Equilibrium in thin security markets under restricted participation
aa r X i v : . [ q -f i n . E C ] A ug EQUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTEDPARTICIPATION
MICHAIL ANTHROPELOS AND CONSTANTINOS KARDARAS
This is an incomplete version of the paper. Please do not refer to any of the results until the finalised versionbecomes available.
Abstract.
We consider a market of financial securities with restricted participation, inwhich traders may not have access to the trade of all securities. The market is assumed thin:traders may influence the market and strategically trade against their price impacts. Weprove existence and uniqueness of the equilibrium even when traders are heterogeneous withrespect to their beliefs and risk tolerance. An efficient algorithm is provided to numericallyobtain the equilibrium prices and allocations given market’s inputs.
Introduction
Motivation.
Participation of traders in security markets is often limited or restricted, sinceseveral factors may prevent individual or institutional investors from accessing certain assets.This is the case, for example, for mutual and pension funds, which by law are not allowed tohold certain instruments (for instance, bonds with low credit rates, over-the-counter deriva-tives, securities in private placements, etc.), or hedge funds that choose not to trade in somesecurities in order to emphasise the speciality of their investment strategies. Furthermore,investors may avoid certain securities due to high transaction costs and margin requirements,or due to the difficulty in processing information related to their payoff; see, for example, therelated discussion in [CG10] and the references therein.Prompted by these aforementioned restricted participation considerations, several theo-retical studies have developed equilibrium models assuming that traders’ portfolio sets arerestricted—see [AC09], [PS97], [CG10] and [HHP06] for exogenously imposed restrictions,and [CGV09] and [CGES04] for endogenously arisen restrictions. These models assume acompetitive market structure: individual traders do not impact prices as part of the transac-tions, and are essentially considered as price-takers. However, several empirical studies (seefor instance [RW15] and the references therein) have shown that large institutional investorscover a large part of the market’s volume; therefore, their orders will influence the prices of
Date : August 20, 2019.
Key words and phrases. thin markets, restricted participation, limited participation, price impact, risksharing, Nash equilibrium. traded securities, and eventually allocations in the portfolios of all traders. Especially in mar-kets with restricted participation, where trading involves less participants, assuming that theinvestors (and in particular the large ones) have no price impact is not consistent with whatwe observe in practice. In this paper, we assume that investors have possible participationconstrains, and recognise the impact they have to equilibrium prices. The assumption that alltraders’ actions have price impact essentially implies that only large investors participate inthe transaction of each security, in an oligopolistic market structure. The latter is effectivelyconsistent with (at least) the primary level of security trading.
Contribution and connections with existing literature.
Our work is related to twostrands of literature on security equilibrium pricing. Firstly, we contribute to the ongoingresearch on thin financial markets, where all traders are assumed strategic; secondly, ourresults are linked to equilibrium models under restricted (or limited) traders’ participation.More precisely, we adapt the standard CARA-normal model and as in [RW15], [Viv11],[MR17], [Ant17] and study the equilibrium pricing and allocation of a bundle of securitieswhere the assumption of traders being price-takers is withdrawn. We suppose that traders’actions impact prices and, therefore, securities allocation choice, and that traders act strate-gically, through the demand schedules they submit in the transaction. The paper contributeson this front of the literature by considering a model where(1) traders are heterogeneous, not only concerning their risk tolerance, but also in theirbeliefs (on expectations and on covariance matrices); and(2) traders have restricted participation, in the sense that they do not necessarily haveaccess to the trade of all securities.In such a setting, we consider a demand-slope game (as in [RW12], [RW15] and [MR17])and prove the existence and the (global) uniqueness of Nash equilibrium for any traders’participation scheme. (The only additional imposed assumption is that at least three tradersparticipate in the trading of each security. The latter is necessary for equilibrium to exist; see,amongst others, [Kyl89] and [RW15].) Our main result, Theorem 1.4, also gives an iterativenumerical algorithm for fast numerical calculation of the equilibrium quantities. Theorem1.4 can be seen as a generalisation of [RW15, Lemma 1] and of [MR17, Proposition 1, items(i)-(iii)], in that traders are heterogeneous with respect to their risk aversions and their beliefs.In the centralised market model of [MR17], traders have different risk aversions but share thesame view on the covariance matrix of the security payoffs.Literature discussing models with limited participation (see, amongst others, [RZ09], [Zig04]and [Zig06]) distinguishes market participants to arbitrageurs and competitive investors. Ar-bitrageurs have access to all tradeable assets and act strategically in a Cournot-type of frame-work, while investors are assumed price-takers. In our paper, all traders act strategically, even
QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 3 if some of them have access to all securities and others do not. Our model is hence moreappropriate when large investors know that they can influence the market even if they arerestricted to trade only a subset of the securities.We further provide a simple example in § Equilibrium Price Impact with Restricted Participation
Traders, securities and notation.
In the market, we consider a finite number oftraders and use the index set I to denote them. There are a finite number of tradeable riskysecurities, and their index set is denoted by K . We model restricted market participationby assuming that trader i ∈ I has access to (effectively, is allowed to trade in) only a subset K i ⊆ K of the securities; in other words, trader i ∈ I may select units of securities in thesubspace of X ≡ R K defined via X i := { x ∈ X | x j = 0 , ∀ j ∈ K \ K i } , i ∈ I. Before giving more details of the model’s structure, we need to establish some necessarydefinitions and notation. For each i ∈ I , we shall denote by π i the projection operatorfrom X on the space X i ; for x ∈ X , π i x has the effect of keeping all coordinate entries of x corresponding to K i intact, while replacing all coordinate entries of x corresponding to K \ K i with zero.Define S (cid:23) as the set of all symmetric linear nonnegative-definite forms on X . On S (cid:23) , definethe partial order (cid:22) via A (cid:22) B ⇐⇒ ( B − A ) ∈ S (cid:23) . Furthermore, for a subspace Y of X , let S Y≻ consist of A ∈ S (cid:23) such that Ax = 0 for all x ∈ X orthogonal to Y , and which are strictly positive definite on Y : if y ∈ Y is such that h y, Ay i = 0, then y = 0. (Throughout the paper, h· , ·i denotes standard Euclidean innerproduct.) Note that B ∈ S X i ≻ can be regarded as a K × K matrix where only the elements B jℓ with ( j, ℓ ) ∈ K i × K i may be nonzero, and that π i B = B = Bπ i holds whenever B ∈ S X i ≻ . For A ∈ S Y≻ and B ∈ S Y≻ , we write A ≺ Y B to mean that ( B − A ) ∈ S Y≻ . We shall denote by B −X i the unique element of S X i ≻ which, on X i , coincides with the unique inverse of B .For k ∈ K , define I k := { i ∈ I | k ∈ K i } to be the set of traders that have access to tradingsecurity k . A minimal requirement for any meaningful equilibrium model is that | I k | ≥
2, forall k ∈ K . When we deal with price impact later on, we shall see that the stronger condition MICHAIL ANTHROPELOS AND CONSTANTINOS KARDARAS | I k | ≥
3, for all k ∈ K , is necessary and sufficient for existence (and uniqueness) equilibrium.(Note that the necessity of the latter assumption on linear Nash demand equilibria is well-known in the literature—see, for instance, [Kyl89] and [Viv11].)The next simple linear algebra result will be used throughout the paper, sometimes tacitly. Lemma 1.1.
Suppose that | I k | ≥ , for all k ∈ K . For fixed j ∈ I , if D i ∈ S X i ≻ for all i ∈ I \ { j } , then (cid:16)P i ∈ I \{ j } D i (cid:17) ∈ S X≻ .Proof. Set D − j := P i ∈ I \{ j } D i . Let z ∈ X , and assume h z, D − j z i = 0. Then, h z, D i z i = 0 forall i ∈ I \ { j } . Since D i ∈ S X i ≻ , we have z ℓ = 0 for all ℓ ∈ K i , whenever i ∈ I \ { j } . Therefore, z ℓ = 0 for all ℓ ∈ S i ∈ I \{ j } K i . But, S i ∈ I \{ j } K i = K , since we assume that | I k | ≥ k ∈ K . (cid:3) Preferences and demand.
Trader i ∈ I has preferences numerically represented viathe linear-quadratic functional(1.1) X i ∋ x U i ( x ) ≡ u i + h g i , x i − D x, B −X i i x E , where x represents units of securities held from the set K i , u i ∈ R is the baseline utility oftrader i ∈ I , B i ∈ S X i ≻ and g i ∈ X i . Remark . A special case of preferences numerically represented by the functional in (1.1),is when the latter coincides with the certainty equivalent of expected constant absolute riskaversion (CARA) utility, and payoffs have a joint Gaussian distribution—see, for instance,[Kyl89], [Vay99] and [MR17]. We elaborate more on this in the next paragraph, in order toenforce the point that, in our modelling framework, heterogeneity in multiple levels is allowed;more precisely, our model shall allow for: • heterogeneity in the traders’ risk aversions; • heterogeneous in the traders’ subjective beliefs regarding the expectations and covari-ance structure of the securities; and • traders’ initial endowments which may not be spanned by the securities.To the best of our knowledge, this is the first work in a setting with restricted participationwhich allows all of the above.Let S ≡ ( S k ; k ∈ K ) denote the vector of securities, and E i denote the random initialposition of agent i ∈ I . Assume that δ i > i ∈ I , and thevector ( E i , S ) has a joint Gaussian law under the agent’s subjective probability P i . Let C i bethe covariance matrix under P i of S , where only the components of K i are regarded, and theother entries are equal to zero. Assume that C i ∈ S X i ≻ . Furthermore, let c i ∈ X i be the vectorwhose entry k ∈ K i is the covariance under P i between E i and S k , and f i ∈ X i denote the QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 5 vector whose entry k ∈ K i is the expectation under P i of S k . Finally, let the baseline utilityof agent i ∈ I equal u i := − δ i log E i (cid:20) exp (cid:18) − E i δ i (cid:19)(cid:21) ∈ R Then, a position x ∈ X i leads to certainty equivalent equal to − δ i log E i (cid:20) exp (cid:18) − E i + h x, S i δ i (cid:19)(cid:21) = u i + h x, f i − (1 /δ i ) c i i − δ i h x, C i x i , which is exactly of the form (1.1), with g i := f i − δ i c i , B i := δ i C −X i i . Note that the above measures utility in monetary terms , B −X i captures jointly the trader’srisk tolerance level δ i and the covariance matrix of the securities.1.3. Price impact.
Under a competitive market setting, each trader i ∈ I is assumed to be aprice-taker; therefore, for any given vector of security prices p ∈ X , the aim is the maximisationof the utility U i ( q ) − h q, p i , over demand vectors q ∈ X i . However, and as emphasised in theintroductory section, there are several security markets where such a price-taking assumptionis problematic. Especially under a restricted participation environment as the one dealt withhere, the possibility that large investors may influence the market is more intense, and theneed arises to take into account the strategic behaviour of participating traders in the market.We consider and analyse the concept of Bayesian Nash market equilibrium in linear bidschedules, as has appeared in [RW15] and [MR17], among others. It is assumed that alltraders are strategic, and that no noise traders or traders without price impact are involvedin the transactions. We extend the one-round full participation game appearing in [RW15], inthat, in our setting, traders have heterogeneous preferences (both in risk aversions, as well ason expectations and covariances), and do not necessarily have access to all the securities. Onthe other hand, the decentralised market of [MR17] has richer structure on possible restrictionsfor trading, but all traders have the same subjective views on covariances of the securities.It is important to point out that, although traders may have access to only a subset of allsecurities, their actions will impact the equilibrium prices of all the securities.Below, we give the line of argument for the individual trader’s optimal allocation givena perceived price impact. We follow [Wer11] and [RW15], assuming that traders perceive alinear price impact of the orders they submit; more precisely, a net order of ∆ q ∈ X i for trader i ∈ I will move prices by Λ i ∆ q , where Λ i ∈ S ≻ is the so-called price impact (similar to Kyle’slambda [Kyl89]), and will be eventually endogenously determined in equilibrium. Let e p ∈ X be a vector of pre-transaction security prices. Under the previous linear price impact setting,an allocation q ∈ X i for trader i ∈ I will cost h q, p i = h q, e p + Λ i q i , where p = e p + Λ i q will be MICHAIL ANTHROPELOS AND CONSTANTINOS KARDARAS the actual transaction security prices. This means that the post-transaction utility of trader i ∈ I will equal U i ( q ) − h q, p i = u i + h q, g i − e p i − D q, B −X i i q E − h q, Λ i q i . Given Λ i , each trader i ∈ I wants to maximise the above utility by choosing demand vectors q from the subspace X i , as there is no demand for securities that trader has no access to.Therefore, with pre-transaction prices e p , the optimisation problem that trader i ∈ I faces is(1.2) q i = argmax q ∈X i (cid:18) h q, g i − e p i − D q, B −X i i q E − h q, Λ i q i (cid:19) . Since the above maximisation problem is strictly concave on X i , we may use first-order con-ditions for optimality, which give that g i − π i e p − B −X i i q i − π i Λ i q i = 0; in other words, g i − π i ( e p + Λ i q i ) = B −X i i q i + π i Λ i q i ⇐⇒ g i − π i p = ( B −X i i + π i Λ i π i ) q i , where the fact that π i q i = q i holds (since q i ∈ X i ) was used. Note that the above first orderconditions are consistent with [MR17, optimisation relation (5)], adjusted to our restrictedparticipation setting. Since (cid:0) B −X i i + π i Λ i π i (cid:1) ∈ S X i ≻ , upon defining(1.3) X i := (cid:16) B −X i i + π i Λ i π i (cid:17) − ∈ S X i ≻ , i ∈ I, and noting that X i π i = X i , we obtain that(1.4) q i = X i g i − X i p. To recapitulate: given a perceived linear price impact Λ i ∈ S ≻ , and with X i given by (1.3),the relationship between the optimal allocation q i ∈ X i of trader i ∈ I with actual transactionprices p ∈ X is given by (1.4). In view of (1.4), the matrix X i ∈ S X i ≻ of (1.3) has theinterpretation of a negative demand slope for trader i ∈ I .1.4. Equilibrium with restricted participation and price impact.
Given the abovebest response individual traders’ problem, we shall discuss now how price impact is formed inequilibrium.Assuming that each trader i ∈ I perceives linear price impact Λ i ∈ S ≻ , and given therelationship between the optimal allocation q i ∈ X i of trader i ∈ I with transaction prices p ∈ X given by (1.4) and X i ∈ S X i ≻ given by (1.3), the equilibrium prices b p that will clear themarket satisfy: 0 = X i ∈ I q i = X i ∈ I X i g i − X i ∈ I X i ! b p. Given that P i ∈ I X i ∈ S ≻ holds by Lemma 1.1, it follows that b p = X i ∈ I X i ! − X i ∈ I X i g i ! . QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 7
Given these equilibrium prices, the equilibrium allocation ( b q i ; i ∈ I ) will be given by substi-tuting the above expression b p for p in (1.4).Within equilibrium, each trader’s perceived market impact should coincide with their actualones; in this regard, see also [RW15, Lemma 1]. Assume that all traders, except trader i ∈ I ,have price impacts (Λ j ; j ∈ I \ { i } ), leading to ( X j ; j ∈ I \ { i } ) as in (1.3). If trader i ∈ I wishes to move allocation from b q i to (cid:0)b q i + ∆ q (cid:1) ∈ X i , the aggregate position of all other tradershas to change by − ∆ q , which would imply that new prices would equal b p + ∆ p , where, by(1.4), − ∆ q + X j ∈ I \{ i } b q j = X j ∈ I \{ i } X j g j − X j ∈ I \{ i } X j ( b p + ∆ p ) . Given that X j ∈ I \{ i } b q j = X j ∈ I \{ i } X j g j − X j ∈ I \{ i } X j b p, we obtain that ∆ p = X − − i ∆ q, where X − i := X j ∈ I \{ i } X j . It follows that Λ i = X − − i has to hold in equilibrium, for all i ∈ I . With the above understand-ing, and recalling (1.3), we give the following definition of equilibrium. Definition 1.3.
A collection ( X ∗ i ; i ∈ I ) ∈ ( S ≻ ) I will be called equilibrium negative demandslopes if(1.5) X ∗ i = (cid:16) B −X i i + π i ( X ∗− i ) − π i (cid:17) −X i , i ∈ I, where X ∗− i := P j ∈ I \{ i } X ∗ j , for all i ∈ I .Given equilibrium negative demand slopes ( X ∗ i ; i ∈ I ) as above, the equilibrium priceimpacts (Λ ∗ i ; i ∈ I ) ∈ ( S ≻ ) I are given by Λ ∗ i = ( X ∗− i ) − , i ∈ I .1.5. Main result.
Recall that we assume that, in order to have a meaningful equilibriumdiscussion, there are at least two traders for every security: | I k | ≥ k ∈ K . AsLemma 2.2 shows, if | I k | = 2 holds for some k ∈ K , then there exists no Nash equilibrium inthe sense of Definition 1.3. Therefore, the stronger condition | I k | ≥ k ∈ K is necessary for Nash equilibrium; the next result shows that this condition is also sufficient for existenceof Nash equilibrium, and that it is unique. Theorem 1.4.
Whenever | I k | ≥ holds for all k ∈ K , a unique equilibrium ( X ∗ i ; i ∈ I ) inthe sense of Definition 1.3 exists. Moreover, for any initial collection ( X i ; i ∈ I ) ∈ Q i ∈ I S X i ≻ ,if one defines inductively the updating sequence X ni := (cid:16) B −X i i + π i (cid:0) X n − − i (cid:1) − π i (cid:17) −X i , i ∈ I, n ∈ N , MICHAIL ANTHROPELOS AND CONSTANTINOS KARDARAS it holds that lim n →∞ X ni = X ∗ i , ∀ i ∈ I. Note that the above result not only guarantees the existence of a unique Nash equilibrium,but also provides an iterative algorithm to numerically calculate the equilibrium demandsand price impacts, where the only inputs are the participation’s restrictions and matrices( B i ; i ∈ I ).1.6. A limiting equilibrium.
We consider the case where a trader approaches risk neutrality,in the sense that the quadratic part of the trader’s utility is getting arbitrarily close to zero. Weshow below that, as the preferences of this trader approach risk-neutrality, the correspondingsequence of equilibria converges to a well-defined limit. The proof of Proposition 1.5 thatfollows is given in § Proposition 1.5.
Let I = { , . . . , m } , where m ≥ . Consider a fixed ( B i ; i ∈ I \ { } ) , aswell as a nondecreasing sequence ( B n ; n ∈ N ) with the property that lim n →∞ ( B n ) −X = 0 . If ( X n ; n ∈ N ) stands for the sequence of equilibria corresponding to ( B n ; n ∈ N ) , then ( X n ; n ∈ N ) monotonically converges to a limit X ∞ ∈ Q i ∈ I S X i ≻ . Furthermore, ( X ∞ i ; i ∈ I ) solves the system X ∞ i = (cid:16) B −X i i + π i (cid:0) X ∞− i (cid:1) − π i (cid:17) −X i , i ∈ I \ { } ,X ∞ = (cid:16) π (cid:0) X ∞− (cid:1) − π (cid:17) −X . In the context of Proposition 1.5, assume further that K = K , i.e., that trader 0 has accessto the whole market. We could interpret this trader as an asymptotically risk-neutral marketmaker that gives prices to all the securities in the market. Assuming that market maker’spreferences are (close to) risk neutral is common in the literature—see, e.g., [Kyl85], [FJ02],[BGS05] and the references therein. In this case, we obtain X ∞ = (cid:16)(cid:0) X ∞− (cid:1) − (cid:17) − = X ∞− . But then X ∞− i = 2 X ∞− − X ∞ i holds for all i ∈ I \{ } , by which we obtain that ( X ∞ i ; i ∈ I \{ } )solves the system ( X ∞ i ) −X i = B −X i i + π i (cid:0) X ∞− − X ∞ i (cid:1) − π i , i ∈ I \ { } . QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 9
An example on social inefficiency.
For the purposes of § K = { , } , I = { , , , } . We use directly the notation andset-up of Remark 1.2, and assume that δ i = 1 holds for all i ∈ I . Traders in I − = { , , } will be identical, and such that g i = 0 for i ∈ I − , and C i = C for i ∈ I − , where C = ! . The covariance matrix of trader 0 is given by C = ρρ ! . Lastly, set g = ( γ , γ ).1.7.1. Restricted participation.
Here, K i = K for i ∈ I − , while K = { } . Write ( X ri ; i ∈ I ) for the solution to the system of equations, where the superscript “ r ” denotes restricted participation. By symmetry, X r = X ri holds for all i ∈ I − . Noting that X r ( i, j ) = 0 whenever( i, j ) = (1 , /X r (1 ,
1) = 1 + (1 / /X r (1 , ,X r = (cid:0) C + ( X + 2 X r ) − (cid:1) − . In fact, one may directly check that these equations have the solution X r = / ! , X r = / / ! . Therefore, prices at restricted participation equilibrium are given by p r = ( X r +3 X r ) − X r g =( γ / , q r = − X r p r = − ( γ / ,
0) for agents in I − , and a position q r = − g = ( γ / ,
0) for agent 0. It follows that the aggregate utility differential at restrictedparticipation equilibrium will equal X i ∈ I (cid:18) h q ri , g i i − δ ri h q ri , C i q ri i (cid:19) = h q r , g i − h q r , C q r i − h q r , q r i = γ . Full participation.
Here, K i = K for all traders i ∈ I . Write ( X fi ; i ∈ I ) for the solutionto the system of equations, where the superscript “ f ” denotes full participation. By symmetry, X f = X fi for all i ∈ I − , and we have the equations X f = (cid:16) C + (3 X f ) − (cid:17) − X f = (cid:16) C + ( X f + 2 X f ) − (cid:17) − Substituting the first to the second, we obtain( X f ) − = C + (( C + (3 X f ) − ) − + 2 X f ) −
10 MICHAIL ANTHROPELOS AND CONSTANTINOS KARDARAS
If we can solve the latter one, we then also have the value of X f . We have C = V ρ
00 1 − ρ ! V, where V = 1 √ − ! . Note that V is unitary and symmetric V ( V = id ), and therefore trivially diagonalises theidentity matrix C . It then follows that V X f V will be also diagonal. To east notation below,define the matrix D ρ := / (1 + ρ ) 00 1 / (1 − ρ ) ! , so that C − = V D ρ V , and for functions h : (0 , ∞ ) → R write h ( D ρ ) for the 2 × h (1 / (1 + ρ )) , h (1 / (1 − ρ ))). Then, one solves the above matrixequation for X f to obtain X f = V h ( D ρ ) V, where h ( x ) = 14 − x + s(cid:18) − x (cid:19) + 23 x, x ∈ (0 , ∞ ) . It then also follows from X f = (cid:16) C + (3 X f ) − (cid:17) − that X f = V h ( D ρ ) V, where h ( x ) = ( x − + (3 h ( x )) − ) − = 3 xh ( x ) x + 3 h ( x ) , x ∈ (0 , ∞ ) . Prices at full participation equilibrium are given by p f = ( X f + 3 X f ) − X f g = V ( h ( D ρ ) + 3 h ( D ρ )) − h ( D ρ ) V g , which gives, for agents in I − , a position of q fi = q f , where q f = − X f p f = − V h ( D ρ )( h ( D ρ ) + 3 h ( D ρ )) − h ( D ρ ) V g = V η ( D ρ ) V g , where η ( x ) = − h ( x )( h ( x ) + 3 h ( x )) − h ( x ) = − h ( x )1 + 3 h ( x ) /h ( x ) = − xh ( x )2 x + 3 h ( x ) . The position of agent 0 is q f = − q f = − V η ( D ρ ) V g .The aggregate utility differential at Nash equilibrium in full participation will equal X i ∈ I (cid:18)D q fi , g i E − δ i D q fi , C i q fi E(cid:19) = D q f , g E − D q f , C q f E − D q f , q f E = g ′ V κ ( D ρ ) V g , where κ ( x ) = − η ( x ) − η ( x ) x − η ( x ) , x ∈ (0 , ∞ ) . QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 11
Comparison.
It follows that the difference between utilities in full and restricted par-ticipation equilibrium equals g ′ V κ ( D ρ ) V g − g ′ ! g = g ′ V κ ( D ρ ) − !! V g . To see whether this may become negative, we calculate the smallest eigenvalue of κ ( D ρ ) − ! = κ ((1 + ρ ) − ) − / − / − / κ ((1 − ρ ) − ) − / ! , which equals κ ((1 + ρ ) − ) + κ ((1 − ρ ) − ) − / p ( κ ((1 + ρ ) − ) − κ ((1 − ρ ) − )) + 1 /
92A plot shows that this is (symmetric in ρ , of course, and) negative for values of ρ close to zero.It follows that when ρ is not equal to zero, it may be not be socially optimal to move fromrestricted participation to full participation when price impact is involved.2. Proofs
The proof of Theorem 1.4 will be given in a series of subsections, starting with § § § The fixed point equation.
Let F : Q i ∈ I S X i ≻ Q i ∈ I S X i ≻ be defined via(2.1) F i ( X ) = (cid:16) B −X i i + π i ( X − i ) − π i (cid:17) −X i , i ∈ I. According to Definition 1.3, the equilibrium negative demand slopes are given as the fixedpoints of F , i.e., solutions to the equation(2.2) X = F ( X ) ⇐⇒ X i = (cid:16) B −X i i + π i ( X − i ) − π i (cid:17) −X i , i ∈ I. The following lemma provides upper bounds for the functional F . Lemma 2.1.
For each i ∈ I , it holds that F i ( X ) ≺ X i B i , as well as F i ( X ) ≺ X i π i X − i π i (cid:22) X − i . Proof.
For the first part of the lemma, we readily have that B −X i i ≺ X i B −X i i + π i ( X − i ) − π i ,which implies the order F i ( X ) = (cid:16) B −X i i + π i ( X − i ) − π i (cid:17) −X i ≺ X i B i , ∀ i ∈ I. For the second order, we first show that(2.3) ( π i X − i π i ) −X i (cid:22) π i ( X − i ) − π i holds for each i ∈ I . Indeed, upon rearranging the columns and rows of X − i bringing thesubmatrix corresponding to K i on the left top, write X − i and X − − i in block format as X − i = A CC ′ B ! , X − − i = D FF ′ E ! where A and D are ( K i × K i )-dimensional. Since A CC ′ B ! D FF ′ E ! = AD + CF ′ AF + CEC ′ D + BF ′ C ′ F + BE ! , the fact that X − i X − − i is the identity matrix gives: AF + CE = 0 ⇒ F = − A − CED = A − − A − CF ′ = A − + A − CEC ′ A − (cid:23) A − . We then get (2.3), since( π i X − i π i ) −X i = A −
00 0 ! (cid:22) D
00 0 ! = π i ( X − i ) − π i . Then, it follows from (2.3) that ( π i X − i π i ) −X i ≺ B −X i + π i ( X − i ) − π i . The latter gives that F i ( X ) = (cid:16) B −X i i + π i ( X − i ) − π i (cid:17) −X i ≺ X i π i X − i π i . (cid:3) We can already see that there is no hope for equilibrium in the case where there exists atleast one asset that can be traded by at most two traders. This result is consistent with thecorresponding no-equilibrium result in two-trader markets (see for instance [Kyl89], [Vay99]and [Viv11]).
Lemma 2.2. If | I k | = 2 holds for some k ∈ K , there exists no Nash equilibrium.Proof. Suppose that X ∗ is Nash equilibrium, so that X ∗ = F ( X ∗ ), and that I k = { i, j } holdsfor some k ∈ K and i, j ∈ I with i = j . If e k ∈ R K stands for the zero vector with entry 1only in the k th coordinate, then (since π i e k = e k = π j e k ), we get from Lemma 2.1 that X ∗ i ( k, k ) = h e k , X ∗ i e k i < (cid:10) e k , X ∗− i e k (cid:11) = X ∗− i ( k, k ) = X ∗ j ( k, k ) . A symmetric argument shows that X ∗ j ( k, k ) < X ∗ i ( k, k ), which leads to contradiction. (cid:3) QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 13
Existence of fixed points.
Taking into account Lemma 2.2, we assume hereafter that | I k | ≥ k ∈ K . Under that assumption and based on the characterisation of theequilibrium negative demand slopes through (2.2), we first show that there always exists suchan equilibrium. The next step toward this goal is to show that functional F defined in (2.1)is nondecreasing. For this, we need to extend the order (cid:22) on Q i ∈ I S X i ≻ , by defining the order X ≡ ( X i ; i ∈ I ) (cid:22) ( Y i ; i ∈ I ) ≡ Y ⇐⇒ X i (cid:22) Y i , ∀ i ∈ I. Since X (cid:22) Y implies X − i (cid:22) Y − i , for all i ∈ I , i.e., π i ( Y − i ) − π i (cid:22) π i ( X − i ) − π i , it follows that F i ( X ) = (cid:16) B −X i i + π i ( X − i ) − π i (cid:17) −X i (cid:22) (cid:16) B −X i i + π i ( Y − i ) − π i (cid:17) −X i = F i ( Y ) , for all i ∈ I . Therefore,(2.4) X ≡ ( X i ; i ∈ I ) (cid:22) ( Y i ; i ∈ I ) ≡ Y = ⇒ F ( X ) (cid:22) F ( Y ) , which means that F is nondecreasing. Furthermore, Lemma 2.1 gives(2.5) F ( X ) (cid:22) B , ∀ X ≡ ( X i ; i ∈ I ) ∈ Y i ∈ I S X i ≻ , where B ≡ ( B i ; i ∈ I ).Define now the two sets:(2.6) L := ( X ∈ Y i ∈ I S X i ≻ | X (cid:22) F ( X ) ) , U := ( X ∈ Y i ∈ I S X i ≻ | F ( X ) (cid:22) X ) and note that L ∩ U coincides with the set of fixed points of F . From (2.5), we obtain that t B ∈ U , for all t ∈ [1 , ∞ ). The next result is complementary. Lemma 2.3.
There exists Z ∈ Q i ∈ I S X i ≻ with Z (cid:22) B and with the property that rZ i ≺ X i F i ( r Z ) for all i ∈ I and r ∈ (0 , ; in particular, r Z ∈ L , for all r ∈ (0 , .Proof. Pick α > απ i (cid:22) B i , ∀ i ∈ I. Let Z i = απ i , for all i ∈ I , and note that Z (cid:22) (1 / B (cid:22) B . The fact that | I k \ { i } | ≥ k ∈ K implies Z − i = α X i ∈ I \{ i } π j (cid:23) α id X . Fix r ∈ (0 , B −X i i (cid:22) (4 α ) − π i , we have B −X i i + π i ( rZ − i ) − π i (cid:22) α (cid:18)
12 + 1 r (cid:19) π i = 2 + r rα π i . for all i ∈ I . Therefore, we obtain that F i ( r Z ) = (cid:16) B −X i i + π i ( rZ − i ) − π i (cid:17) −X i (cid:23)
22 + r rαπ i ≻ X i rZ i for all i ∈ I , which in particular shows that r Z (cid:22) F ( r Z ), i.e., r Z ∈ L . (cid:3) Lemma 2.4.
Suppose that X ∈ U and Y ∈ L are such that Y (cid:22) X , and form sequences ( X n ; n ∈ N ) and ( Y n ; n ∈ N ) via X n := F ◦ n ( X ) and Y n := F ◦ n ( Y ) for n ∈ N . Then: (1) The sequence ( X n ; n ∈ N ) is nondecreasing, the sequence ( Y n ; n ∈ N ) is nonincreas-ing, and Y n (cid:22) X n holds for all n ∈ N ; in particular, the limits X ∗ := lim n →∞ X n ∈ Q i ∈ I S X i ≻ and Y ∗ := lim n →∞ Y n ∈ Q i ∈ I S X i ≻ exist, and satisfy Y (cid:22) Y ∗ (cid:22) X ∗ (cid:22) X . (2) It holds that X ∗ = F ( X ∗ ) and Y ∗ = F ( Y ∗ ) , i.e., X ∗ and Y ∗ are equilibrium priceimpacts. (3) Whenever Z ∈ Q i ∈ I S X i ≻ is such that Z = F ( Z ) and Y (cid:22) Z (cid:22) X , then Y ∗ (cid:22) Z (cid:22) X ∗ Proof.
Applying F iteratively to the inequality Y (cid:22) X , and using the facts that X ∈ U and Y ∈ L and the monotonicity property (2.4), the claims of statement (1) immediately follow.Since the monotone limits X ∗ := lim n →∞ X n and Y ∗ := lim n →∞ Y n exist and are Q i ∈ I S X i ≻ -valued, continuity of F gives X ∗ = lim n →∞ X n = lim n →∞ F ( X n − ) = F ( lim n →∞ X n − ) = F ( X ∗ ) , and similarly that Y ∗ = F ( Y ∗ ), which is statement (2).Finally, take Z ∈ L ∩ U with Y (cid:22) Z (cid:22) X . Using the results and notation of statements(1) and (2) with ( Y , Z ) replacing ( Y , X ), we obtain Y ∗ (cid:22) Z ∗ = Z . Similarly, with ( Z , X )replacing ( Y , X ), we obtain Z = Z ∗ (cid:22) X ∗ . (cid:3) The next result shows in particular that a globally maximal solution to (2.2) exists.
Lemma 2.5.
Let X = B . Form a sequence ( X n ; n ∈ N ) via X n = F ◦ n ( X ) , for all n ∈ N .Then, the limit X ∗ := lim n →∞ X n ∈ Q i ∈ I S X i ≻ exists, it holds that X ∗ = F ( X ∗ ) , and for any Y ∈ L it holds that Y (cid:22) X ∗ .Proof. Recall that X = B ∈ U and that there exist Z ∈ L with Z (cid:22) X by Lemma 2.3.Therefore, Lemma 2.4 gives that X ∗ exists and F ( X ∗ ) = X ∗ . If Y ∈ L , then Y (cid:22) F ( Y ) (cid:22) B = X , and Lemma 2.4 gives that Y (cid:22) X ∗ . (cid:3) Uniqueness of the fixed point.
Lemma 2.5 in fact shows that a maximal solution X ∗ of (2.2) exists; combined with Lemma 2.4, it follow that, whenever Y ∈ L , there exists a minimal solution of (2.2) that is dominated below by Y and above by X ∗ . By Lemma 2.3,there exists Z ∈ Q i ∈ I S X i ≻ such that r Z ∈ L , for all r ∈ (0 , r Z coincides with X ∗ for all r ∈ (0 , r ∈ (0 , X ∗ = ( X ∗ i ; i ∈ I ) to (2.2) of Lemma 2.5,we take Z ∈ Q i ∈ I S X i ≻ as in Lemma 2.3, fix r ∈ (0 , X ≡ ( X i ; i ∈ I ) be the minimalfixed point that is bounded below by rZ . We shall show that, necessarily, X = X ∗ . QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 15
Define H := X ∗ − X ; from statement (3) of Lemma 2.5, H i := ( X ∗ i − X i ) ∈ S (cid:23) , for all i ∈ I .Furthermore, pick small enough ǫ > r Z (cid:22) X − ǫ H ,which is possible by statement(3) of Lemma 2.5, which implies that rZ i ≺ X i F i ( r Z ) (cid:22) X i holds for all i ∈ I . Consider themapping [ − ǫ, ∋ t X ( t ) := ( X + t H ) ∈ Y i ∈ I S X i ≻ , and note that X (0) = X and X (1) = X ∗ .It follows directly from definition of functional F , that the mapping( − ǫ, ∋ t Φ( t ) ≡ F ( X ( t )) ∈ Y i ∈ I S X i ≻ is twice continuously differentiable. The next result shows that it is, in fact, “ concave ”. Lemma 2.6.
With Φ( t ) := F ( X ( t )) for t ∈ ( − ǫ, , it holds that − ∂ Φ( t ) ∂t ∈ Y i ∈ I S (cid:23) , ∀ t ∈ ( − ǫ, . It follows that the mapping ( − ǫ, ∋ t
7→ − ∂ Φ( t ) /∂t is nondecreasing in the order of ( S (cid:23) ) I .Proof. For all t ∈ ( − ǫ,
1) and i ∈ I , it holds that ∂∂t X − i ( t ) − = − X − i ( t ) − (cid:18) ∂∂t X − i ( t ) (cid:19) X − i ( t ) − = − X − i ( t ) − H − i X − i ( t ) − . Since Φ i ( t ) = (cid:0) ( B i ) −X i + π i X − i ( t ) − π i (cid:1) −X i ∈ S X i ≻ holds for t ∈ ( − ǫ,
1) and i ∈ I , it followsthat ∂ Φ i ( t ) ∂t = − Φ i ( t ) (cid:18) ∂∂t (cid:0) Φ i ( t ) −X i (cid:1)(cid:19) Φ i ( t )= − Φ i ( t ) (cid:18) π i ∂∂t X − i ( t ) − π i (cid:19) Φ i ( t )= Φ i ( t ) π i X − i ( t ) − H − i X − i ( t ) − π i Φ i ( t )= Φ i ( t ) X − i ( t ) − H − i X − i ( t ) − Φ i ( t ) , where the last line follows because Φ i ( t ) π i = Φ i ( t ) = π i Φ i ( t ). Call t ∋ ( − ǫ, D i ( t ) := X − i ( t ) − H − i X − i ( t ) − ∈ S (cid:23) , ∀ i ∈ I. Since X − i ( t ) D i ( t ) X − i ( t ) = H − i is a constant matrix as a function of t ∈ ( − ǫ, (cid:18) ∂∂t X − i ( t ) (cid:19) D i ( t ) X − i ( t ) + X − i ( t ) (cid:18) ∂∂t D i ( t ) (cid:19) X − i ( t ) + X − i ( t ) D i ( t ) (cid:18) ∂∂t X − i ( t ) (cid:19) = 0 , i.e., H − i D i ( t ) X − i ( t ) + X − i ( t ) (cid:18) ∂∂t D i ( t ) (cid:19) X − i ( t ) + X − i ( t ) D i ( t ) H − i ( t ) = 0 , which gives ∂∂t D i ( t ) = − X − i ( t ) − H − i D i ( t ) − D i ( t ) H − i X − i ( t ) − = − X − i ( t ) − H − i X − − i ( t ) H − i X − i ( t ) − = − D i ( t ) X − i ( t ) D i ( t ) . Therefore, since ∂ Φ i ( t ) /∂t = Φ i ( t ) D i ( t )Φ i ( t ), we obtain ∂ Φ i ( t ) ∂t = ∂ Φ i ( t ) ∂t D i ( t )Φ i ( t ) + Φ i ( t ) (cid:18) ∂D i ( t ) ∂t (cid:19) Φ i ( t ) + Φ i ( t ) D i ( t ) ∂ Φ i ( t ) ∂t = 2Φ i ( t ) D i ( t )Φ i ( t ) D i ( t )Φ i ( t ) − i ( t ) D i ( t ) X − i ( t ) D i ( t )Φ i ( t )= − i ( t ) D i ( t )( X − i ( t ) − Φ i ( t )) D i ( t )Φ i ( t ) . From Lemma 2.1, we have that Φ i ( t ) (cid:22) X − i ( t ). Also, since Φ i ( t ) D i ( t ) ∈ S (cid:23) , it is clear that − ∂ Φ i ( t ) ∂t ∈ S (cid:23) , ∀ t ∈ ( − ǫ,
1) and ∀ i ∈ I. For − ǫ ≤ t < t ≤
1, the above implies that ∂ Φ i ( t ) ∂t (cid:12)(cid:12)(cid:12) t = t − ∂ Φ i ( t ) ∂t (cid:12)(cid:12)(cid:12) t = t = Z t t − ∂ Φ i ( t ) ∂t d t ∈ S (cid:23) , ∀ i ∈ I, completing the argument. (cid:3) Lemma 2.7.
For s ∈ [ − ǫ, , it holds that X ( s ) ∈ U , i.e., F ( X ( s )) (cid:22) X ( s ) .Proof. Note that H = X ∗ − X = F ( X (1)) − F ( X (0)) = Z ∂F ( X ( t )) ∂t d t. In view of the fact that ( − ǫ, ∋ t
7→ − ∂F ( X ( t )) /∂t is nondecreasing as follows from Lemma2.6, we have H = Z ∂F ( X ( t )) ∂t d t (cid:22) s Z s ∂F ( X ( t )) ∂t d t = 1 s ( F ( X ( s )) − F ( X (0))) , s ∈ (0 , . Therefore, ∂F ( X ( s )) ∂s (cid:12)(cid:12)(cid:12) s =0 = lim s ↓ F ( X ( s )) − F ( X (0)) s (cid:23) H . Using again the fact that ( − ǫ, ∋ t
7→ − ∂F ( X ( t )) /∂t is nondecreasing, which implies that ∂F ( X ( t )) /∂t (cid:23) H holds for all t ∈ ( − ǫ, X − F ( X ( s )) = F ( X (0)) − F ( X ( s )) = Z s ∂F ( X ( t )) ∂t d t (cid:23) − s H , ∀ s ∈ [ − ǫ, , which shows that F ( X ( s )) (cid:22) X + s H = X ( s ) holds for s ∈ [ − ǫ, (cid:3) QUILIBRIUM IN THIN SECURITY MARKETS UNDER RESTRICTED PARTICIPATION 17
We are now ready to complete the proof of uniqueness. Recall that ǫ > r Z (cid:22) X ( − ǫ ). Since additionally r Z ∈ L and X ( − ǫ ) ∈ U , Lemma 2.4 gives existenceof a fixed point Y with the property r Z (cid:22) Y (cid:22) X ( − ǫ ). Since X is the smallest fixed pointdominated below by r Z , this would give X (cid:22) Y which together with Y (cid:22) X − ǫ H and H ∈ Q i ∈ I S (cid:23) gives H = 0, i.e., X = X ∗ .2.4. Convergence to solutions through iteration.
Now that uniqueness has been estab-lished, we can show that the iterative procedure will always converge to the unique root andhence finish the proof of Theorem 1.4.
Lemma 2.8.
For an arbitrary X ∈ Q i ∈ I S X i ≻ , form a sequence ( X n ; n ∈ N ) by induction,asking that X n = F ( X n − ) , for all n ∈ N . Then, it holds that lim n →∞ X n = X ∗ . Furthermore, if X ∈ L , the sequence ( X n ; n ∈ N ) is nondecreasing, while if X ∈ U , thesequence ( X n ; n ∈ N ) is nonincreasing.Proof. If X ∈ L , the inequality X (cid:22) F ( X ) = X and the monotonicity of F of the form(2.4) show by induction that ( X n ; n ∈ N ) is nondecreasing. Similarly, if X ∈ U , the inequality X = F ( X ) (cid:22) X and the monotonicity of F show that ( X n ; n ∈ N ) is nonincreasing.Given an arbitrary X ∈ Q i ∈ I S X i ≻ , recall that Lemma 2.1 implies that t B ∈ U for all t ∈ [1 , ∞ ) and that Lemma 2.3 guarantees the existence of Z ∈ Q i ∈ I S X i ≻ , such that Z (cid:22) B and r Z ∈ L for all r ∈ (0 , r ∈ (0 ,
1] sufficiently small and ˆ t ∈ [1 , ∞ ) sufficiently largesuch that W := ˆ r Z (cid:22) X (cid:22) ˆ t B =: Y holds. Then, define the sequences ( W n ; n ∈ N ) and ( Y n ; n ∈ N ) by iteration via F , that is W n := F ( W n − ) and Y n := F ( Y n − ), for each n ∈ N . Since W (cid:22) X (cid:22) Y , monotonicityof F and induction gives W n (cid:22) X n (cid:22) Y n , for each n ∈ N . Also, since W ∈ L and Y ∈ U ,the sequence ( W n ; n ∈ N ) is nondecreasing and, in fact, bounded above by Y , while thesequence ( Y n ; n ∈ N ) is nonincreasing and bounded below by W . It follows that bothsequences have limits W ∞ and Y ∞ , respectively, with W ∞ (cid:22) Y ∞ . Continuity of F givesthat W ∞ = F ( W ∞ ) and Y ∞ = F ( Y ∞ ), exactly as in the proof of Lemma 2.5. By uniquenessof the solution to (2.2), it follows that W ∞ = X ∗ = Y ∞ , from which it further follows thatlim n →∞ X n = X ∗ . (cid:3) The proof of Proposition 1.5.
As discussed above, the main input of our market modelis the traders’ covariance matrices, properly scaled with their risk tolerance coefficients. Thenext auxiliary result, related to [MR17, Proposition 1, item (iv)], implies that the equilibriumprice impacts are monotonically increasing with respect to these matrices.
Lemma 2.9.
Let B = ( B i ; i ∈ I ) ∈ Q i ∈ I S X i ≻ and B = ( B i ; i ∈ I ) ∈ Q i ∈ I S X i ≻ be such that B (cid:22) B . If X = ( X i ; i ∈ I ) ∈ Q i ∈ I S X i ≻ and X = ( X i ; i ∈ I ) ∈ Q i ∈ I S X i ≻ stand for theassociated unique equilibria, then X (cid:22) X .Proof. Set F to be as in (2.2) with B in place of B there, and note that X i = (( B i ) −X i + π i ( X − i ) − π i ) −X i (cid:22) (( B i ) −X i + π i ( X − i ) − π i ) −X i = F i ( X ) , i ∈ I, which shows that X (cid:22) F ( X ). Then, statement (3) of Lemma 2.5 shows that X (cid:22) X . (cid:3) We are now in position to complete the proof of Proposition 1.5. By monotonicity fromLemma 2.9 and the nondecreasing assumption of ( B n ; n ∈ N ), we have that ( X n ; n ∈ N ) isalso nondecreasing in Q i ∈ I S X i ≻ . Furthermore, from Lemma 2.1 we have that X ni (cid:22) B i , ∀ i ∈ I \ { } and also that X n (cid:22) X n − (cid:22) X i ∈ I \{ } B i . It follows that ( X n ; n ∈ N ) has a monotone limit X ∞ ∈ Q i ∈ I S X i ≻ . Since lim n →∞ ( B n ) −X = 0,condition (1.5) in the limit gives that X ∞ = lim n →∞ X n = lim n →∞ (cid:0) ( B n ) −X + π ( X n − ) − π (cid:1) −X = (cid:0) π ( X ∞− ) − π (cid:1) −X . The same limiting argument shows that X ∞ i = (cid:16) B −X i i + π i (cid:0) X ∞− i (cid:1) − π i (cid:17) −X i , i ∈ I References [AC09] Z. Aouani and B. Cornet,
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E-mail address : [email protected] Constantinos Kardaras, Statistics Department, London School of Economics.
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