Exploiting arbitrage requires short selling
aa r X i v : . [ q -f i n . M F ] N ov EXPLOITING ARBITRAGE REQUIRES SHORT SELLING
ECKHARD PLATEN AND STEFAN TAPPE
Abstract.
We show that in a financial market given by semimartingales anarbitrage opportunity, provided it exists, can only be exploited through shortselling. This finding provides a theoretical basis for differences in regulationfor financial services providers that are allowed to go short and those withoutshort sales. The privilege to be allowed to short sell gives access to potentialarbitrage opportunities, which creates by design a bankruptcy risk. Introduction
There has been always a need for design and regulation of the activities of fi-nancial market participants. In particular, the granting of the privilege of shortselling raises the question whether there is a theoretical link between the poten-tial exploitation of arbitrage opportunities in the market by some financial servicesproviders that engage in short selling. Hedge funds and investment banks are typi-cal examples. Exploiting arbitrage opportunities can be interpreted as a service tothe market because it removes usually inconsistencies between asset prices. Thereare other financial services providers that are not engaging in short selling, in-cluding certain pension funds and some investment funds like index funds. Thesefocus on producing targeted payouts and appear not to chase potential arbitrageopportunities.When it comes to practical implementation, it will become crucial to properlydefine short selling. If a very narrow definition would be adopted, many institutionsmay end up being short sellers. Very often, instead of buying physical securities orfor reducing currency risk, derivatives are used which can lead to short selling. Thischallenge is tackled in this paper through the introduction of the concept of primarysecurities, see below, which can be traded but may not be allowed to be short in aportfolio.Besides as an aside, e.g. in [18], the literature seems not to have clarified in detailthe crucial link between short-selling and exploiting arbitrage. The goal of this paperis to fill this gap and show that in a financial market an arbitrage opportunity,provided it exists, can only be exploited through short selling. This rather intuitiveinsight is highly relevant for the design of financial markets and regulation. It arisesthe question, which are the institutions that in a well-designed market should beallowed to remove or exploit arbitrage opportunities that may exist in the market.Various financial institutions should under appropriate market design probably notbe allowed to short sell to avoid potential bankruptcy. A pension fund that maybe not allowed to go short could still produce targeted payoffs but cannot extractany arbitrage that may exist in the market. The government could grant certain
Date : 25 November, 2020.2010
Mathematics Subject Classification.
Key words and phrases. arbitrage opportunity, short selling, supermartingale deflator, self-financing portfolio.The authors like to thank Michael Schmutz and Andreas Haier from the Swiss Financial MarketSupervisory Authority FINMA for fruitful discussions. The authors are also grateful to KostasKardaras for invaluable comments and remarks.
ECKHARD PLATEN AND STEFAN TAPPE privileges to such a pension fund, e. g. some tax advantages because it cannotexploit arbitrage and is highly unlikely to go bankrupt when only targeting pensionpayouts. An investment bank or a hedge fund can extract arbitrage but can gobankrupt and when allowed to grow too large may constitute even a significant riskto the economy. The stability of a financial market may become a paramount goalwhen revising its design and regulation at some future time due to systemic failures.Based on the in the current paper derived fact that arbitrage can be exploited onlywhen taking short positions, such revision may lead to a far more secure marketenvironment for the economic core purposes of the financial market.This paper does not aim to discuss necessary features of a well designed marketand respective regulation, however, it clarifies theoretically the need for short sellingto have the ability to exploit arbitrage opportunities. This general theoretical insightcould be useful for the design of markets and regulation in the future, where theprivilege of short selling gives access to potential arbitrage.Before we provide the mathematical formulation of our results (see Theorems1.1–1.3), let us outline the stochastic framework which we consider in this pa-per; more details and the precise definitions can be found in Section 2. Let S = { S , . . . , S d } be a financial market consisting of strictly positive primary securityaccounts which does not need to have a tradeable numéraire S d = 1 . These accountscould be cum-dividend stocks, savings accounts and combinations of derivatives andother securities offering collateral. Let P sf ( S ) be the set of all self-financing port-folios, and let P δ ≥ ( S ) be the set of all such portfolios without short selling. Wefix a deterministic finite time horizon T ∈ (0 , ∞ ) . Then a portfolio S δ is an ar-bitrage portfolio if S δ = 0 and S δT ∈ L \ { } , where L is the convex cone ofall nonnegative random variables. This definition essentially says that an arbitrageportfolio generates under limited liability from zero initial capital some strictlypositive wealth. Often, one assumes that an arbitrage portfolio is nonnegative (see,for example [19]) or admissible (see, for example [3] or [5]). If such an arbitrageportfolio does not exist, then the market satisfies No Arbitrage (NA), which meansthat K ∩ L = { } , where K denotes the convex cone of all outcomes of self-financing portfolios starting at zero, which are nonnegative or admissible. For thesake of generality, in our upcoming first result we do not assume that an arbitrageportfolio has to be nonnegative or admissible.1.1. Theorem.
For every arbitrage portfolio S δ ∈ P sf ( S ) we have S δ / ∈ P δ ≥ ( S ) . The above theorem states the fact that exploiting arbitrage requires short sell-ing. Similar results also hold true for weaker forms of arbitrage opportunities,such as arbitrages of the first kind or free lunches with vanishing risk. A sequence ( S δ n ) n ∈ N ⊂ P sf ( S ) of self-financing portfolios is called an arbitrage of the first kind if S δ n ↓ and there exists a random variable ξ ∈ L +0 \ { } such that S δ n T ≥ ξ forall n ∈ N . Typically, the sequence ( S δ n ) n ∈ N is nonnegative; see, for example [26]or [15]. If a nonnegative arbitrage of the first kind does not exist, then the marketsatisfies No Arbitrage of the First Kind (NA ). For the sake of generality, in ourupcoming second result we do not assume that an arbitrage of the first kind has tobe nonnegative.1.2. Theorem.
Let ( S δ n ) n ∈ N ⊂ P sf ( S ) be an arbitrage of the first kind. Then thereexists an index n ∈ N such that S δ n / ∈ P δ ≥ ( S ) for all n ≥ n . A sequence ( S δ n ) n ∈ N ⊂ P sf ( S ) of self-financing portfolios with S δ n = 0 for all n ∈ N is called a free lunch with vanishing risk if there exist a random variable ξ ∈ L ∞ + \ { } and a sequence ( ξ n ) n ∈ N ⊂ L ∞ such that k ξ n − ξ k L ∞ → and S δ n T ≥ ξ n for each n ∈ N . Typically, the sequence ( S δ n ) n ∈ N consists of admissible XPLOITING ARBITRAGE REQUIRES SHORT SELLING 3 portfolios; see, for example [3] or [5]. If an admissible free lunch with vanishingrisk does not exist, then the market satisfies
No Free Lunch with Vanishing Risk (NFLVR). For the sake of generality, in our upcoming third result we do not assumethat a free lunch with vanishing risk has to be nonnegative.1.3.
Theorem.
Let ( S δ n ) n ∈ N ⊂ P sf ( S ) be a free lunch with vanishing risk. Thenthere exists an index n ∈ N such that S δ n / ∈ P δ ≥ ( S ) for all n ≥ n . Although there are papers dealing with financial markets under short sale con-straints (see, for example [14, 18, 23, 11, 1]), it is a bit surprising that, best toour knowledge, the literature has not emphasized so far the important fact thatonly through short selling one can exploit arbitrage opportunities. This is highlyimportant for designing and regulating a financial market. Financial institutionsthat short sell can exploit potential arbitrage opportunities and, therefore, have tobe regulated in a different way than those that never go short and cannot exploitarbitrage. Usually, the latter group of financial institutions should have the privilegeto generate pension payouts to protect vulnerable members of the society. Banksand particularly investment banks and hedge funds are allowed to short sell, whichgives these businesses access to potential arbitrage and makes them more risky thanthose regulated under no short sale constraints. The above theorem makes clear thatwhen imposing short sale constraints on a financial service, then one removes accessto arbitrage opportunities. Because of the much lower risk of default these typesof businesses should get privileges in producing payout streams that are essentialto vulnerable members of the society. Financial services providers that are allowedto go short can default by chasing arbitrage opportunities. It would be wise notto support by regulation any overreliance of society on the production of essentialpayout streams, like pensions, by short selling institutions. This finding provides anargument for differences in regulation between financial services providers that areallowed to go short and those with short sale prohibitions. The implementation ofthis finding in market design and regulation raises important questions which go farbeyond the scope of this paper. Currently, both types of businesses are not clearlydifferentiated and would also interact closely in future. The privilege to be allowedto short sell securities, which gives access to potential arbitrage opportunities, maybe inconsistent with potential government bailouts. Since short selling can causebankruptcies, society may have to diversify this risk and limit the size of short sell-ing institutions to avoid major problems for the economy from a potential defaultof a short selling institution. Potentially, short selling institutions may have to belimited in size, whereas not short selling institutions may be allowed to benefit fromeconomies of scale that short selling ones cannot reach in an appropriately designedmarket.Let us briefly outline the essential steps for the proofs of Theorems 1.1–1.3.Without loss of generality we may assume that the considered portfolios are non-negative. Indeed, since the primary security accounts are assumed to be positive, aportfolio which becomes negative can only be realized by short selling; see Lemma2.1 for the formal statement. A crucial concept for the proofs is that of an equivalentsupermartingale deflator (ESMD). First, we show that the existence of an ESMDimplies the statements of Theorems 1.1–1.3 (see Propositions 2.3–2.5). In a secondstep we show that in the present setting an ESMD always exists. By performing achange of numéraire, this essentially follows from [18]; see Proposition 2.8We mention that supermartingale deflators and the related concept of a super-martingale measure have been used in various contexts in the finance literature;in particular in connection with short sale constraints. Such papers include, forexample, [14, 18, 23, 11] and [1]; of these the papers [18] and [23] present versionsof the fundamental theorem of asset pricing with short sales prohibitions. Further
ECKHARD PLATEN AND STEFAN TAPPE references, where supermartingales and supermartingale deflators are studied witha focus to applications in finance, include, for example, [27, 17] and the recent paper[8].The remainder of this paper is organized as follows. In Section 2 we provide theproofs of our results, and in Section 3 we present several examples and illustratehow our results link to asset and money market bubbles.2.
Proofs of the results
In this section we provide the proofs of Theorems 1.1–1.3. First, we introducethe precise mathematical framework. Let T ∈ (0 , ∞ ) be a fixed finite time horizon,and let (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) be a stochastic basis satisfying the usual conditions;see [10, Def. I.1.3]. Furthermore, we assume that F is P -trivial. Then every F -measurable random variable is P -almost surely constant. As mentioned in Section1, we consider a financial market S = { S , . . . , S d } consisting of strictly positivesemimartingales, where d ∈ N denotes a positive integer. More precisely, we assumethat S i , S i − > for each i = 1 , . . . , d . Denoting by S the R d -valued semimartingale S = ( S , . . . , S d ) , let L ( S ) be the space of all S -integrable processes δ = ( δ , . . . , δ d ) in the sense of vector integration; see [25]. Every process δ ∈ L ( S ) is called a strategy for S . We will also use the notation ∆( S ) for the space of all strategies for S . For astrategy δ ∈ ∆( S ) we define the portfolio S δ := δ · S , where we use the short-handnotation δ · S := d X i =1 δ i S i . A strategy δ ∈ ∆( S ) and the corresponding portfolio S δ are called self-financing for S if S δ = S δ + δ · S , where δ · S is the stochastic integral according to [25].Let P sf ( S ) be the set of all self-financing portfolios S δ such that S δ ≥ impliesthat the strategy δ is locally bounded. We denote by P +sf ( S ) the convex cone of allnonnegative self-financing portfolios S δ ≥ , we denote by P admsf ( S ) the convex coneof all admissible self-financing portfolios S δ ≥ − a for some a ∈ R + , and we denoteby P δ ≥ ( S ) the set of all self-financing portfolios S δ ∈ P sf ( S ) such that δ ≥ , whichmeans that δ i ≥ for all i = 1 , . . . , d . A self-financing portfolio S δ is called an arbitrage portfolio if S δ = 0 and S δT ∈ L \ { } .2.1. Lemma.
For every portfolio S δ ∈ P sf ( S ) such that S δ / ∈ P +sf ( S ) we have S δ / ∈ P δ ≥ ( S ) .Proof. By assumption the inequality S δ ≥ is not satisfied. Since S δ = δ · S and S ≥ , we deduce that δ ≥ cannot be satisfied. (cid:3) Definition.
Let X be a family of semimartingales, and let Z be a semimartin-gale such that Z, Z − > . We call Z an equivalent supermartingale deflator (ESMD) for X if XZ is a supermartingale for all X ∈ X . The following Propositions 2.3–2.5 essentially show the statements Theorems1.1–1.3, provided that an ESMD Z for P δ ≥ ( S ) exists.2.3. Proposition.
Suppose that an ESMD Z for P δ ≥ ( S ) ∩ P +sf ( S ) exists. Thenfor every arbitrage portfolio S δ ∈ P +sf ( S ) such that δ is locally bounded we have S δ / ∈ P δ ≥ ( S ) .Proof. Let S δ ∈ P δ ≥ ( S ) ∩ P +sf ( S ) be a self-financing portfolio such that S δ = 0 and ξ := S δT ∈ L . Since Z is an ESMD for P δ ≥ ( S ) ∩ P +sf ( S ) , the process S δ Z is a XPLOITING ARBITRAGE REQUIRES SHORT SELLING 5 nonnegative supermartingale. By Doob’s optional stopping theorem we obtain E [ ξZ T ] = E [ S δT Z T ] ≤ E [ S δ Z ] = 0 , and hence, we deduce that E [ ξZ T ] = 0 . Since ξ ≥ and P ( Z T >
0) = 1 , this shows ξ = 0 . Consequently, the portfolio S δ cannot be an arbitrage portfolio. (cid:3) Proposition.
Suppose that an ESMD Z for P δ ≥ ( S ) ∩ P +sf ( S ) exists. Let ( S δ n ) n ∈ N ⊂ P +sf ( S ) be an arbitrage of the first kind such that δ n is locally bounded for each n ∈ N . Thenthere exists an index n ∈ N such that S δ n / ∈ P δ ≥ ( S ) for all n ≥ n .Proof. We have S δ n ↓ and there exists a random variable ξ ∈ L +0 \ { } suchthat S δ n T ≥ ξ for all n ∈ N . Suppose, contrary to the assertion above, there isa subsequence ( S δ nk ) k ∈ N such that S δ nk ∈ P δ ≥ ( S ) for all k ∈ N . Let k ∈ N be arbitrary. Since Z is an ESMD for P δ ≥ ( S ) ∩ P +sf ( S ) , the process S δ nk Z is anonnegative supermartingale, and hence by Doob’s optional stopping theorem weobtain E [ ξZ T ] ≤ E [ S δ nk T Z T ] ≤ E [ S δ nk Z ] = S δ nk Z . Since S δ nk ↓ , we deduce that E [ ξZ T ] = 0 . Since ξ ≥ and P ( Z T >
0) = 1 , weobtain the contradiction ξ = 0 . (cid:3) Proposition.
Suppose that an ESMD Z for P δ ≥ ( S ) ∩ P +sf ( S ) exists. Let ( S δ n ) n ∈ N ⊂ P +sf ( S ) be a free lunch with vanishing risk such that δ n is locally bounded for each n ∈ N .Then there exists an index n ∈ N such that S δ n / ∈ P δ ≥ ( S ) for all n ≥ n .Proof. There exist a random variable ξ ∈ L ∞ + \ { } and a sequence ( ξ n ) n ∈ N ⊂ L ∞ such that k ξ n − ξ k L ∞ → and S δ n T ≥ ξ n for each n ∈ N . Note that also k ξ + n − ξ k L ∞ → , and hence k ξ + n k L ∞ → k ξ k L ∞ . Since k ξ k L ∞ > , there exists anindex n ∈ N such that k ξ + n k L ∞ > for each n ≥ n . Now, let n ≥ n be arbitrary.Noting that S δ n ≥ , we have S δ n T ≥ ξ + n . Since ξ + n ∈ L ∞ + \ { } , the portfolio S δ n isan arbitrage portfolio. Therefore, by Proposition 2.3 we obtain S δ n / ∈ P δ ≥ ( S ) . (cid:3) Now, our goal is to prove that an ESMD Z for P δ ≥ ( S ) ∩ P +sf ( S ) exists. We define X i := S i /S d for i = 1 , . . . , d − and the R d -valued semimartingale ¯ X := ( X, ,where X := ( X , . . . , X d − ) . Furthermore, we define X := { X , . . . , X d − } and thediscounted market ¯ X := { ¯ X , . . . , ¯ X d } . The following are two well-known resultsabout the change of numéraire technique.2.6. Lemma. [26, Prop. 5.2]
For an R d -valued predictable process δ the followingstatements are equivalent: (i) We have δ ∈ ∆ sf ( S ) . (ii) We have δ ∈ ∆ sf ( ¯ X ) .If the previous conditions are fulfilled, then we have S δ = S d ¯ X δ . (2.1)For a pair ( x, θ ) ∈ R × ∆( X ) the corresponding wealth process is defined as X x,θ := x + θ · X. Lemma. [26, Lemma 5.1]
Then there is a bijection between R × ∆( X ) and ∆ sf ( ¯ X ) , which is defined as follows: ECKHARD PLATEN AND STEFAN TAPPE (1)
For δ ∈ ∆ sf ( ¯ X ) we assign δ ( x, θ ) := ( X δ , δ , . . . , δ d − ) ∈ R × ∆( X ) . (2.2)(2) For ( x, θ ) ∈ R × ∆( X ) we assign ( x, θ ) δ = ( θ, X x,θ − − θ · X − ) ∈ ∆ sf ( ¯ X ) . (2.3) Furthermore, for all ( x, θ ) ∈ R × ∆( X ) and the corresponding self-financing strategy δ ∈ ∆ sf ( ¯ X ) we have ¯ X δ = X x,θ . (2.4)2.8. Proposition.
There exists an ESMD Z for P δ ≥ ( S ) ∩ P +sf ( S ) .Proof. According to [18, Thm. 1.3] there exists an ESMD Y for ¯ X such that Y = 1 .We set Z := Y ( S d ) − . Let S δ ∈ P δ ≥ ( S ) ∩ P +sf ( S ) be arbitrary. By Lemma 2.6 wehave ¯ X δ ∈ P δ ≥ ( ¯ X ) and the identity (2.1). Furthermore, by Lemma 2.7 we have(2.4), where ( x, θ ) ∈ R × ∆( X ) is given by (2.2). Using integration by parts, weobtain Y ¯ X δ = Y X x,θ = x + θ · ( Y X ) + ( X x,θ − − θ · X − ) · Y = x + δ · ( Y ¯ X ) , where Y ¯ X denotes the R d -valued supermartingale with components Y ¯ X i , i =1 , . . . , d ; cf. [18, p. 2685]. Since δ ≥ is locally bounded, we deduce that Y ¯ X δ is a supermartingale. Taking into account (2.1), this shows that ZS δ is a super-martingale. Consequently, the process Z is ESMD for P δ ≥ ( S ) ∩ P +sf ( S ) . (cid:3) Now, we are ready to provide the proofs of the results stated in Section 1.
Proof of Theorem 1.1.
Let S δ ∈ P sf ( S ) be an arbitrage portfolio. By Lemma 2.1 wemay assume that S δ ∈ P +sf ( S ) . Therefore, applying Propositions 2.3 and 2.8 showsthat S δ / ∈ P δ ≥ ( S ) . (cid:3) Proof of Theorem 1.2.
Let ( S δ n ) n ∈ N ⊂ P sf ( S ) be an arbitrage of the first kind. ByLemma 2.1 we may assume that S δ n ∈ P +sf ( S ) for all n ∈ N . Therefore, applyingPropositions 2.4 and 2.8 completes the proof. (cid:3) Proof of Theorem 1.3.
Let ( S δ n ) n ∈ N ⊂ P sf ( S ) be a free lunch with vanishing risk.By Lemma 2.1 we may assume that S δ n ∈ P +sf ( S ) for all n ∈ N . Therefore, applyingPropositions 2.5 and 2.8 completes the proof. (cid:3) Examples
In this section we present examples, where arbitrage portfolios can explicitly becomputed, and verify that, in accordance with Theorem 1.1, these portfolios indeedrequire short selling. We start with a simple example.3.1.
Example.
Consider the market S = { S , S } consisting of two primary secu-rity accounts such that S = S and S t > S t for all t > . For example, the assetscould be S t = e t for t ∈ R + and S = 1 . Then δ = (1 , − is a self-financing strat-egy, and we obtain the nonnegative arbitrage portfolio S δ = S − S . In accordancewith Theorem 1.1, we see that this arbitrage portfolio requires short selling. In the next example we consider a simple one-period model in discrete time. Asshown in [10, page 14], the discrete time setting can be regarded as a particularcase of the continuous time setting, which we have introduced in Section 2.
XPLOITING ARBITRAGE REQUIRES SHORT SELLING 7
Example.
This example can be found in [6, Exercise 1.1.1] . The probabilityspace (Ω , F , P ) is defined as follows. Let Ω := { ω , ω , ω } with pairwise differentelements ω , ω , ω , the σ -algebra is given by the power set F := P (Ω) , and con-cerning the probability measure P we assume that P ( { ω i } ) > for all i = 1 , , .The filtration ( F t ) t =0 , is given by F := { Ω , ∅} and F := F . In this examplewe have three primary security accounts. The values at time t = 0 are given by thevector π ∈ R defined as π := , and the values at the terminal time T = 1 are given by the random vector S : Ω → R defined as S ( ω ) := , S ( ω ) := and S ( ω ) := . Hence, the market S consists of one savings account and two risky security accounts.Note that in this example every arbitrage portfolio S δ ∈ P sf ( S ) is nonnegative; thatis, we have S δ ∈ P +sf ( S ) . It is easily verified that the market S admits arbitrage, andthat all arbitrage portfolios are specified by the set ξ ξ − ξ : ξ > . Hence, in accordance with Theorem 1.1, each such arbitrage portfolio requires shortselling. More precisely, in order to exploit arbitrage, we have to go short in thesecond risky primary security account, whereas for the first risky primary securityaccount and for the savings account we have long positions.
So far, we have considered financial markets, where NA is not satisfied; that is,at least one nonnegative arbitrage portfolio exists. Now, we are interested in modelssatisfying NA, but still admitting arbitrage portfolios which can go negative, sayadmissible arbitrage portfolios. In order to construct such examples, strict localmartingales play a crucial role, providing a link to asset and money market bubbles.Important references on this topic include, for example, [9, 2, 12, 13, 7] and thearticle [22] with an overview of the respective literature.3.3.
Definition.
The market S has a bubble if there are an index i ∈ { , . . . , d } and a self-financing portfolio S ϑ ∈ P +sf ( S ) such that S ϑ < S i and S ϑT ≥ S iT . Lemma.
Suppose that the market S has a bubble. Then there exists an arbitrageportfolio.Proof. There are an index i ∈ { , . . . , d } and a self-financing portfolio S ϑ ∈ P +sf ( S ) such that S ϑ < S i and S ϑT ≥ S iT . If S ϑ = 0 , then S δ with δ := ϑ is already anarbitrage portfolio. Now, we assume that S ϑ > . Setting α := S i S ϑ > , by [21, Lemma 7.21] the process δ := αϑ − e i is also a self-financing strategy forthe market S , and we have S δ = αS ϑ − S i . It follows that S δ = 0 and S δT > , showing that S δ is an arbitrage portfolio. (cid:3) ECKHARD PLATEN AND STEFAN TAPPE
A semimartingale Z with Z, Z − > is called an equivalent local martingaledeflator (ELMD) for S if S i Z ∈ M loc for each i = 1 , . . . , d .3.5. Lemma.
Suppose there exists an ELMD Z for S such that S i Z ∈ M for each i = 1 , . . . , d . Then the market S does not have a bubble.Proof. According to [20, Prop. 4.12] the process Z is also an ELMD for P +sf ( S ) . Let i ∈ { , . . . , d } and S ϑ ∈ P +sf ( S ) be such that S ϑT ≥ S iT . By [10, Lemma I.1.44] andDoob’s optional stopping theorem for nonnegative supermartingales we obtain ( S ϑ − S i ) Z = E [ S ϑ Z ] − E [ S i Z ] ≥ E [ S ϑT Z T ] − E [ S iT Z T ] = E [( S ϑT − S iT ) Z T ] ≥ , and hence S ϑ ≥ S i . (cid:3) The following result contains sufficient conditions for the converse statement ofthe previous result.3.6.
Lemma.
Suppose there exists an ELMD Z for S such that for some i ∈{ , . . . , d } we have S i Z / ∈ M . Furthermore, we assume that S iT Z T ∈ L andthat there exists a self-financing portfolio S ϑ ∈ P +sf ( S ) such that S ϑt = E [ S iT Z T | F t ] Z t , t ∈ [0 , T ] . (3.1) Then the market S has a bubble.Proof. By (3.1) we have S ϑ Z ∈ M and S ϑT = S iT . Furthermore, since S i Z / ∈ M ,we have E [ S i Z ] > E [ S iT Z T ] . Therefore, we obtain S ϑ Z = E [ S ϑT Z T ] = E [ S iT Z T ] < S i Z , and hence S ϑ < S i . (cid:3) We call every predictable càdlàg process B of locally finite variation with B, B − > a savings account .3.7. Proposition.
Suppose that S d = B for some savings account B such that B and B − are bounded. Furthermore, we assume there exists a local martingale D ∈ M loc with D, D − > such that the multiplicative special semimartingale Z = DB − is an ELMD for S . Then the following statements are true: (1) Nonnegative arbitrage portfolios, nonnegative arbitrages of the first kind andnonnegative free lunches with vanishing risk do not exist. (2) If D ∈ M , then an admissible free lunch with vanishing risk does not exist. (3) If D is the unique local martingale such that DB − is an ELMD for S , thenthe existence of an admissible arbitrage portfolio (or, more generally, of anadmissible free lunches with vanishing risk) implies that D / ∈ M .Proof. These statements follow from [20, Thm. 7.4 and Thm. 7.6]. (cid:3)
Remark.
By Lemmas 3.5 and 3.6 the condition
D / ∈ M is necessary andessentially also sufficient for the existence of a money market bubble. In the following situation we can show that the condition
D / ∈ M implies theexistence of a money market bubble.3.9. Proposition.
Let W be an R -valued Wiener process, and assume that thefiltration ( F t ) t ∈ [0 ,T ] is the right-continuous, completed filtration generated by W .Suppose that the market is of the form S = { S, } , containing a continuous semi-martingale S > such that S − ∈ M loc with S − / ∈ M and the paths of h S − , S − i are strictly increasing. Then the market S has a bubble. XPLOITING ARBITRAGE REQUIRES SHORT SELLING 9
Proof.
Setting Z := S − , we have Z ∈ M loc . Since Z is nonnegative, it is also asupermartingale, and hence we have Z T ∈ L . We define the martingale M ∈ M as M t := E [ Z T | F t ] , t ∈ [0 , T ] . By the martingale representation theorem (see [10, Thm. III.4.33]) there exist
H, K ∈ L ( W ) such that Z = Z + H · W and M = M + K · W . Since thepaths of h Z, Z i = H · λ are strictly increasing, we have H = 0 up to an evanescentset. We define the predictable process θ := KH .
Since the Wiener process W is continuous, we have H, θH ∈ L ( W ) . Therefore,by the first associativity theorem for vector stochastic integrals with respect tolocal martingales (proven as [25, Thm. 4.6] by using [25, Lemma 4.4]) we have θ ∈ L ( Z ) and K · W = ( θH ) · W = θ · ( H · W ) = θ · Z. Therefore, we have M = M + θ · Z. We define the R -valued predictable process ξ = ( η, θ ) as η := θ · Z − θZ. Introducing the discounted market ¯ S = { , Z } , we have ξ ∈ ∆ sf (¯ S ) and M = M + (1 , Z ) ξ = M + η + θZ. By Lemma 2.6 we have ξ ∈ ∆ sf ( S ) . Furthermore, we obtain M S = M S + ( S, ξ = M S + ηS + θ. Now, we define the R -valued predictable process ϑ as ϑ := ξ + M e By [21, Lemma 7.21] we have ϑ ∈ ∆ sf ( S ) and ( S, ϑ = ( S, ξ + M S. Therefore, we have ( S, ϑ = M S, which means that ( S, ϑt = E [ Z T | F t ] Z t , t ∈ [0 , T ] . Consequently, by Lemma 3.6 the market S has a bubble. (cid:3) Let us mention two well-known examples of a market S = { S, } , where S isgiven as the power of a Bessel process and Proposition 3.9 applies:(1) The process S > is a Bessel process of dimension 3, which is given by thesolution to the SDE dS t = 1 S t dt + dW t . (3.2) Its inverse Z := S − is a strict local martingale, which is a solution to theSDE dZ t = − Z t dW t . (2) The process S > is a squared Bessel process of dimension 4, which isgiven by the solution to the SDE dS t = 4 dt + 2 p S t dW t . Its inverse Z := S − is a strict local martingale, which is a solution to theSDE dZ t = − Z t dW t . Now, we will see an example of a financial market satisfying NA, but admitting anadmissible arbitrage portfolio which can be computed explicitly.3.10.
Example.
Consider the market S = { S, } , where S is a Bessel process ofdimension 3 given by the solution of the SDE (3.2). The existence of arbitrage in thismodel has been pointed out in [4] , and explicit constructions of arbitrage portfoliosare provided in [16] and [24] . Let us have a look at Example 4.6 in [16] , and assumethat S = 1 and T = 1 . Denoting by Φ : R → [0 , the distribution function of thestandard normal distribution N(0 , , we define the function F : [0 , × R → (0 , ∞ ) as F ( t, x ) := Φ( x/ √ − t )Φ(1) , where for t = 1 this expression is understood to be zero. Then, according to [16,Example 4.6] , an arbitrage portfolio S δ is given by the self-financing strategy δ =( θ, η ) defined by the continuous processes θ t := ∂∂x F ( t, S t ) , t ∈ [0 , , (3.3) η := θ · S − θS, (3.4) and we have S δt = F ( t, S t ) − , t ∈ [0 , . In particular, we see that this arbitrage portfolio is admissible; that is, we have S δ ∈ P admsf ( S ) . However, by Proposition 3.7 this arbitrage portfolio cannot be non-negative; that is, we have S δ / ∈ P +sf ( S ) . Moreover, in accordance with Theorem 1.1,the arbitrage portfolio S δ requires short selling; that is, we have S δ / ∈ P δ ≥ ( S ) . Moreprecisely, as (3.3) and (3.4) show, we have a long position θ in the risky primarysecurity account S , and we have a short position η in the savings account on aninterval [0 , ǫ ] for some ǫ > . References [1] D. Coculescu and M. Jeanblanc. Some no-arbitrage rules under short-sales constraints, andapplications to converging asset prices.
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University of Technology Sydney, School of Mathematical and Physical Sciences,Finance Discipline Group, PO Box 123, Broadway, NSW 2007, Australia
Email address : [email protected] Karlsruhe Institute of Technology, Institute of Stochastics, Postfach 6980,76049 Karlsruhe, Germany
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