1 Completeness of a General Semimartingale Market under Constrained Trading

1

Completeness of a General Semimartingale Market under Constrained Trading

Tomasz R. Bielecki^∗, Monique Jeanblanc^†, and Marek Rutkowski

1 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

2 D´epartement de Math´ematiques, Universit´e d’ ´Evry Val d’Essonne, 91025 ´Evry Cedex, France

3 School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia and Faculty of Mathematics and

Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warszawa, Poland

1.1 Introduction

In this note, we provide a rather detailed and comprehensive study of the ba- sic properties of self-financing trading strategies in a general security market model driven by discontinuous semimartingales. Our main goal is to analyze the issue of replication of a generic contingent claim using a self-financing trading strategy that is additionally subject to an algebraic constraint, re- ferred to as the balance condition. Although such portfolios may seem to be artificial at the first glance, they appear in a natural way in the analysis of hedging strategies within the reduced-form approach to credit risk.

Let us mention in this regard that in a companion paper by Bielecki et al.

[1] we also include defaultable assets in our portfolio, and we show how to use constrained portfolios to derive replicating strategies for defaultable contin- gent claims (e.g., credit derivatives). The reader is also referred to Bielecki et al. [1], where the case of continuous semimartingale markets was studied, for some background information regarding the probabilistic and financial set-up, as well as the terminology used in this note. The main emphasis is put here on the relationship between completeness of a security market model with unconstrained trading and completeness of an associated model in which only trading strategies satisfying the balance condition are allowed.

∗The research of the first author was supported in part by NSF Grant 0202851 and by Moody’s Corporation grant 5-55411.

†The research of the second author was supported in part by Moody’s Corporation grant 5-55411.

1.2 Trading in Primary Assets

Let Y_t¹, Y_t², . . . , Y_t^k represent cash values at time t of k primary assets. We postulate that the pricesY¹, Y², . . . , Y^kfollow (possibly discontinuous) semi- martingales on some probability space (Ω,F,P), endowed with a filtrationF satisfying the usual conditions. Thus, for example, general L´evy processes, as well as jump-diffusions are covered by our analysis. Note that obviously F^Y ⊆ F, where F^Y is the filtration generated by the prices Y¹, Y², . . . , Y^k of primary assets. As it is usually done, we set X₀₋ =X₀ for any stochastic process X, and we only consider semimartingales with c`adl`ag sample paths.

We assume, in addition that at least one of the processesY¹, Y², . . . , Y^k, say Y¹, is strictly positive, so that it can be chosen as a numeraire asset. We consider trading within the time interval [0, T] for some finite horizon date T >0. We emphasize that we do not assume the existence of a risk-free asset (a savings account).

1.2.1 Unconstrained Trading Strategies

Letφ= (φ¹, φ², . . . , φ^k) be a trading strategy; in particular, each processφⁱ is predictable with respect to the reference filtration F. The component φⁱ_t represents the number of units of theith asset held in the portfolio at timet.

Then the wealth Vt(φ) at timet of the trading strategy φ= (φ¹, φ², . . . , φ^k) equals

Vt(φ) =

k

X

i=1

φⁱ_tY_tⁱ, ∀t∈[0, T], (1.1) andφis said to be aself-financing strategyif

V_t(φ) =V₀(φ) +

k

X

i=1

Z t 0

φⁱ_udY_uⁱ, ∀t∈[0, T]. (1.2) Let Φ be the class of all self-financing trading strategies. By combining the last two formulae, we obtain the following expression for the dynamics of the wealth process of a strategyφ∈Φ

dV_t(φ) = V_t(φ)−

k

X

i=2

φⁱ_tY_tⁱ

(Y_t¹)⁻¹dY_t¹+

k

X

i=2

φⁱ_tdY_tⁱ.

The representation above shows that the wealth process V(φ) depends only onk−1 components ofφ. Note also that, in our setting, the process

Vt(φ)− Pk

i=2φⁱ_tY_tⁱ

(Y_t¹)⁻¹ is predictable.

Remark 1.Let us note that Protter [4] assumes that the component of a strat- egyφthat corresponds to the savings account (which is a continuous process)

is merely optional. The interested reader is referred to Protter [4] for a thor- ough discussion of other issues related to the regularity of sample paths of processesφ¹, φ², . . . , φ^k andV(φ).

Choosing Y¹ as a numeraire asset, and denoting V_t¹(φ) = Vt(φ)(Y_t¹)⁻¹, Y_t^i,1 = Y_tⁱ(Y_t¹)⁻¹, we get the following well-known result showing that the self-financing feature of a trading strategy is invariant with respect to the choice of a numeraire asset.

Lemma 1.(i)For any φ∈Φ, we have V_t¹(φ) =V₀¹(φ) +

k

X

i=2

Z t 0

φⁱ_udY_u^i,1, ∀t∈[0, T]. (1.3) (ii)Conversely, letX be anFT-measurable random variable, and let us assume that there existsx∈RandF-predictable processesφⁱ, i= 2,3, . . . , ksuch that

X =Y_T¹

x+

k

X

i=2

Z T 0

φⁱ_tdY_t^i,1

.

Then there exists an F-predictable process φ¹ such that the strategy φ = (φ¹, φ², . . . , φ^k) is self-financing and replicates X. Moreover, the wealth pro- cess of φ satisfies Vt(φ) = V_t¹Y_t¹, where the process V¹ is given by formula (1.4)below.

Proof. The proof of part (i) is given, for instance, in Protter citeProtter. We shall thus only prove part (ii). Let us set

V_t¹=x+

k

X

i=2

Z t 0

φⁱ_udY_u^i,1, ∀t∈[0, T], (1.4) and let us define the processφ¹ as

φ¹_t=V_t¹−

k

X

i=2

φⁱ_tY_t^i,1= (Y_t¹)⁻¹

Vt−

k

X

i=2

φⁱ_tY_tⁱ

,

whereV_t=V_t¹Y_t¹. From (1.4), we havedV_t¹=Pk

i=2φⁱ_tdY_t^i,1, and thus dVt=d(V_t¹Y_t¹) =V_t−¹dY_t¹+Y_t−¹ dV_t¹+d[Y¹, V¹]t

=V_t−¹dY_t¹+

k

X

i=2

φⁱ_t Y_t−¹ dY_t^i,1+d[Y¹, Y^i,1]_t .

From the equality

dY_tⁱ=d(Y_t^i,1Y_t¹) =Y_t−^i,1dY_t¹+Y_t−¹ dY_t^i,1+d[Y¹, Y^i,1]_t,

it follows that

dVt=V_t−¹ dY_t¹+

k

X

i=2

φⁱ_t dY_tⁱ−Y_t−^i,1dY_t¹

= V_t−¹ −

k

X

i=2

φⁱ_tY_t−^i,1 dY_t¹+

k

X

i=2

φⁱ_tdY_tⁱ, and our aim is to prove that

dVt=

k

X

i=1

φⁱ_tdY_tⁱ.

The last equality holds if φ¹_t =V_t¹−

k

X

i=2

φⁱ_tY_t^i,1=V_t−¹ −

k

X

i=2

φⁱ_tY_t−^i,1, (1.5) i.e., if∆V_t¹=Pk

i=2φⁱ_t∆Y_t^i,1, which is the case from the definition (1.4) ofV¹. Note also that from the second equality in (1.5) it follows that the processφ¹ is indeedF-predictable. Finally, the wealth process ofφsatisfiesV_t(φ) =V_t¹Y_t¹

for everyt∈[0, T], and thusV_T(φ) =X. ut

1.2.2 Constrained Trading Strategies

In this section, we make an additional assumption that the price processY^kis strictly positive. Let φ= (φ¹, φ², . . . , φ^k) be a self-financing trading strategy satisfying the following constraint:

k

X

i=l+1

φⁱ_tY_t−ⁱ =Zt, ∀t∈[0, T], (1.6) for some 1 ≤ l ≤ k−1 and a predetermined, F-predictable process Z. In the financial interpretation, equality (1.6) means that the portfolio φshould be rebalanced in such a way that the total wealth invested in securities Y^l+1, Y^l+2, . . . , Y^k should match a predetermined stochastic process (for in- stance, we may assume that it is constant over time or follows a deterministic function of time). For this reason, the constraint (1.6) will be referred to as thebalance condition.

Our first goal is to extend part (i) in Lemma 1 to the case of constrained strategies. LetΦl(Z) stand for the class of all self-financing trading strategies satisfying the balance condition (1.6). They will be sometimes referred to as constrained strategies. Since any strategyφ∈Φl(Z) is self-financing, we have

∆Vt(φ) =

k

X

i=1

φⁱ_t∆Y_tⁱ=Vt(φ)−

k

X

i=1

φⁱ_tY_t−ⁱ ,

and thus we deduce from (1.6) that V_t−(φ) =

k

X

i=1

φⁱ_tY_t−ⁱ =

l

X

i=1

φⁱ_tY_t−ⁱ +Zt.

Let us writeY_t^i,1=Y_tⁱ(Y_t¹)⁻¹, Y_t^i,k=Yⁱ(Y_t^k)⁻¹, Z_t¹=Zt(Y_t¹)⁻¹.The follow- ing result extends Lemma 1.7 in Bielecki et al. [1] from the case of continuous semimartingales to the general case. It is apparent from Proposition 1 that the wealth processV(φ) of a strategyφ∈Φl(Z) depends only onk−2 components ofφ.

Proposition 1.The relative wealth V_t¹(φ) =V_t(φ)(Y_t¹)⁻¹ of a strategy φ∈ Φl(Z)satisfies

V_t¹(φ) =V₀¹(φ) +

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φⁱ_u

dY_u^i,1− Y_u−^i,1 Y_u−^k,1dY_u^k,1

+ +

Z t 0

Z_u¹

Y_u−^k,1dY_u^k,1. (1.7)

Proof. Let us consider discounted values of price processes Y¹, Y², . . . , Y^k, withY¹taken as a numeraire asset. By virtue of part (i) in Lemma 1, we thus have

V_t¹(φ) =V₀¹(φ) +

k

X

i=2

Z t 0

φⁱ_udY_u^i,1. (1.8) The balance condition (1.6) implies that

k

X

i=l+1

φⁱ_tY_t−^i,1=Z_t¹, and thus

φ^k_t = (Y_t−^k,1)⁻¹

Z_t¹−

k

X

i=l+1

φⁱ_tY_t−^i,1

. (1.9)

By inserting (1.9) into (1.8), we arrive at the desired formula (1.7). ut Let us takeZ = 0, so thatφ∈Φl(0). Then the balance condition becomes Pk

i=l+1φⁱ_tY_t−ⁱ = 0, and (1.7) reduces to dV_t¹(φ) =

l

X

i=2

Z t 0

φⁱ_tdY_t^i,1+

k−1

X

i=l+1

φⁱ_t

dY_t^i,1− Y_t−^i,1 Y_t−^k,1dY_t^k,1

. (1.10)

1.2.3 Case of Continuous Semimartingales

For the sake of notational simplicity, we denote byY^i,k,1the process given by the formula

Y_t^i,k,1= Z t

0

dY_u^i,1− Y_u−^i,1 Y_u−^k,1dY_u^k,1

(1.11) so that (1.7) becomes

V_t¹(φ) =V₀¹(φ) +

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φⁱ_udY_u^i,k,1+ +

Z t 0

Z_u¹

Y_u−^k,1dY_u^k,1. (1.12)

In Bielecki et al. [1], we postulated that the primary assets Y¹, Y², . . . , Y^k follow strictly positive continuous semimartingales, and we introduced the auxiliary processesYb_t^i,k,1=Y_t^i,ke^−αi,k,1t , where

α^i,k,1_t =hlnY^i,k,lnY^1,kit= Z t

0

(Y_u^i,k)⁻¹(Y_u^1,k)⁻¹dhY^i,k, Y^1,kiu. In Lemma 1.7 in Bielecki et al. [1] (see also Vaillant [5]), we have shown that, under continuity ofY¹, Y², . . . , Y^k, the discounted wealth of a self-financing trading strategy φ that satisfies the constraint Pk

i=l+1φⁱ_tY_tⁱ = Zt can be represented as follows:

V_t¹(φ) =V₀¹(φ) +

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φb^i,k,1_u dYb_u^i,k,1+ +

Z t 0

Z_u¹ Yu^k,1

dY_u^k,1, (1.13)

where we write φb^i,k,1_t =φⁱ_t(Y_t^1,k)⁻¹e^αi,k,1t . The following simple result recon- ciles expression (1.12) established in Proposition 1 with representation (1.13) derived in Bielecki et al. [1].

Lemma 2.Assume that the pricesY¹, Yⁱ andY^k follow strictly positive con- tinuous semimartingales. Then we have

Y_t^i,k,1= Z t

0

(Y_u^1,k)⁻¹e^αi,k,1u dbY_u^i,k,1 and

dY_t^i,k,1= (Y_t^1,k)⁻¹ dY_t^i,k−Y_t^i,kdα^i,k,1_t .

Proof. In the case of continuous semimartingales, formula (1.11) becomes Y_t^i,k,1=

Z t 0

dY_u^i,1− Y_u^i,1 Yu^k,1

dY_u^k,1

= Z t

0

dY_u^i,1−Y_u^i,kd(Y_u^1,k)⁻¹ . On the other hand, an application of Itˆo’s formula yields

dbY_t^i,k,1=e^−αi,k,1t dY_t^i,k−(Y_t^1,k)⁻¹dhY^i,k, Y^1,kit

and thus

(Y_t^1,k)⁻¹e^αi,k,1t dYb_u^i,k,1= (Y_t^1,k)⁻¹ dY_t^i,k−(Y_t^1,k)⁻¹dhY^i,k, Y^1,kit

. One checks easily that for any two continuous semimartingales, sayX andY, we have

Y_t⁻¹ dX_t−Y_t⁻¹dhX, Yi_t

=d(X_tY_t⁻¹)−X_tdY_t⁻¹,

provided that Y is strictly positive. To conclude the derivation of the first formula, it suffices to apply the last identity to processesX =Y^i,k andY = Y^1,k. For the second formula, note that

dY_t^i,k,1= (Y_t^1,k)⁻¹e^αi,k,1t dYb_t^i,k,1= (Y_t^1,k)⁻¹e^αi,k,1t d(Y_t^i,ke^−αi,k,1t )

= (Y_t^1,k)⁻¹ dY_t^i,k−Y_t^i,kdα^i,k,1_t ,

as required. ut

It is obvious that the processes Y^i,k,1 and Yb^i,k,1 are uniquely specified by the joint dynamics ofY¹, Yⁱ andY^k. The following result shows that the converse is also true.

Corollary 1.The price Y_tⁱ at timet is uniquely specified by the initial value Y₀ⁱ and either

(i)the joint dynamics of processes Y¹, Y^k andYb^i,k,1, or (ii)the joint dynamics of processes Y¹, Y^k andY^i,k,1. Proof. SinceYb_t^i,k,1=Y_t^i,ke^−αi,k,1t , we have

α_t^i,k,1=hlnY^i,k,lnY^1,ki_t=hlnYb^i,k,1,lnY^1,ki_t, and thus

Y_tⁱ =Y_t^kYb_t^i,k,1e^αi,k,1t =Y_t^kYb_t^i,k,1e^{hlnYbi,k,1,lnY1,kit}.

This completes the proof of part (i). For the second part, note that the process Y^i,1 satisfies

Y_t^i,1=Y₀^i,1+Y_t^i,k,1+ Z t

0

Y_u^i,1 Yu^k,1

dY_u^k,1. (1.14) It is well known that the SDE

Xt=X0+Ht+ Z t

0

XudYu,

whereH andY are continuous semimartingales (withH₀= 0) has the unique, strong solution given by the formula

X_t=E_t(Y)

X₀+ Z t

0

E_u⁻¹(Y)dH_u− Z t

0

E_u⁻¹(Y)dhY, Hi_u

.

Upon substitution, this proves (ii). ut

1.3 Replication with Constrained Strategies

The next result is essentially a converse to Proposition 1. Also, it extends part (ii) of Lemma 1 to the case of constrained trading strategies. As in Section 1.2.2, we assume that 1≤l≤k−1, andZ is a predetermined,F-predictable process.

Proposition 2.Let an FT-measurable random variable X represent a con- tingent claim that settles at time T. Assume that there exist F-predictable processes φⁱ, i= 2,3, . . . , k−1 such that

X =Y_T¹

x+

l

X

i=2

Z T 0

φⁱ_tdY_t^i,1+

k−1

X

i=l+1

Z T 0

φⁱ_tdY_t^i,k,1+ Z T

0

Z_t¹ Y_t−^k,1dY_t^k,1

. (1.15) Then there exist the F-predictable processes φ¹ and φ^k such that the strategy φ= (φ¹, φ², . . . , φ^k)belongs to Φl(Z)and replicates X. The wealth process of φequals, for everyt∈[0, T],

Vt(φ) =Y_t¹

x+

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φⁱ_udY_u^i,k,1+ Z t

0

Z_u¹ Y_u−^k,1dY_u^k,1

. (1.16) Proof. As expected, we first set (note thatφ^k isF-predictable)

φ^k_t = 1 Y_t−^k

Zt−

k−1

X

i=l+1

φⁱ_tY_t−ⁱ

(1.17)

and

V_t¹=x+

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φⁱ_udY_u^i,k,1+ Z t

0

Z_u¹

Y_u−^k,1dY_u^k,1. Arguing along the same lines as in the proof of Proposition 1, we obtain

V_t¹=V₀¹+

k

X

i=2

Z t 0

φⁱ_udY_u^i,1.

Now, we define

φ¹_t=V_t¹−

k

X

i=2

φⁱ_tY_t^i,1= (Y_t¹)⁻¹

Vt−

k

X

i=2

φⁱ_tY_tⁱ

,

whereVt=V_t¹Y_t¹.As in the proof of Lemma 1, we check that φ¹_t =V_t−¹ −

k

X

i=2

φⁱ_tY_t−^i,1,

and thus the process φ¹ is F-predictable. It is clear that the strategy φ = (φ¹, φ², . . . , φ^k) is self-financing and its wealth process satisfiesV_t(φ) =V_tfor every t ∈ [0, T]. In particular, V_T(φ) =X, so that φ replicates X. Finally, equality (1.17) implies (1.6), and thusφ∈Φl(Z). ut Note that equality (1.15) is a necessary (by Proposition 1) and sufficient (by Proposition 2) condition for the existence of a constrained strategy repli- cating a given contingent claimX.

1.3.1 Modified Balance Condition

It is tempting to replace the constraint (1.6) by a more convenient condition:

k

X

i=l+1

φⁱ_tY_tⁱ=Zt, ∀t∈[0, T], (1.18) whereZ is a predetermined,F-predictable process. If a self-financing trading strategyφsatisfies the modified balance condition (1.18) then for the relative wealth process we obtain (cf. (1.7))

V_t¹(φ) =V₀¹(φ) +

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φⁱ_u

dY_u^i,1− Y_u^i,1 Y_u−^k,1dY_u^k,1

+

+ Z t

0

Z_u¹

Y_u−^k,1dY_u^k,1. (1.19)

Note that in many cases the integrals above are meaningful, so that a counter- part of Proposition 1 with the modified balance condition can be formulated.

To get a counterpart of Proposition 2, we need to replace (1.15) by the equality X =Y_T¹ x+

l

X

i=2

Z T 0

φⁱ_tdY_t^i,1+

k−1

X

i=l+1

Z T 0

φⁱ_t

dY_t^i,1− Y_t^i,1 Y_t−^k,1dY_t^k,1

+

+ Z T

0

Z_t¹ Y_t−^k,1dY_t^k,1

!

, (1.20)

whereφ³, φ⁴, . . . , φ^k areF-predictable processes. We define V_t¹=x+

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φⁱ_u

dY_u^i,1− Y_u^i,1 Y_u−^k,1dY_u^k,1

+

+ Z t

0

Z_u¹

Y_u−^k,1dY_u^k,1, and we set

φ^k_t = 1 Y_t^k

Z_t−

k−1

X

i=l+1

φⁱ_tY_tⁱ

, φ¹_t=V_t¹−

k

X

i=2

φⁱ_tY_t^i,1.

Suppose, for the sake of argument, that the processes φ¹ and φ^k defined above are F-predictable. Then the trading strategy φ = (φ¹, φ², . . . , φ^k) is self-financing on [0, T], replicatesX, and satisfies the constraint (1.18). Note, however, that the predictability of φ¹ andφ^k is far from being obvious, and it is rather difficult to provide non-trivial and practically appealing sufficient conditions for this property.

1.3.2 Synthetic Assets

Let us fix i, and let us analyze the auxiliary processY^i,k,1 given by formula (1.11). We claim that this process can be interpreted as the relative wealth of a specific self-financing trading strategy associated with Y¹, Y², . . . , Y^k. Specifically, we will show that for any i = 2,3, . . . , k−1 the process ¯Y^i,k,1, given by the formula

Y¯_t^i,k,1=Y_t¹Y_t^i,k,1=Y_t¹ Z t

0

dY_u^i,1− Y_u−^i,1 Y_u−^k,1dY_u^k,1

,

represents the price of asynthetic asset.For brevity, we shall frequently write Y¯ⁱ instead of ¯Y^i,k,1.Note that the process ¯Yⁱ is not strictly positive (in fact, Y¯₀ⁱ= 0).

Equivalence of Primary and Synthetic Assets

Our goal is to show that trading in primary assets is formally equivalent to trading in synthetic assets. The first result shows that the process ¯Yⁱ can be obtained from primary assetsY¹, Yⁱ andY^k through a simple self-financing strategy. This justifies the name synthetic asset given to ¯Yⁱ.

Lemma 3.For any fixed i = 2,3, . . . , k−1, let an FT-measurable random variableY¯_Tⁱ be given as

Y¯_Tⁱ =Y_T¹Y_T^i,k,1=Y_T¹ Z T

0

dY_t^i,1− Y_t−^i,1 Y_t−^k,1dY_t^k,1

. (1.21)

Then there exists a strategyφ∈Φ₁(0)that replicates the claimY¯_Tⁱ. Moreover, we have, for every t∈[0, T],

Vt(φ) =Y_t¹Y_t^i,k,1=Y_t¹ Z t

0

dY_u^i,1− Y_u−^i,1 Y_u−^k,1dY_u^k,1

= ¯Y_tⁱ. (1.22) Proof. To establish the existence of a strategyφwith the desired properties, it suffices to apply Proposition 2. We fix iand we start by postulating that φⁱ = 1 andφ^j = 0 for any 2≤j ≤k−1, j 6=i. Then equality (1.21) yields (1.15) withX= ¯Y_Tⁱ, x= 0, l= 1 andZ= 0. Note that the balance condition becomes

k

X

j=2

φ^j_tY_t−^j =Y_t−ⁱ +φ^k_tY_t−^k = 0.

Let us defineφ¹ andφ^k by setting φ^k_t =−Y_t−ⁱ

Y_t−^k , φ¹_t =V_t¹−Y_t^i,1−φ^k_tY_t^k,1. Note that we also have

φ¹_t =V_t−¹ −Y_t−^i,1−φ^k_tY_t−^k,1=V_t−¹.

Hence,φ¹andφ^kareF-predictable processes, the strategyφ= (φ¹, φ², . . . , φ^k) is self-financing, and it satisfies (1.6) withl= 1 andZ= 0, so thatφ∈Φ₁(0).

Finally, equality (1.22) holds, and thusV_T(φ) = ¯Y_Tⁱ. ut Note that to replicate the claim ¯Y_Tⁱ = ¯Y_T^i,k,1, it suffices to invest in primary assets Y¹, Yⁱ and Y^k. Essentially, we start with zero initial endowment, we keep at any time one unit of theith asset, we rebalance the portfolio in such a way that the total wealth invested in theith andkth assets is always zero, and we put the residual wealth in the first asset. Hence, we deal here with a specific strategy such that the risk of the ith asset is perfectly offset by rebalancing the investment in the kth asset, and our trades are financed by taking positions in the first asset.

Note that the processY^i,1satisfies the following SDE (cf. (1.14)) Y_t^i,1=Y₀^i,1+ ¯Y_t^i,1+

Z t 0

Y_u−^i,1

Y_u−^k,1dY_u^k,1, (1.23) which is known to possess a unique strong solution. Hence, the relative price Y_t^i,1 at timet is uniquely determined by the initial valueY₀^i,1 and processes Y¯^i,1 andY^k,1. Consequently, the priceY_tⁱ at timet of the ith primary asset is uniquely determined by the initial valueY₀ⁱ, the pricesY¹, Y^k of primary assets, and the price ¯Yⁱof theith synthetic asset. We thus obtain the following result.

Lemma 4.Filtrations generated by the primary assetsY¹, Y², . . . , Y^k and by the price processesY¹, Y², . . . , Y^l,Y¯^l+1, . . . ,Y¯^k−1, Y^k coincide.

Lemma 4 suggests that for any choice of the underlying filtrationF(such that F^Y ⊆ F), trading in assets Y¹, Y², . . . , Y^k is essentially equivalent to trading inY¹, Y², . . . , Y^l,Y¯^l+1, . . . ,Y¯^k−1, Y^k. Let us first formally define the equivalence of market models.

Definition 1.We say that the two unconstrained models, M and Mf say, are equivalent with respect to a filtration F if both models are defined on a common probability space and every primary asset in M can be obtained by trading in primary assets in Mf and vice versa, under the assumption that trading strategies are F-predictable.

Note that we do not assume that modelsMandMfhave the same number of primary assets. The next result justifies our claim of equivalence of primary and synthetic assets.

Corollary 2.Models M = (Y¹, Y², . . . , Y^k;Φ) and M¯ = (Y¹, Y², . . . , Y^l, Y¯^l+1, . . . ,Y¯^k−1, Y^k;Φ)are equivalent with respect to any filtrationFsuch that F^Y ⊆F.

Proof. In view of Lemma 3, it suffices to show that the price process of each primary asset Yⁱ for i = l, l+ 1, . . . , k−1 can be mimicked by trading in Y¹,Y¯ⁱ andY^k. To see this, note that for any fixedi=l, l+ 1, . . . , k−1, we have (see the proof of Lemma 3)

Y¯_tⁱ=V_t(φ) =φ¹_tY_t¹+Y_tⁱ+φ^k_tY_t^k with

dY¯_tⁱ=dVt(φ) =φ¹_tdY_t¹+dY_tⁱ+φ^k_tdY_t^k. Consequently,

Y_tⁱ=−φ¹_tY_t¹+ ¯Y_tⁱ−φ^k_tY_t^k and

dY_tⁱ =−φ¹_tdY_t¹+dY¯_tⁱ−φ^k_tdY_t^k.

This shows that the strategy (−φ¹,1,−φ^k) inY¹,Y¯ⁱ andY^k is self-financing

and its wealth equalsYⁱ. ut

Replicating Strategies with Synthetic Assets

In view of Lemma 3, the replicating trading strategy for a contingent claim X, for which (1.15) holds, can be conveniently expressed in terms of primary securities Y¹, Y², . . . , Y^l andY^k, and synthetic assets ¯Y^l+1,Y¯^l+2, . . . ,Y¯^k−1. To this end, we represent (1.15)-(1.16) in the following way:

X=Y_T¹

x+

l

X

i=2

Z T 0

φⁱ_tdY_t^i,1+

k−1

X

i=l+1

Z T 0

φⁱ_tdY¯_t^i,1+ Z T

0

Z_t¹ Y_t−^k,1dY_t^k,1

(1.24) where ¯Y_t^i,1= ¯Y_tⁱ/Y_t¹=Y_t^i,k,1, and

Vt(φ) =Y_t¹

x+

l

X

i=2

Z t 0

φⁱ_udY_u^i,1+

k−1

X

i=l+1

Z t 0

φⁱ_udY¯_u^i,1+ Z t

0

Z_u¹ Y_u−^k,1dY_u^k,1

. (1.25) Corollary 3.Let X be an FT-measurable random variable such that (1.24) holds for some F-predictable process Z and some F-predictable processes φ², φ³, . . . , φ^k−1. Letψⁱ=φⁱ fori= 2,3, . . . , k−1,

ψ_t^k = Z_t¹ Y_t−^k,1 = Z_t

Y_t−^k , and

ψ¹_t =V_t¹−

l

X

i=2

ψ_tⁱY_t^i,1−

k−1

X

i=l+1

ψ_tⁱY¯_t^i,1−ψ_t^kY_t^k,1

=V_t−¹ −

l

X

i=2

ψⁱ_tY_t−^i,1−

k−1

X

i=l+1

ψⁱ_tY¯_t−^i,1−ψ^k_tY_t−^k,1.

Then ψ = (ψ¹, ψ², . . . , ψ^k) is a self-financing trading strategy in assets Y¹, . . . , Y^l,Y¯^l+1, . . . ,Y¯^k−1, Y^k. Moreover, ψsatisfies ψ_t^kY_t−^k =Zt, t∈[0, T], and it replicates X.

Proof. In view of (1.24), it suffices to apply Proposition 2 withl=k−1. ut

1.4 Model Completeness

We shall now examine the relationship between the arbitrage-free property and completeness of a market model in which trading is restricted a priori to self-financing strategies satisfying the balance condition.

1.4.1 Minimal Completeness of an Unconstrained Model

Let M= (Y¹, Y², . . . , Y^k;Φ) be an arbitrage-free market model. Unless ex- plicitly stated otherwise, Φ stands for the class of all F-predictable, self- financing strategies. Note, however, that the number of traded assets and their selection may be different for each particular model. Consequently, the dimension of a strategy φ ∈ Φ will depend on the number of traded assets in a given model. For the sake of brevity, this feature is not reflected in our notation.

Definition 2.We say that a model M is complete with respect to F if any boundedFT-measurable contingent claimX is attainable inM. Otherwise, a model Mis said to be incompletewith respect toF.

Definition 3.A modelM= (Y¹, Y², . . . , Y^k;Φ) isminimally completewith respect to Fif M is complete, and for anyi= 1,2, . . . , k the reduced model Mⁱ = (Y¹, Y², . . . , Yⁱ⁻¹, Yⁱ⁺¹, . . . , Y^k;Φ)is incomplete with respect to F, so that for eachithere exists a bounded,FT-measurable contingent claim, which is not attainable in the model Mⁱ. In this case, we say that the degree of completenessof Mequalsk.

Let us stress that trading strategies in the reduced model Mⁱ are pre- dictable with respect toF, rather than with respect to the filtration generated by price processesY¹, Y², . . . , Yⁱ⁻¹, Yⁱ⁺¹, . . . , Y^k. Hence, when we move from MtoMⁱ, we reduce the number of traded asset, but we preserve the original information structureF. Minimal completeness of a modelMmeans that all primary assets Y¹, Y², . . . , Y^k are needed if we wish to generate the class of all (bounded)FT-measurable claims throughF-predictable trading strategies.

The following lemma is thus an immediate consequence of Definition 3.

Lemma 5.Assume that a modelMis complete, but not minimally complete, with respect to F. Then there exists at least one primary asset Yⁱ, which is redundant inM, in the sense that it corresponds to the wealth process of some trading strategy in the reduced modelMⁱ.

Complete models that are not minimally complete do not seem to describe adequately the real-life features of financial markets (in fact, it is frequently argued that the real-life markets are not even complete). Also, from the theo- retical perspective, there is no advantage in keeping a redundant asset among primary securities. For this reasons, in what follows, we shall restrict our at- tention to market modelsMthat are either incomplete or minimally complete.

Lemma 6 shows that the degree of completeness is a well-defined notion, in the sense that it does not depend on the choice of traded assets, provided that the model completeness is preserved.

Lemma 6.Let a model M = (Y¹, Y², . . . , Y^k;Φ) be minimally complete with respect to F. Let Mf = (Ye¹,Ye², . . . ,Ye^k;Φ), where the processes Yeⁱ = V(φi), i = 1,2, . . . , k represent the wealth processes of some trading strate- gies φ1, φ2, . . . , φk ∈Φ. If a model Mfis complete with respect to Fthen it is also minimally complete with respect toF, and thus its degree of completeness equalsk.

Proof. The proof relies on simple algebraic considerations. By assumption, for every i= 1,2, . . . , k, we have

dYe_tⁱ=

k

X

j=1

φ^ij_t dY_t^j,

for some familyφ^ij, i, j= 1,2, . . . , kofF-predictable stochastic processes. By assumption, the marketMfis complete. To check that it is minimally complete, it suffices to show that the marketMf¹ = (Ye², . . . ,Ye^k;Φ) is incomplete (the same proof will work for any reduced model Mfⁱ). Suppose, on the contrary, thatMf¹is complete with respect toF. In particular, the price of each primary asset Y^l, l = 1,2, . . . , k can be replicated in Mf¹ by means of some trading strategy ψ^l = (ψ^l2, . . . , ψ^lk). In other words, there exists a family ψ^li, l = 1,2, . . . , k, i= 2,3, . . . , k ofF-predictable stochastic processes such that

dY_t^l=

k

X

i=2

ψ_t^lidYe_tⁱ. (1.26)

Since ψ¹, ψ², . . . , ψ^k are F-predictable processes with values in R^k−1, it is rather clear that there exists a family α¹, α², . . . , α^k−1 of F-predictable pro- cesses such that we have, for everyt∈[0, T],

ψ¹_t = (ψ¹²_t , ψ_t¹³, . . . , ψ_t^1k) =

k

X

j=2

α^j_t(ψ_t^j2, ψ^j3_t , . . . , ψ_t^jk) =

k

X

j=2

α^j_tψ_t^j.

Consequently, using (1.26), we obtain dY_t¹=

k

X

i=2

ψ_t¹ⁱdYe_tⁱ=

k

X

i=2 k

X

j=2

α^j_tψ^ji_t dYe_tⁱ=

k

X

j=2

α^j_t

k

X

i=2

ψ_t^jidYe_tⁱ=

k

X

j=2

α^j_tdY_t^j.

We conclude thatY¹is redundant in M, and thus the reduced modelM¹is complete. This contradicts the assumption thatMis minimally complete. ut By combining Lemma 6 with Corollary 2, we obtain the following result.

Corollary 4.A modelM= (Y¹, Y², . . . , Y^k;Φ)is minimally complete if and only if a model M¯ = (Y¹, Y², . . . , Y^l,Y¯^l+1, . . . ,Y¯^k−1, Y^k;Φ) has this prop- erty.

As one might easily guess, the degree of a model completeness depends on the relationship between the number of primary assets and the number of independent sources of randomness. In the two models examined in Sections 1.5.1 and 1.5.2 below, we shall deal withk= 4 primary assets, but the number of independent sources of randomness will equal two and three for the first and the second model, respectively.

1.4.2 Completeness of a Constrained Model

Let M= (Y¹, Y², . . . , Y^k;Φ) be an arbitrage-free market model, and let us denote byM_l(Z) = (Y¹, Y², . . . , Y^k;Φ_l(Z)) the associated model in which the

classΦis replaced by the classΦ_l(Z) of constrained strategies. We claim that if Mis arbitrage-free and minimally complete with respect to the filtration F=F^Y, where Y = (Y¹, Y², . . . , Y^k), then the constrained modelM_l(Z) is arbitrage-free, but it is incomplete with respect toF. Conversely, if the model M_l(Z) is arbitrage-free and complete with respect to F, then the original model M is not minimally complete. To prove these claims, we need some preliminary results.

The following definition extends the notion of equivalence of security mar- ket models to the case of constrained trading.

Definition 4.We say that the two constrained models are equivalent with respect to a filtrationFif they are defined on a common probability space and the class of all wealth processes ofF-predictable constrained trading strategies is the same in both models.

Corollary 5.The constrained model

Ml(Z) = (Y¹, Y², . . . , Y^k;Φ_l(Z)) is equivalent to the constrained model

M¯_k−1(Z) = (Y¹, Y², . . . , Y^l,Y¯^l+1, . . . ,Y¯^k−1, Y^k;Φ_k−1(Z)).

Proof. It suffices to make use of Corollaries 2 and 3. ut Note that the model ¯M_k−1(Z) is easier to handle thanMl(Z). For this reason, we shall state the next result for the modelMl(Z) (which is of our main interest), but we shall focus on the equivalent model ¯M_k−1(Z) in the proof.

Proposition 3.(i)Assume that the modelMis arbitrage-free and minimally complete. Then for anyF-predictable processZ and anyl= 1,2, . . . , k−1 the constrained modelMl(Z)is arbitrage-free and incomplete.

(ii)Assume that the constrained modelMl(Z)associated withMis arbitrage- free and complete. ThenMis either not arbitrage-free or not minimally com- plete.

Proof. The arbitrage-free property of Ml(Z) is an immediate consequence of Corollary 5 and the fact that Φ_k−1(Z) ⊂ Φ. In view of Corollary 4, it suffices to check that the minimal completeness of ¯Mimplies that ¯M_k−1(Z) is incomplete. By assumption, there exists a bounded,FT-measurable claimX that cannot be replicated in ¯M^k = (Y¹, Y², . . . , Y^l,Y¯^l+1, . . . ,Y¯^k−1;Φ) (i.e., when trading in Y^k is not allowed). Let us consider the following random variable

Y =X+ Z T

0

Zt

Y_t^kdY_t^k.

We claim that Y cannot be replicated in ¯Mk−1(Z). Indeed, for any trading strategyφ∈Φ_k−1(Z), we have