S 0 t . (H (x)) ϑ∈Θ

0

L2 (P )

where F ∈ and x ∈ R are fixed. In financial terms, given an FT -measurable contin2 gent claim F ∈ L (P ), we are looking for a self-financed portfolio with respect to the primitive assets family {S 0 , S}, with initial investment x, that minimizes the expected square of the hedging residual. By Proposition 8 there exists a unique solution ϑ∗ = ϑ∗ (x, H) to the problem (H (x)) called the optimal hedging strategy, with associated value process µ ¶ Z T ¡ ¢ ∗ ∗ 0 ∗ 0 Vt = Vt (x, F ) = ST x + ϑt (x, F ) d S/S t . 0

∗ , ϑ∗ )

The couple (V is called optimal hedging portfolio. Let us consider the following optimization problem: · µ ¶¸2 Z T ¡ ¢ 0 0 min E ST 1 + ϑt d S/S t ϑ∈Θ

(P)

0

which is a particular case of (H (x)) for a zero cash flow F = 0 and with initial wealth x = 1. Problem (P) has a solution e a ∈ Θ ,which leads to a unique terminal wealth RT 1,e a 0 0 VT (e a) = VT = ST (1 + 0 e at d(S/S )t ). Let us consider also the dual quadratic problem of (P): ¸ · 1 dQ 2 min E 0 . (D) Q∈M2 ST dP Under the standing assumption (7) and (8), the set ¾ ½ 1 dQ : Q ∈ M2 D2 : = ST0 dP is a non-empty closed convex set in L2 (P ), and therefore problem (D) admits a unique solution Pe ∈ M2 . It is now possible to extend all results of GLP, Section 4, to the infinite-dimensional case. We summarize them in the following: Theorem 14 Assume (7) and (8). 1. The variance-optimal martingale measure (abbr. VOMM) Pe solution to problem (D) is related to the terminal wealth VT (e a) corresponding to the solution e a of problem (P) by " #2 1 dPe 1 dPe =E VT (e a). (15) ST0 dP ST0 dP 13

2. Pe is equivalent to P ; that is , Pe ∈ Me2 . 3. The self-financed portfolio value process V (e a) is strictly positive: Vt (e a) > 0,

P a.s., t ∈ [0, T ] .

Proof. The proof is exactly as in the case of a finite number of risky assets, for which see GLP, Theorem 4.1, for point 1 and Theorem 4.2 for points 2 and 3. ¤ Since V (e a) is strictly positive, we can use it as a num´eraire that we call the hedging num´eraire. From (15), the VOMM is then related to the hedging num´eraire by VT (e a) ST0 dPe ¤ = £ dP E VT (e a) ST0

(16)

Following GLP, we will solve problem (H (x)) by transforming it into a simpler one corresponding to the martingale case thanks to the artificial extension method. Let us consider the hedging num´erare e a and the associated e a-extended assets family {V (e a), S 0 , S}. We can define the equivalent e a-martingale measure Pe(e a) given by the relation dPe (e a) VT (e a)2 = dP E [VT (e a)]2

(17)

and we call it the variance-optimal e a-martingale measure. Let us consider the quadratic optimization problem · ¸2 Z T F e a J (x, F ) = min EPe(ea) −x− φt (e a) dXt (e a) (Hea (x)) VT (e a) φ(e a)∈Φ(e a) 0 A straightforward extension of Proposition 5.1 in GLP is that problems (H (x)) and (Hea (x)) are equivalent in the following sense: if θ∗ and φ∗ (e a) are the unique solutions of, respectively, problem (H (x)) and problem (Hea (x)), then they have the same value process, i.e. µ ¶ µ ¶ Z t Z t 0 ∗ 0 ∗ St V0 + ϑs d(S/S )s = Vt (e a) V0 + φs (e a)dXs (e a) , t ∈ [0, T ]. (18) 0

0

Moreover, the relation (5.2) in GLP, between their minimal quadratic risks, is still verified, i.e. J (x, F ) = E [VT (e a)]2 J ea (x, F ) . (19) Now since Pe(e a) ∈ Me2 (e a), the continuous process X(e a) is a locally square integrable martingale under Pe(e a). Furthermore, being F square integrable under P , the claim F/VT (e a) is e square integrable under P (e a). The infinite-dimensional GKW-projection theorem (Proposition 6) implies that there exists a U-valued predictable process φF (e a) satisfying ·¿Z À ¸ EPe(ea) φF (e a) dX (e a) <∞ T

14

e a), orthogonal to X(e and a real-valued square integrable Pe(e a)-martingale R(e a) under Pe(e a), such that · ¸ Z T F F eT (e = EPe(ea) + φF (e a) dX (e a) + R a) . (20) VT (e a) VT (e a) 0 Clearly, the solution φ∗ (e a) to problem (Hea (x)) is given by the integrand in the decomposition (20), i.e. φ∗ (e a) = φF (e a), and the associated minimal quadratic risk of problem (Hea (x)) is given by · ¸ ¶2 µ h i2 F e a eT (e a) . − x + EPe(ea) R (21) J (x, F ) = EPe(ea) VT (e a) We now summarize how to “theoretically” solve our initial infinite-dimensional MVHproblem (H (x)): compute the hedging num´eraire e a and consider the MVH-problem (Hea (x)) corresponding to the price process X(e a), the strategies set Φ(e a) and the probability Pe(e a), which is a martingale measure for the new integrator; the GKW-projection theorem gives its unique solution φ∗ (e a). Now, in order to find the optimal strategy ϑ∗ , solve with respect to ϑ the following stochastic equation: ¶ ¶ µ µ Z t Z t ∗ 0 0 φs (e a)dXs (e a) , t ∈ [0, T ]. (22) ϑs d(S/S )s = Vt (e a) V0 + St V0 + 0

0

Observe that Proposition 12 ensures the existence of a unique solution for this equation. We conclude this section by considering the problem min J (x, F ) ,

R

x∈

(H)

which corresponds to the projection of F on the closed subspace {GT (x, Θ) : x ∈ R} of L2 (P ) (to see this use the same argument as in GLP, p. 195). The solution x∗ (F ) to problem (H) is called the approximation price for F (see Schweizer [29]) and it is obviously a generalization of the usual arbitrage-free price for F . From (19) and (21) one can easily deduce that the approximation price for F is given by · ¸ F x∗ (F ) = EPe(ea) VT (e a) and so, by Proposition 11,

· ∗

x (F ) = EPe

¸ F . ST0

(23)

This shows that, even in a large financial market, the VOMM can be interpreted as a viable price system corresponding to a mean-variance criterion, and also extends Theorem 5.2 of GLP.

15

6

Finite-dimensional MVH problems

For all n ≥ 1, we denote by Fn = {Ftn : t ∈ [0, T ]} the (completed) filtration generated by n the n-dimensional primitive assets family {S 0 , S }, F n = FTn , by P n the restriction on F n of the probability measure P and we set Mn2 = {Qn probability measure on F n : Qn ¿ P n , ¾ 1 dQn n 2 0 n ∈ L (P ) , S /S is a local Q -martingale ST0 dP and

n n n n Mn,e 2 = {Q ∈ M2 : Q ∼ P } . n

Assumption (8) ensures that, for all n ≥ 1, the set Mn,e 2 is not empty. Recall that S = 1 n (S , · · · , S ), n ≥ 1. In this section we consider for all n ≥ 1 the following n-dimensional mean-variance hedging (n-MVH) problem: · min E F −

ϑn ∈Θn

ST0

µ Z x+ 0

T

ϑnt d

¡ n 0¢ S /S t

¶¸2 ,

(Hn (x))

n

where we have denoted by Θn the set of all Rn -valued S /S 0 -integrable Fn -predictable RT n processes ϑn such that ST0 0 ϑnt d(S /S 0 )t ∈ L2 (F n , P ) and for all Qn ∈ Mn2 , the process R n n ϑ d(S /S 0 ) is a Qn -martingale, F ∈ L2 (P ) and x ∈ R being fixed. All the objects that we have introduced in the previous two sections have their ndimensional counterparts, their notations and financial interpretations will be self-evident. The aim of this section is to study the asymptotical behavior, as n → ∞, of the sequence n,∗ (ϑ )n≥1 , where ϑn,∗ is the unique solution to problem (Hn (x)). Let us consider the following finite-dimensional dual problem associated to the assets n family {S 0 , S }, n ≥ 1: · ¸ 1 dQn 2 min E 0 . (Dn ) Qn ∈Mn ST dP n 2 Under the standing assumptions (7) and (8), problem (Dn ) admits a unique solution Pen ∈ Mn2 , which we call n-dimensional variance-optimal martingale measure (abbr. n-VOMM). Now some other notations from Delbaen and Schachermayer [9]: for all n ≥ 1, we denote by K0n the subspace of L∞ (F n , P ) spanned by the stochastic integrals of the form ³¡ ¢ ¡ n ¢ ´ n fn = h0n S /S 0 τ2 − S /S 0 τ1 n

where τ1 ≤ τ2 are Fn -stopping times such that the stopped process (S /S 0 )τ2 is bounded and hn is a bounded Rn -valued Fτn1 -measurable function. cn the closure of K n in L2 (P ) and by K cn the closure of the span of K n We denote by K 0 0 0 2 n n c and the constants in L (P ), i.e. K = span(K0 , 1). 16

On the other hand we denote by K0 the subspace of L∞ (F n , P ) given by the union of c0 the closure of K0 in L2 (P ) and by K b the closure of all K0n , i.e. K0 := ∪n≥1 K0n , by K b = span(K0 , 1). Obviously a probability the span of K0 and the constants in L2 (P ), i.e. K n n measure Q on F (resp. a probability measure Q on F) is a local martingale measure for n S /S 0 (resp. for S/S 0 ) if and only if EQn [fn ] = 0 for every fn ∈ K0n (resp. EQ [f ] = 0 for every f ∈ K0 ). We recall the following characterizations of the VOMM Pe and the n-VOMM Pen (here we identify any measure Q with the linear functional EQ [·] and linear functionals on L2 (P ) with elements of L2 (P )): Lemma 15 (Lemma 2.1(c) [9]) Assume (7) and (8). b vanishing on K c0 and equaling 1 on the constant function 1. Pe is the unique element of K 1; cn vanishing on K cn and equaling 1 on 2. For all n ≥ 1, Pen is the unique element of K 0 the constant function 1. We have then the following Proposition 16 Assume (7) and (8). The sequence Pen converges in L2 (P ), as n → ∞, to the VOMM Pe solution to problem (D). Proof. It is an immediate application of the projection theorem for Hilbert spaces. ¤ Now, we consider the following n-MVH problem: · µ ¶¸2 Z T ¡ n 0¢ 0 n min E ST 1 + ϑt d S /S t n n ϑ ∈Θ

and we set GnT

(Pn )

0

½ µ Z 0 (x, Θ ) := ST x + n

0

T

ϑns d

¡ n 0¢ S /S s

¶

¾ n

n

:ϑ ∈Θ

.

We remark that for all n ≥ 1 ¡ ¢ GnT (x, Θn ) ⊆ Gn+1 x, Θn+1 ⊆ GT (x, Θ) . T It is well known that, under the assumptions (7) and (8), GnT (0, Θn ) is closed in L2 (P ) and so each problem (Pn ) has a unique solution e an ∈ Θn , which leads to a unique terminal RT n n 1,e an n n 0 wealth VT (e a ) = VT = ST (1 + 0 e at d(S /S 0 )t ). As in GLP and in the previous section, the n-VOMM Pen satisfies the following properties (see Theorems 4.1 and 4.2 in GLP): 1. the n-VOMM Pen is related to the terminal wealth VTn (e an ) corresponding to the solun tion e a of problem (Pn ) by " #2 1 dPen 1 dPen =E VTn (e an ) (24) ST0 dP n ST0 dP n 17

2. Pen ∈ Mn,e 2 3. Vtn (e an ) > 0,

P n a.s., t ∈ [0, T ].

We can so use the self-financed portfolio value process V n (e an ) as a num´eraire that we call n-dimensional hedging num´eraire. We define the n-dimensional variance-optimal e an -martingale measure by the relation VTn (e an )2 dPen (e an ) = £ ¤2 dP n En VTn (e an ) and consider the n-dimensional analogue of problem (Hea (x)): · min

φn (e an )∈Φn (e an ) n

EPen (ean )

F −x− n VT (e an )

Z 0

T

n φnt (e an ) dXt (e an )

¸2 n

(Hena (x)) n

n

where X(e an ) = X /V n (e an ) and Φn (e an ) is the set of all Rn+1 -valued X(e an ) -integrable preR n T dictable processes such that VTn (e an ) 0 φnt (e an )dXt (e an ) ∈ L2 (F n , P ) and for all Qn (e an ) ∈ R n Mn2 (e an ) (which has an obvious meaning), the process φn (e an )dX(e an ) is a local Qn (e an )martingale. n We recall that, for all n ≥ 1, the solution to problem (Hnea (x)) is given by the Rn+1 valued predictable integrand φn,∗ (e an ) satisfying the integrability condition À ¸ ·¿Z n n,∗ n n a ) <∞ EPen (ean ) φ (e a ) dX (e T

in the following GKW-decomposition · ¸ Z T F F n eTn (e = EPen (ean ) + φn,∗ (e an ) dX (e an ) + R an ) , n n n n VT (e a ) VT (e a ) 0

(25)

en (e where R an ) is a real-valued square integrable (Fn , Pen (e an ))-martingale, orthogonal to n n n n X(e an ) under Pe (e a ), and then the associated minimal quadratic risk of problem (Hena (x)) is given by ¸ ¶2 µ · h i2 F n n n e − x + E R (e a ) . (26) J ea (x, F ) = EPen (ean ) T Pen (e an ) VTn (e an ) Proposition 17 Let ϑn,∗ and φn,∗ be solutions to problems (respectively) (Hn (x)) and n (Hena (x)) for all n ≥ 1. Then we have the following assertions: RT ∗ n 0 2 ∗ 0 ϑn,∗ t d(S /S )t converges to 0 ϑt d(S/S )t in L (P ) as n → ∞, where ϑ is the solution to problem (H (x)); RT RT n a)dXt (e a) in L2 (P ) as n → ∞, 2. VTn (e an ) 0 φn,∗ an ) converges to VT (e a) 0 φ∗t (e an )dXt (e t (e where φ∗ is the solution to problem (Hea (x)). 1.

RT 0

18

RT n 0 Proof. 1. It suffices to note that for all n ≥ 1, ST0 0 ϑn,∗ t d(S /S )t is the orthogonal R T projection of ST0 0 ϑ∗t d(S/S 0 )t onto the subspace GnT (0, Θn ) closed in L2 (P ). 2. By Proposition 3.2 of GLP and Proposition 12 we have, respectively, that for all n≥1 µ ¶ µ ¶ Z T Z T ¢ n n,∗ ¡ n n,∗ 0 0 n n n n ST x + ϑt d S /S t = VT (e a ) x+ φt (e a ) dXt (e a ) 0

and ST0

0

µ Z x+ 0

T

ϑ∗t d

¡ ¢ S/S 0 t

¶

µ Z = VT (e a) x +

0

T

¶ φ∗t (e a) dXt (e a)

.

Then assertion 2. follows easily. ¤

Remark 18 Firstly, we point out that, with respect to the Emery topology, used to prove Proposition 12, the L2 (P )-convergence works only for the finite-dimensional optimal hedging strategies. Then, to establish the correspondence between the sets Θ and Φ(a) of all strategies, one has to deal with convergence in S(P ).

7

Conclusions

In this paper, we have generalized the artificial extension method, developed in a market with a finite number of risky assets by GLP, to a large financial market. The most delicate task was using the SI theory (by De Donno and Pratelli [7]) for a sequence of semimartingales in order to define the “good” set of trading strategies and to extend the invariance property (under the change of num´eraire) of the set of all attainable contingent claims. Indeed, since in the infinite-dimensional setting the strategies are allowed to take values in the space U of non necessarily bounded operators on E, it is not possible to multiply the strategies and the price process S. This observation has two consequences: firstly, we cannot define the value process in the usual way but directly as a stochastic integral (Definition 7); secondly, to pass from the solution of the artificial MVH problem to the original one, we have to solve equation (22), whose solution is known explicitly in the finite case, but it is not in this infinite-dimensional setting. We have also studied (Section 6) the asymptotic behavior of the solutions of the finitedimensional MVH problems and shown their convergence in L2 (P ) sense to the optimal strategy ϑ∗ (which suggests a practical method to approximate it). A further generalization of this approach could be to a market with a continuum of stochastic processes modelling the dynamics of forward rates, i.e. to a process taking values in the space C([0, T ]) of all continuous real functions defined on the time interval [0, T ]. Applications to more concrete models, e.g. interest rate or “large” stochastic volatility models, will be the subject of a future work.

19

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22