Arbitrage-free exchange rate ensembles over a general trade network
aa r X i v : . [ q -f i n . E C ] J un S. Palasek
Arbitrage-free exchange rate ensemblesover a general trade network S tan P alasek Princeton [email protected]
June 2014
Abstract
It is assumed that under suitable economic and information-theoretic conditions, market exchangerates are free from arbitrage. Commodity markets in which trades occur over a complete graph areshown to be trivial. We therefore examine the vector space of no-arbitrage exchange rate ensemblesover an arbitrary connected undirected graph. Consideration is given for the minimal information fordetermination of an exchange rate ensemble. We conclude with a topical discussion of exchanges in whichour analyses may be relevant, including the emergent but highly-regulated (and therefore not a completegraph) market for digital currencies.
I. I ntroduction
Consider a set of goods with a fixed exchangerate defined for each pair traded among ra-tional investors. Provided that there is com-plete information and the goods have objectiveworth (as might, for instance, a currency), it isnecessary that the rates be such that no marketparticipant can make a strict profit by execut-ing a trade which ends in the same denomina-tion with which it began. Otherwise, becausethe quantity of each good is conserved and theinvestors by assumption have identical pref-erences, executing the trade would leave theothers taking an effective loss. We will referto this assumption as the “no-arbitrage” con-dition. [1, p. 322] Although history shows thatit is at times possible to profit from differen-tial cross and direct exchange rates due to,for instance, asymmetric information, [2, 3] wewill for now neglect these uncommon compli-cations.Ellerman notes that the arbitrage-free prop-erty of such an exchange is equivalent bothto the system being path-independent and tothe exchange rates taking on their trivial cross-rate values. [4] These truths will become ev- ident given the formulation presented here,though we will proceed in the manner ofMirowski and consider only undirected net-works in which a reciprocal exists for each po-tential trade. [5] Under the presumption thatall relevant economies are free from arbitrage,we will further examine the structure of thespace of plausible exchange ensembles froma graph-theoretic perspective. Of particularinterest are markets without a universal cur-rency (ie. graphs without a vertex connected toall others). In such an economy it is meaning-less to speak of “price” since no denominationis universal (although we will prove a theoremthat provides an equivalent alternative). Theresulting ambiguity over the “value” of a com-modity is of interest to postmodern philoso-phers who have themselves studied this sys-tem. [6] II. T he space of arbitrage - freeensembles Definition 1.
Let G be a connected nth-ordergraph with vertices g , g , . . . , g n which representsthe exchanges between the n goods that may oc-cur. Define an exchange matrix associated with G . Palasekas any n × n matrix ˜ E of positive reals where ˜ E i , j is the exchange rate of g j to g i or equivalently theprice of g i in units of g j with the additional prop-erty that if ( i , j ) / ∈ G, then E i , j = . Furthermore,for an exchange matrix ˜ E, define its additive ex-change matrix E to be the n × n matrix of realswith E i , j = ln ˜ E i , j . Definition 2.
Define a no-arbitrage matrix over aconnected graph G as an exchange matrix ˜ E suchthat for any closed walk upon G over the verticesg n , g n , . . . , g n t , g n , we have ˜ E n t , n · t − ∏ i = ˜ E n i , n i + = or equivalently for the additive exchange matrix,E n t , n + t − ∑ i = E n i , n i + =
0. (2)Note that Definition 2 considers (1) and (2)over all closed walks on G , even those withrepeated vertices (not counting the repetitionof g i ). However, because a closed walk withrepeated vertices can trivially be fragmentedinto a sequence of closed walks without re-peated vertices and (1) and (2) apply associa-tively, it is equivalent to consider the equationsover non-repeating closed walks (“cycles”). Definition 3.
Let M ( G ) be the set of all additiveno-arbitrage matrices over the connected graph G. Lemma 4. M ( G ) is a subspace of the vector spaceof n × n matrices over standard matrix operations.Proof. Choose any E , E ′ ∈ M ( G ) and c ∈ R and consider the matrix E + c · E ′ . Evaluatingit in the left-hand side of (2), the terms sepa-rate due to linearity, again yielding zero. Fur-thermore, since both matrices are over G , theycontain zero entries wherever an edge is notin G ; therefore so does their linear combina-tion. Thus all the conditions are satisfied for E + c · E ′ ∈ M ( G ) so the space is closed underlinear combination. Lemma 5.
Let G be an nth-order connected graphand E an additive no-arbitrage matrix on it. Thenfor any i , j,1. E i , i =
2. E i , j = − E j , i .Proof. If ( i , j ) / ∈ G , then by Definition 1 E i , j = ( i , j ) ∈ G , then we can apply Def-inition 2 to the closed walk ( i , j , i ) to obtain E i , j + E j , i = i = j , the first state-ment follows from the second.Note that if i is not reflexively connected toitself, the first statement of the lemma holdsnot for any economically relevant reason, butrather because we arbitrarily defined the en-tries of E corresponding to non-existent edgesof G to be 0. This convention is nonethelessuseful as it preserves M ’s additive and scalar-multiplicative closure which we needed in theproof of Lemma 4. Definition 6.
Let G be a connected graph of ordern and ≤ i k , j k ≤ n for k =
1, 2, . . . , t. A col-lection of ordered pairs ( i , j ) , ( i , j ) , . . . , ( i t , j t ) is a basis for G’s additive exchange matrix if fixingthe entries E i , j , E i , j , . . . , E i t , j t of the no-arbitrageadditive exchange matrix E of G uniquely and min-imally determines the rest of the matrix. We willrefer to t as the dimension. Lemma 7.
Let G be the complete graph of order n.Then for any k, the entries ( k , 1 ) , ( k , 2 ) , . . . , ( k , k − ) , ( k , k + ) , . . . , ( k , n − ) , ( k , n ) are a basis forthe exchange matrix of G.Proof. Fix values for E k ,1 through E k , n exclud-ing E k , k . Consider i , j satisfying ( i , j ) ∈ G .Case 1: If i = j = k , then E i , j is determined bypart 1 of Lemma 5. Case 2: Suppose i = k = j .Then E i , j = E k , j is among the fixed values.Switching the roles of i and j is determinedlikewise up to a sign by the second part ofLemma 5. Case 3: Suppose i , j = k . Since When we refer to a complete graph here and elsewhere, it is irrelevant whether one includes or excludes the reflexiveedges connecting each vertex to itself, ie. whether ones or zeros constitute the main diagonal of the adjacency matrix. Themain diagonal of the exchange matrix will in either case be zeros due either to no-arbitrage over the reflexive loops orthe convention of setting to zero the entries corresponding to absent edges. . Palasek the graph is complete, ( j , k ) and ( k , i ) are in G .Therefore g i , g j , g k , g i form a closed walk andby Definition 2, E i , j + E j , k + E k , i =
0. Usingthe second part of Lemma 5, this rearranges to E i , j = E k , j − E k , i which is again determined interms of the fixed values. Thus we’ve shownthat every entry of E is determined uniquely.Next we illustrate that a smaller collectionof entries does not determine the exchangematrix uniquely. Let E i , j = E k , j − E k , i (3)for all 1 ≤ i , j , k ≤ n . Then (2) becomes E k , n − E k , n t + t − ∑ i = ( E k , n i + − E k , n i ) = E k ,1 , E k ,2 , . . . , E k , k − , E k , k + , . . . , E k , n − , E k , n . Weconclude that no smaller set could determinethe exchange matrix uniquely; thus both con-ditions of Definition 6 are met.In the next lemma, unlike before, we willadditionally refer to the standard linear alge-bra definition of basis. However the resultwill allow us to identify the Definition 6 un-derstanding of basis and dimension preciselywith a basis for and the dimension of the space M ( G ) . Definition 8.
Let G be a connected graph of ordern and ( i , j ) , ( i , j ) , . . . , ( i t , j t ) a basis for it. Fork =
1, 2, . . . , t, define ǫ k to be the matrix uniquelydetermined by letting E i k , j k = and setting therest of the basis entries to zero. Lemma 9.
Let G be a connected graph of order nwith ( i , j ) , ( i , j ) , . . . , ( i t , j t ) and ǫ , ǫ , . . . , ǫ k as in Definition 8. Then ∑ tk = a k ǫ k is the uniqueno-arbitrage exchange matrix over G with a k at theentry ( i k , j k ) for all k =
1, 2, . . . , t.Proof.
It follows from Definition 8 that the k thmatrix’s ( i k , j k ) entry is one and for m = k , its ( i m , j m ) entry is zero. In mathematical terms, ( ǫ k ) i m , j m = δ k , m . (5) Therefore, t ∑ k = a k ǫ k ! i m , j m = t ∑ k = a k ( ǫ k ) i m , j m (6) = t ∑ k = a k δ k , m (7) = a m . (8)By Lemma 4, ∑ tk = a k ǫ k ∈ M ( G ) . By Defini-tion 8, it is unique. Theorem 10.
Let G be a connected graphof order n with ( i , j ) , ( i , j ) , . . . , ( i t , j t ) and ǫ , ǫ , . . . , ǫ k as in Definition 8. Then the col-lection of matrices ǫ , ǫ , . . . , ǫ t form a basis of M ( G ) .Proof. We must show that every no-arbitrageadditive exchange matrix over G has a uniqueexpression as a linear combination of the ǫ k . By Lemma 9, this is equivalent tothe proposition that every no-arbitrage ad-ditive exchange matrix over G can be ex-pressed with a unique choice of constantsto fill the entries ( i , j ) , ( i , j ) , . . . , ( i t , j t ) .This proposition is precisely the meaning of ( i , j ) , ( i , j ) , . . . , ( i t , j t ) being a basis as givenby Definition 6. Corollary 11.
Let G be the complete graph andH a connected tree (acyclic graph), both of order n.Then dim M ( G ) = dim M ( H ) = n −
1. (9)
Proof.
The equality for G follows immediatelyfrom Lemma 7 and Theorem 10.By definition of a tree, the only closedwalks on H upon which we need to in-voke (2) are those which double backon themselves, ie. those of the form ( i , i , . . . , i k − , i k , i k − , . . . , i , i ) for some k ∈ N . Linearity of 2 implies that it is neces-sary and sufficient to check this conditionwith k =
2, yielding anticommutativity as inLemma 5 as the sole restriction. Thus the ad-ditive exchange has a degree of freedom foreach edge, of which there are n − M ( H ) = n −
1. 3 . Palasek
Theorem 12.
If G is a connected graph, then dim M ( G ) = | G | −
1. (10)
Proof.
Let n = | G | , the number of vertices inthe graph. G must contain a normal spanningtree [7, p. 16, Prop. 1.5.6]; call it H n − in recog-nition of the fact that it has n − k · k . Next, construct a sequenceof graphs H n − , H n , . . . , H k G k− , H k G k with thefollowing properties for n − ≤ i < k G k :1. H i is a spanning subgraph of H i + k H i + k = k H i k + H k G k = G In other words, the sequence of graphs from H k G k to H n − is the transformation of G intoone of its spanning trees, successively remov-ing edges at each step. Since H n − is by def-inition a subgraph of G , such a sequence ev-idently exists. We will proceed by inductionover the H i to show that each of the M ( H i ) and in particular M ( H k G k ) has dimension n − i = n − i satisfying n − ≤ i < k G k ,dim M ( H i ) = n −
1. Let the edge ( k , m ) be thesingleton element of H i + \ H i . There are threecases. Case 1: Suppose k = m . The only newcycle is the reflexive loop ( k , m ) = ( k , k ) , theexchange matrix entry over which is triviallydetermined to be 0. Thus the collection of en-tries that form a basis of H i in the Definition 6sense likewise form one of H i + . By Theorem10 they have equal dimension. Case 2: Sup-pose there is exactly one path from g k to g m consisting of distinct vertices, call it C . Thenthe addition of the edge ( k , m ) creates four cy-cles on H i + that did not exist on H i : ( k , m , k ) , C ∪ ( k ) , and their respective reversals. Analo-gously to Lemma 5, the no-arbitrage conditionon ( k , m , k ) implies E k , m = − E m , k . (11) Thus all the edges in H i + are anticommuta-tive, so if a cycle satisfies (2) then its rever-sal must as well. We therefore see that theonly nontrivial new constraint introduced bythe edge ( k , m ) is the no-arbitrage conditionon one direction of C ∪ ( k ) . Furthermore, by(11), there is only one new unique variable tobe determined.Consider the equation we referenced im-posed by 2 on C ∪ ( k ) . Because H i is a span-ning subgraph of H i + and a basis of size n − H i ’s exchange matrix determines each ofthe variables, that basis determines all of thevariables in the no-arbitrage equation exceptfor E k , m , which in turn is determined by theequation. We therefore see that H i + has a ba-sis of size n − M ( H i + ) = n −
1. Case 3:Suppose there is more than one path from g k to g m consisting of distinct vertices. Wewill illustrate that additional paths impose nogreater restriction than the single one we con-sidered in the second case. Let C and C ′ betwo non-identical paths from g k to g m on H i . H i + now has two cycles that we did not con-sider in Case 2: C ′ ∪ ( k ) and its reverse. Againby anticommutativity, the no-arbitrage equa-tions imposed by these paths are equivalent.Furthermore, observe that C ∪ ( − C ′ ) is a cy-cle on H i where − C ′ is the reversal of the path C ′ . Since the basis for H i provided for no-arbitrage over this cycle as it does not assumethe edge ( k , m ) , its no-arbitrage equation canbe assumed. ∑ ( p , q ) ∈ C ∪ ( − C ′ ) E p , q = ∑ ( p , q ) ∈ C E p , q = − ∑ ( p , q ) ∈− C ′ E p , q (13)Thus the no-arbitrage condition is equivalentover any two paths so the third case reducesto the second. It follows by induction thatdim M ( H i ) = n − n − ≤ i ≤ k G k . Let-ting i = k G k proves the theorem.4 . Palasek III. D iscussion
The cited literature’s consideration of ex-change rates over arbitrary networks (see, forinstance, [4, 6]) seems to rely on the premisesboth that complete graphs are too trivial tobe of interest and that nontrivial underlyingnetworks may exist ex papyro . However, theconditions formulated both implicitly and ex-plicitly in the introduction may considerablylimit the scope of the class of systems whichour analyses may encompass. First, we re-quired that all traders have the same prefer-ences, ie. that they demand the same price fora given asset; otherwise we would not havea fixed exchange matrix. Second, in order toobtain a nontrivial (here, non-sparse) graph,trades must occur between a variety of pairsof assets; if there is a single commodity thatis behaving as a universal currency, then ev-ery cross rate is immediately determined tri-angularly. The empirical existence of idiosyn-cratic consumer preferences, investment objec-tives, and information makes the first condi-tion implausible (American reality televisionprovides an explicit counterexample, see [8]).Furthermore, the second condition obsolescedalong with the bartering system in moderneconomies. Hence currency markets are theonly logical applications where our consider-ations might be relevant. Neglecting differen-tial preferences for foreign goods, a given cur-rency has the same value across individuals(my euro buys as many goods as your euro).Now we must contend only with the po-tential of the underlying network to be notsparse enough . Certainly there are active ex-changes between virtually every pair of con-ventional currencies and, as we showed inLemma 7, such complete no-arbitrage net-works are uniquely determined by the setof n − When included in the vertex set, these “cur-rencies” may fascinatingly not yield a com- plete graph. In May 2014, for instance, thePeople’s Bank of China began urging banksto be wary of transactions in Bitcoin, a par-ticularly prominent electronic currency. In-deed, China’s largest banks banned “activitiesrelated to Bitcoin trading.” [10] The edge inthe market graph between the yuan and Bit-coin is therefore not present. One might fur-ther extend the network, data permitting, toinclude the elicit goods that make up a sig-nificant fraction of transactions involving elec-tronic currencies. [11] One might expect thatthe popularity of resorting to alternate pay-ment methods indicates an aversion to tradingvia traditional centralized currencies, perhapssignaling additional absent edges.We have so far considered only static mar-ket equilibria. Allowing the discussion to re-main in the context of currencies, we willbriefly discuss considerations for dynamics onexchange networks. Ellerman notes that theno-arbitrage condition of equation (2) is for-mally equivalent to Kirchhoff’s junction ruleof electrodynamics. [4] We must remember,however, that the economically-relevant sys-tem is multiplicative as in (1). It is thereforeunrealistic to expect exchange rate dynamicsakin to the charging of a circuit in response tochanges in consumer demand or central banksupply. Rather, we might imagine a set of ba-sis entries ( e ∗ i , j , e ∗ i , j , . . . , e ∗ i n − , j n − ) perturbedaccording to e ∗ i k , j k e ∗ i k , j k + δ e ∗ i k , j k (14)along with the observation from Definition 1that δ ˜ E i , j = exp ( E i , j ) δ E . (15)Letting A be a third-order tensor mapping abasis to its unique arbitrage-free exchange ma-trix, linearity gives us δ E = A δ e ∗ i , j ... δ e ∗ i n − , j n − (16) Although the Internal Revenue Service might disagree with this terminology [9], electronic currencies nonethelessconform to the conditions we outlined for them to fit within the scope of the model presented here. . Palasek whence we may proceed to solve the dynamicequations. Though Theorem 12 simplifies thecalculation by guaranteeing that it is sufficientto know the changes to just n − R eferences [1] Harrison, M. & Waldron, P. (2011). Math-ematics for Economics and Finance.
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