Asset Price Bubbles in market models with proportional transaction costs
aa r X i v : . [ q -f i n . M F ] J a n Asset Price Bubbles in market models with proportionaltransaction costs
Francesca Biagini ∗ Thomas Reitsam ∗† January 20, 2020
Abstract
We study asset price bubbles in market models with proportional transaction costs λ ∈ ( , ) and finite time horizon T in the setting of [49]. By following [28], we definethe fundamental value F of a risky asset S as the price of a super-replicating portfoliofor a position terminating in one unit of the asset and zero cash. We then obtain adual representation for the fundamental value by using the super-replication theoremof [50]. We say that an asset price has a bubble if its fundamental value differs fromthe ask-price ( + λ ) S . We investigate the impact of transaction costs on asset pricebubbles and show that our model intrinsically includes the birth of a bubble. Keywords: financial bubbles, fundamental value, super-replication, transaction costs, consistentprice systems
Mathematics Subject Classification (2010):
JEL Classification:
G10, C60
In this paper we study financial asset price bubbles in market models with proportionaltransaction costs and finite time horizon. In the economic literature there are severalcontributions discussing the impact of transaction costs on the formation of asset pricebubbles. It is apparent that bubbles may also appear in markets with big transactioncosts, see [5], [22] and also [23], [44], [54] for the speficic case of the real estate market.Several approaches can be found in the literature to explain bubbles, like asymmetricinformation, see [2], [3], heterogenous beliefs, see [27], [53], and noise trading such aspositive feedback activity [15], [55], [58], in combination with limits to arbitrage, see [1],[14], [56], [57]. In [53], the authors include transaction costs in an equilibrium modelwith heterogeneous beliefs and show that small transaction costs may reduce speculative ∗ Workgroup Financial and Insurance Mathematics, Department of Mathematics, Ludwig-MaximiliansUniversit¨at, Theresienstrae 39, 80333 Munich, Germany. Emails: [email protected], [email protected]. † The financial support of the Verein zur Versicherungswissenschaft M¨unchen e.V. is gratefully acknowl-edged marketprice of the asset and its fundamental value . While the market price can be observed,it is less obvious how to define the fundamental value. In the martingale approach, see[12], [31], [37], [45], the fundamental value of a given asset S is given by its expectation offuture cash flows with respect to an equivalent local martingale measure. This approachhas been criticized in [24] for its sensitivity with respect to model’s choice. Anotherapproach defines the fundamental value of an asset by its super-replication prices, see [28],[29]. Other approaches explain in a mathematical model the impact of microeconomicinteractions on asset price formation, see [10], [36]. In [36], the fundamental value isexogenously given and asset price bubbles are endogenously determined by the impact ofliquidity risk. In [10], microeconomic dynamics may at an aggregate level determine ashift in the martingale measure. Further references on asset price bubbles are [7], [8], [9],[30], [34], [35], [52]. For a comprehensive overview see also [47] and the entry “Bubblesand Crashes” of [42].The aim of this paper is to introduce the notion of asset price bubble in market modelswith proportional transaction costs. In [24], the authors suggest a robust definition ofasset price bubbles which can be interpreted as a bubble under proportional transactioncosts. However, to the best of our knowledge a thorough study of this topic is still missingin the literature.In market models with proportional transaction costs λ ∈ ( , ) we distinguish betweenthe ask price ( + λ ) S and the bid price ( − λ ) S for a given asset price S . It is well-knownthat in the frictionless case the no-arbitrage condition no free lunch with vanishing risk(NFLVR) is equivalent to the existence of an equivalent local martingale measure, see [16].In the presence of proportional transaction costs, the existence of consistent (local) pricesystems (Definition 2.1) for each λ > andNLABPs conditions in the robust sense. Roughly speaking, a consistent (local) pricesystem can be thought as a dual market model without transaction costs where the tradinghappens parallel. For a detailed overview of the theory of proportional transaction costs,we refer to the books [38] and [51].Due to the presence of transaction costs, positions in cash and in the asset are asymmetric.By following [28], we define the fundamental value F of a given asset S as the price of asuper-replicating portfolio for a position terminating in one unit of the asset and zero cash.More precisely, we are interested in super-replicating a position in the asset, and not inthe liquidation value of the portfolio. First we study some properties of the fundamental no unbounded profit with bounded risk no local arbitrage with bounded portfolios F for any time t ∈ [ , T ] based on thesuper-replication results from [11] and [50] and show time independence of the consistent(local) price system in the dual representation, see Theorem 3.9. In particular, in Theorem3.13 we prove that the fundamental value admits a c`adl`ag modification. An asset pricebubble is defined as the difference of the ask price with respect to the fundamental value.Introducing a concept of asset price bubble for the bid price is more complex and we referto the discussion in Section 6. For the frictionless case, if the NFLVR condition is satisfiedin the model of [28] one can apply the duality result from [43] and obtain that there is abubble if and only if the price process S is a strict local martingale under all equivalentlocal martingale measures. In particular, if there is at least one equivalent local martingalemeasure such that S is a true martingale, there is no bubble in the market model. In ourmodel there is no bubble in the market model if there exists a consistent price system inthe non-local sense for any λ >
0, see Proposition 3.11.Further, we discuss this theoretical setting in several examples. In particular, the impactof proportional transaction costs is investigated by comparing our model to the frictionlessframework of [28]. It is immediate to see that no bubble in the frictionless market modelmeans no bubble in the analogous market model with transaction costs. On the otherside, if there is a bubble in the market model with proportional transaction costs, thereis also a bubble in the frictionless market model. However, if there is a bubble in thefrictionless market model, the introduction of transaction costs can possibly eliminate theasset price bubble. Finally, we note that our definition of asset price bubble intrinsicallyincludes bubbles’ birth.The paper is organized as follows. In Section 2, we outline the setting for market modelswith proportional transaction costs and extend the notion of admissible strategies. InSection 3, we introduce the definition of the fundamental value and of asset price bubbles,and establish a dual representation for the fundamental value. Further, we prove the mainresults, Theorem 3.9 and Theorem 3.13. In Section 4, we illustrate our results throughconcrete examples. In Section 5, the impact of proportional transaction costs on bubbles’formation and size is investigated. In Section 6, we briefly discuss bubbles for the bid price.In the Appendix, we state the super-replication results from [50] with small modifications.
Let T > ( Ω , F , (F t ) ≤ t ≤ T , P ) be a filtered probabilityspace where the filtration F ∶ = (F t ) ≤ t ≤ T satisfies the usual conditions of right-continuityand saturatedness, with F = { ∅ , Ω } and F T = F . We consider a financial market modelconsisting of a risk-free asset B , normalized to B ≡
1, and a risky asset S . Throughout thepaper we assume that S = ( S t ) ≤ t ≤ T is an F -adapted stochastic process, with c`adl`ag andpositive paths. For trading the risky asset in the market model, proportional transactioncosts 0 < λ < S at time t the trader has to pay ( + λ ) S t and for selling one share of S at time t the trader receives ( − λ ) S t . The interval [( − λ ) S t , ( + λ ) S t ] is called bid-ask-spread . Further, we assume that S t ∈ L + (F t , P ) forall t ∈ [ , T ] . This assumption is needed in the proof of Lemma 3.6 and thus also for themain result, Theorem 3.9. 3 efinition 2.1. For λ > ≤ σ < τ ≤ T , we call CPS ( σ, τ ) (resp.CPS loc ( σ, τ ) ) the family of pairs ( Q , ˜ S Q ) such that Q is a probability measure on F τ , Q ∼ P ∣ F τ , ˜ S Q is a martingale (resp. local martingale) under Q on ⟦ σ, τ ⟧ , and ( − λ ) S t ≤ ˜ S Q t ≤ ( + λ ) S t , for σ ≤ t ≤ τ. (2.1)A pair ( Q , ˜ S Q ) in CPS ( σ, τ ) (resp. CPS loc ( σ, τ ) ) is called consistent price system (resp. consistent local price system ). By Q( σ, T ) (resp. Q loc ( σ, T ) ) we denote the set of measures Q such that there exists a pair ( Q , ˜ S Q ) ∈ CPS ( σ, T ) (resp. ( Q , ˜ S Q ) ∈ CPS loc ( σ, T ) ). Fur-ther, we write L p (F σ , Q) ∶ = ⋂ Q ∈Q( σ,T ) L p (F σ , Q ) and L p (F σ , Q loc ) ∶ = ⋂ Q ∈Q loc ( σ,T ) L p (F σ , Q ) .By L p + (F σ , Q loc ) (resp. L p + (F σ , Q) ) we denote the space of [ , ∞) -valued random variables X ∈ L p (F σ , Q loc ) (resp. X ∈ L p (F σ , Q) ).A consistent (local) price system can be thought as a frictionless market with betterconditions for traders, see [25]. The existence of a λ -consistent (local) price system forevery 0 < λ < Assumption 1.
We assume that S admits a consistent local price system for every 0 < λ ′ ≤ λ .In the sequel we will sometimes need a stronger assumption, namely, the existence ofconsistent price systems (in the non-local sense) for every 0 < λ ′ ≤ λ . Assumption 2.
We assume that S admits a consistent price system for every 0 < λ ′ ≤ λ . Remark 2.2.
We first note for every ( Q , ˜ S Q ) ∈ CPS loc ( , T ) , then ˜ S Q t ∈ L + (F t , Q ) because ˜ S Q is a Q -supermartingale.Furthermore, Assumption 1 guarantees that for any t ∈ [ , T ] , S t ∈ L (F t , Q loc ) , as (2.1) implies S t ≤ − λ ˜ S Q t , ∀ t ∈ [ , T ] , for any ( Q , ˜ S Q ) ∈ CPS loc ( , T ) . By following [26], [39], for λ > K λt the solvency cone at time t , defined as K λt = cone {( + λ ) S t e − e , − e + ( − λ ) S t e } , (2.2)4here e = ( , ) , e = ( , ) are the unit vectors in R , and by (− K λt ) ○ the correspondingpolar cone, given by (− K λt ) ○ = {( w , w ) ∈ R + ∣ ( − λ ) S t ≤ w w ≤ ( + λ ) S t } = { w ∈ R ∣ ⟨ x, w ⟩ ≤ , ∀ x ∈ (− K λt )} . (2.3) Definition 2.3.
We define Z( σ, τ ) (resp. Z loc ( σ, τ ) ) as the set of processes Z = ( Z t , Z t ) σ ≤ t ≤ τ such that Z is a P -martingale on ⟦ σ, τ ⟧ and Z is a P -martingale (resp. local P -martingale) on ⟦ σ, τ ⟧ and such that Z t ∈ (− K λt ) ○ /{ } a.s. for all t ∈ ⟦ σ, τ ⟧ .The following proposition from [26] provides a convenient representation of consistent(local) price systems by elements in Z (resp. Z loc ) and follows directly from the definitionof (− K λt ) ○ in (2.3). Proposition 2.4 (Proposition 3, [26]) . Let Z = ( Z t , Z t ) σ ≤ t ≤ T be a -dimensional stochas-tic process with Z τ ∈ L (F τ , P ) . Define the measure Q ( Z ) ≪ P by d Q ( Z )/ d P ∶ = Z τ / E [ Z τ ] .Then Z ∈ Z( σ, τ ) (resp. Z ∈ Z loc ( σ, τ ) ) if and only if ( Q ( Z ) , ( Z / Z )) is a consistentprice system (resp. consistent local price system) on ⟦ σ, τ ⟧ . Next, we introduce the notion of self-financing strategies and admissibility , by extendingDefinition 3 and 5 of [49] to a general starting value.
Definition 2.5.
Let 0 < λ <
1. A self-financing trading strategy starting with initial en-dowment X σ ∈ L + (F σ , P ) is a pair of F -predictable finite variation processes ( ϕ t , ϕ t ) σ ≤ t ≤ T on ⟦ σ, T ⟧ such that(i) ϕ σ = X σ and ϕ σ = ϕ t = ϕ σ + ϕ , ↑ t − ϕ , ↓ t and ϕ t = ϕ , ↑ t − ϕ , ↓ t , the Jordan-Hahn decompositionof ϕ and ϕ into the difference of two non-decreasing processes, starting at ϕ , ↑ σ = ϕ , ↓ σ = ϕ , ↑ σ = ϕ , ↓ σ =
0, these processes satisfyd ϕ t ≤ ( − λ ) S t d ϕ , ↓ t − ( + λ ) S t d ϕ , ↑ t , σ ≤ t ≤ T. (2.4) Definition 2.6.
Let 0 < λ < X σ ∈ L + (F σ , Q loc ) . Then a self-financing trading strategy ϕ = ( ϕ , ϕ ) is called admissible in a num´eraire-based sense on ⟦ σ, T ⟧ with ϕ σ = X σ if there is M σ ∈ L + (F σ , Q loc ) such that the liquidation value V liqτ satisfies V liqτ ( ϕ , ϕ ) ∶ = ϕ τ + ( ϕ τ ) + ( − λ ) S τ − ( ϕ τ ) − ( + λ ) S τ ≥ − M σ , (2.5)for all ⟦ σ, T ⟧ -valued stopping times τ .(ii) Let X σ ∈ L + (F σ , Q) . Then a self-financing trading strategy ϕ = ( ϕ , ϕ ) is called admissible in a num´eraire-free sense on ⟦ σ, T ⟧ with ϕ σ = X σ if there is ( M σ , M σ ) ∈ L + (F σ , Q) × L ∞+ (F σ , Q) such that V liqτ ( ϕ , ϕ ) ∶ = ϕ τ + ( ϕ τ ) + ( − λ ) S τ − ( ϕ τ ) − ( + λ ) S τ ≥ − M σ − M σ S τ , (2.6)for all ⟦ σ, T ⟧ -valued stopping times τ . 5e denote by V M ( X σ , σ, T, λ ) (resp. V loc M ( X σ , σ, T, λ ) ) the set of all such trading strategiesin the num´eraire-free sense (resp. num´eraire-based sense) ϕ on the interval ⟦ σ, T ⟧ . We alsouse the notation V σ,T ( X σ , λ ) = ⋃ M V M ( X σ , σ, T, λ ) (resp. V loc X σ ,σ,T ( λ ) = ⋃ M V loc M ( X σ , σ, T, λ ) ).For more details on the differential form of (2.4) we refer the interested reader to [49].Note that both accounts, the holdings in the bond ϕ as well as the holdings in the asset ϕ are separately given in the definition of a trading strategy ϕ . Having an inequalityin (2.4) allows for “throwing money away”, see [49]. As it is explained in [49] we couldrequire equality in (2.4) in order to express ϕ in terms of ϕ . However, for our approachit is more convenient to specify both accounts separately. Remark 2.7.
We now discuss the definition of admissible strategies. Since we are in-terested in considering strategies on a random interval with non-zero initial endowment,we need to extend Definitions 3 and 5 of [49], as we now explain for the num´eraire-basedcase. The argument for the num´eraire-free setting is analogous. In a first step we considerthe case of zero initial endowments. Assume that ϕ σ = ( , ) and V liqτ ( ϕ ) ≥ − M for all ⟦ σ, T ⟧ -valued stopping times τ and a constant M > . Then ϕ corresponds to a admissiblestrategy ψ on [ , T ] according to Definition 3 of [49], where ( ψ , ψ ) ≡ ( , ) on ⟦ , σ ⟧ and ψ t = ϕ t for all t ≥ σ . Conversely, any strategy ψ on [ , T ] with ( ψ , ψ ) ≡ ( , ) on ⟦ , σ ⟧ , which is admissible in a num´eraire-based sense in the sense of [49], also satisfiesDefinition 2.6. Suppose now to have a non-zero initial endowment. By translation, anyadmissible strategy on [ , T ] with initial endowments corresponds to an admissible strategyon [ , T ] without initial endowment. This correspondence is more delicate for strategies on ⟦ σ, T ⟧ . Let ϕ σ = ( X σ , ) for some X σ ∈ L + (F σ , Q loc ) with V liqτ ( ϕ ) ≥ − M and define ˜ ϕ t = ( ˜ ϕ t , ˜ ϕ t ) ∶ = ( ϕ t − X σ , ϕ t ) for all σ ≤ t ≤ T . Then V liqτ ( ˜ ϕ ) = V liqτ ( ϕ ) − X σ ≥ − M − X σ = ∶ − M σ .Hence, it is not enough to bound the liquidation value of a strategy by a constant in orderto have a one-to-one correspondence of admissible strategies with and without endowmentson ⟦ σ, T ⟧ . Definition 2.6 allows to obtain from any admissible strategy ψ on [ , T ] anadmissible strategy ϕ ∶ = ψ ∣ ⟦ σ,T ⟧ on ⟦ σ, T ⟧ . Note that in the case of σ = Definition 2.6 andDefinition 3 of [49] coincide.When we consider the definition of admissibility in a num´eraire-based sense on [ , T ] froman economical perspective, the role of M > is to hedge the portfolio by M units of thebond, see [49]. In particular, when we superhedge a portfolio on ⟦ σ, T ⟧ , it seems reason-able to use the information which are available up to time σ , namely, to superhedge theportfolio by M σ units of the bond, where M σ ∈ L + (F σ , Q loc ) . We now comment on the integrability conditions of the lower bound M σ . Remark 2.8.
We discuss the local and the non-local case separately. In Definition 3of [49] the liquidation value of an admissible strategy in the num´eraire-based sense ϕ = ( ϕ t , ϕ t ) t ∈ [ ,T ] is required to be lower bounded by a constant. This guarantees that ( ϕ t + ϕ t ˜ S Q t ) t ∈ [ ,T ] is an optional strong Q -supermartingale for all ( Q , ˜ S Q ) ∈ CPS loc ( , T ) , seeProposition 2 of [49].As explained in Remark 2.7 we wish to extend the definitions of [49] to include admissiblestrategies on an arbitrary interval with arbitrary initial endowment. To this propose weneed to impose condition (2.5) . However, we still obtain an arbitrage-free market model. n the proof of Proposition 2 of [49] the lower bound is used to apply Proposition 3.3 of [4],respectively Theorem 1 of [60]. The conditions of these results are still fulfilled on ⟦ σ, T ⟧ if (2.5) holds, and thus ( ϕ t + ϕ t ˜ S Q t ) σ ≤ t ≤ T is an optional strong Q -supermartingale for all ( Q , ˜ S Q ) ∈ CPS loc ( σ, T ) .In the non-local case, Definition 5 of [49] requires that the liquidation value of an admissiblestrategy in the num´eraire-free sense ϕ = ( ϕ t , ϕ t ) t ∈ [ ,T ] satisfies V τ ( ϕ , ϕ ) ≥ − M ( + S τ ) , (2.7) for all [ , T ] -valued stopping times τ . This guarantees that ( ϕ t + ϕ t ˜ S Q t ) t ∈ [ ,T ] is an optionalstrong Q -supermartingale for all ( Q , ˜ S Q ) ∈ CPS ( , T ) , see Proposition 3 of [49]. Followingthe proof of Proposition 3 of [49], we apply the following conditional version of Fatou’slemma. Let ( X n ) n ∈ N be a sequence of real-valued random variables on ( Ω , F , Q ) convergingalmost surely to X and such that the negative parts ( X − n ) n ∈ N are uniformly Q -integrable.Then E Q [ lim inf n →∞ X n ∣ G ] ≤ lim inf n →∞ E Q [ X n ∣ G ] . In our case, the family {( ϕ τ + ϕ τ ˜ S Q τ ) − ∶ σ ≤ τ ≤ T } is uniformly Q -integrable with respectto Q for all ( Q , ˜ S Q ) ∈ CPS ( σ, T ) , as we have for σ ≤ τ ≤ Tϕ τ + ϕ τ ˜ S Q τ ≥ V liqτ ( ϕ , ϕ ) ≥ − M σ − M σ S τ , because S τ ≤ − λ ˜ S Q τ for any ( Q , ˜ S Q ) ∈ CPS ( , T ) and ˜ S Q is a Q -martingale, and ( M σ , M σ ) ∈ L + ( F σ , Q ) × L ∞+ ( F σ , Q ) by assumption. Therefore, ( ϕ t + ϕ t ˜ S Q t ) σ ≤ t ≤ T is an optional strong Q -supermartingale on ⟦ σ, T ⟧ for all ( Q , ˜ S Q ) ∈ CPS ( σ, T ) and all trading strategies ϕ = ( ϕ t , ϕ t ) σ ≤ t ≤ T are admissible in a num´eraire-free sense. The notion of an asset price bubble consists of two components, namely, the market priceof an asset and its fundamental value . We assume that the market price is given by theprice process S . For the fundamental value of an asset, we here follow the approach of[28] and define the fundamental value by means of the super-replication price of the asset.In frictionless market models, it is equivalent to hold the asset or to have the (market)value of the asset in the money market account. This symmetry fails in the presenceof transaction costs. A trader who wants to buy a share of the asset at time t ∈ [ , T ] has to pay ( + λ ) S t . A trader who wants to liquidate her position in the asset at time t ∈ [ , T ] only receives ( − λ ) S t per share of the asset. Therefore, a natural question arises.Which position should we super-replicate in order to obtain a reasonable definition of thefundamental value in the presence of transaction costs? Definition 3.1.
The fundamental value F = ( F t ) t ∈ [ ,T ] of an asset S at time t ∈ [ , T ] ina market model with proportional transaction costs 0 < λ < F t ∶ = ess inf { X t ∈ L + ( F t , Q loc ) ∶ ∃ ϕ ∈ V loc t,T ( X t , λ ) with ϕ t = ( X t , ) and ϕ T = ( , )} .
7e say there is an asset price bubble in the market model with transaction costs if P ( F σ < ( + λ ) S σ ) > σ with values in [ , T ] . We define the asset pricebubble as the process β = ( β t ) ≤ t ≤ T given by β t ∶ = ( + λ ) S t − F t , t ∈ [ , T ] . (3.1) Remark 3.2.
In Definition 4.2 of [24], the authors provide a robust definition of an as-set price bubble, which can also be interpreted as a bubble under proportional transactioncosts. A difference with respect to Definition 3.1 lies in the chosen specification of tradingstrategies. In [24], in the worst case scenario the strategy begins in cash, but the initialcapital is all in stock, or analogously, the strategy ends in cash, but the trader has to deliverone share of the asset.Specifying both components of the trading strategies in our model allows to consider strate-gies starting in cash and ending in a position in the stock only.
Proposition 3.3.
Under Assumption 1, we have that the fundamental value F = ( F t ) t ∈ [ ,T ] is such that F t ≤ ( + λ ) S t , t ∈ [ , T ] , and F t ∈ L ( F t , Q loc ) , t ∈ [ , T ] . Moreover, the bubble β = ( β t ) t ∈ [ ,T ] has almost surelynon-negative paths.Proof. Consider the buy and hold strategy starting at time t ∈ [ , T ] . With an initialendowment ϕ t = (( + λ ) S t , ) it is possible to buy one share of the asset at time t andkeep it until time T . Then F t ≤ ( + λ ) S t for all t ∈ [ , T ] . Therefore, the bubble has almostsurely non-negative paths. The fact that F σ ∈ L + ( F σ , Q loc ) follows by Remark 2.2.We now comment on Definition 3.1, which could be interpreted as the fundamentalvalue for the ask price . Alternatively, we could consider to super-replicate the position ϕ T = (( − λ ) S T , ) which is the liquidation value of the asset S at time T , instead, or ϕ T = (( + λ ) S T , ) which is the price one has to pay to buy the asset at time T . Atrader who wants to super-replicate (( − λ ) S T , ) is only interested in cash, namely, inthe liquidation value of the asset. However, it is not possible to re-buy at T a share ofthe asset at price ( − λ ) S T . On the other hand, a trader who wants to super-replicate (( + λ ) S T , ) is actually interested in having the asset at T in the portfolio. So, super-replicating (( + λ ) S T , ) might be too expensive. Therefore, we consider the position ϕ T = ( , ) and its corresponding super-replication price as fundamental value. This cor-responds to the price a trader is willing to pay if she had to hold the asset in her portfoliountil the terminal time T , see [32]. As in [28] this definition allows bubble birth, see [7].We give an explicit example in Example 4.7.In a frictionless market model there are well-known super-replication theorems which es-tablish a dual representation, see e.g. [19], [43]. Analogously there are super-replicationtheorems for market models with proportional transaction costs to obtain a dual represen-tation, see e.g. [13], [39], [41], [40]. We refer to the super-replication theorems of [11] and[50]. The formulations of Theorem 1.4 and Theorem 1.5 of [50] can be found in AppendixA. 8 roposition 3.4. Let Assumption 1 hold. We consider an F T -measurable contingentclaim X T = ( X T , X T ) which pays X T many units of the bond and X T many units of therisky asset at time T . Let X σ ∈ L + ( F σ , Q loc ) . If X T − X σ + ( X T ) + ( − λ ) S T − ( X T ) − ( + λ ) S T ≥ − M σ , (3.2) for some F σ -measurable random variable M σ satisfying sup Q ∈Q loc E Q [ M σ ] < ∞ , then thefollowing assertions are equivalent(i) There is a self-financing trading strategy ϕ on ⟦ σ, T ⟧ with ϕ σ = ( X σ , ) and ϕ T = ( X T , X T ) which is admissible in a num´eraire-based sense.(ii) For every ( Q , ˜ S Q ) ∈ CPS loc ( σ, T ) we have E Q [ X T − X σ + X T ˜ S Q T ∣ F σ ] ≤ . (3.3) Proof. ( i ) ⇒ ( ii ) It is possible to apply Proposition 2 of [49] although we consider theinterval ⟦ σ, T ⟧ and initial endowment ϕ σ = ( X σ , ) , see Remark 2.8. Then ( ϕ t + ϕ t ˜ S Q t ) σ ≤ t ≤ T is an optional strong supermartingale and thus E Q [ X T − X σ + X T ˜ S Q T ∣ F σ ] = E Q [ ϕ T − X σ + ϕ T ˜ S Q T ∣ F σ ] ≤ ϕ σ − X σ + ϕ σ ˜ S Q σ = . ( ii ) ⇒ ( i ) For ˜ X T ∶ = ( X T − X σ + M σ − sup Q ∈Q E Q [ M σ ] , X T ) , we have ˜ X T + ( ˜ X T ) + ( − λ ) S T − ( ˜ X T ) − ( + λ ) S T ≥ − sup Q ∈Q E Q [ M σ ] , (3.4)by equation (3.2), and for ( Q , ˜ S Q ) ∈ CPS ( σ, T ) we get E Q [ ˜ X T + ˜ X T ˜ S Q T ] = E Q [ X T − X σ + X T ˜ S Q T ] + E Q [ M σ ] − sup Q ∈Q E Q [ M σ ] ≤ , (3.5)by equation (3.3). Thus we can apply Theorem A.1 (Theorem 1.4 of [50]) which yieldsa strategy ˜ ϕ with ˜ ϕ ≡ ⟦ , σ ⟧ and ˜ ϕ T = ˜ X T which is admissible in a num´eraire-basedsense on [ , T ] . In particular, ϕ = ( ϕ , ϕ ) defined by ϕ t ∶ = ˜ ϕ t + X σ − M σ + sup Q ∈Q E Q [ M σ ] and ϕ t ∶ = ˜ ϕ t for σ ≤ t ≤ T , is an admissible strategy in the num´eraire-based sense on ⟦ σ, T ⟧ according to Definition 2.6.From Proposition 3.4 we obtain a duality representation for the fundamental value. Proposition 3.5.
Let Assumption 1 hold. Then the fundamental value F = ( F t ) t ∈ [ ,T ] ofthe asset S at time t ∈ [ , T ] is given by F t = ess sup ( Q , ˜ S Q ) ∈ CPS loc ( t,T,λ ) E Q [ ˜ S Q T ∣ F t ] , (3.6) for t ∈ [ , T ] . roof. For X T = ( , ) and X t ∈ L + ( F t , Q loc ) , condition (3.2) is satisfied and we get byProposition 3.4 that { X t ∈ L + ( F t , Q loc ) ∶ ∃ ϕ ∈ V loc t,T ( X t , λ ) with ϕ t = ( X t , ) and ϕ T = ( , )} = { X t ∈ L + ( F t , Q loc ) ∶ E Q [ ˜ S Q T ∣ F t ] ≤ X t , for all ( Q , ˜ S Q ) ∈ CPS loc ( t, T, λ )} = ∶ D t . (3.7)By Definition 3.1 and (3.7) we have that F t = ess inf D t , t ∈ [ , T ] . It is left to show that ess inf D t = ess sup ( Q , ˜ S Q ) ∈ CPS loc ( t,T,λ ) E Q [ ˜ S Q T ∣ F t ] . (3.8)For the first direction “ ≤ ” we note that ess sup ( Q , ˜ S Q ) ∈ CPS loc ( t,T,λ ) E Q [ ˜ S Q T ∣ F t ] ∈ D t , wherewe used that ess sup ( Q , ˜ S Q ) ∈ CPS loc ( t,T,λ ) E Q [ ˜ S Q T ∣ F t ] ≤ ( + λ ) S t ∈ L ( F t , Q loc ) .For the reverse direction “ ≥ ” we have that ess inf D t ≥ E Q [ ˜ S Q T ∣ F t ] for all ( Q , ˜ S Q ) ∈ CPS loc ( t, T, λ ) which implies by the definition of the essential supremum thatess inf D t ≥ ess sup ( Q , ˜ S Q ) ∈ CPS loc ( t,T,λ ) E Q [ ˜ S Q T ∣ F t ] . Note that in the above proof t ∈ [ , T ] can be replaced by a stopping time 0 ≤ σ ≤ T .In Proposition 3.5 the essential supremum is taken over the set CPS loc ( t, T, λ ) whichdepends on the initial time t . In contrast, if we consider the frictionless case of [28] andassume that Theorem 3.2 from [43] applies, the fundamental value S ∗ σ of an asset S attime σ is given by S ∗ σ = ess sup Q ∈M loc ( S ) E Q [ S T ∣ F σ ] , where M loc ( S ) denotes the set of equivalent local martingale measures for S . The essentialsupremum is taken over all equivalent local martingale measure of S , independently of theinitial time σ . We now show that a similar independence property also holds for thefundamental value under transaction costs, see Theorem 3.9. In order to prove it, we needsome preliminary results. We start with a local version of Lemma 6 and Corollary 3 of[26]. Lemma 3.6.
Let Assumption 1 hold. For each stopping time ≤ σ ≤ T and each randomvariable f ∈ L ( F σ , P ) such that ( − λ ) S σ < f < ( + λ ) S σ , (3.9) and for each ¯ λ > λ there is an ¯ λ -consistent local price system ( ˇ Q , ˇ S ) ∈ CPS loc ( , T, ¯ λ ) with ˇ S σ = f . roof. The proof is partially based on the proof of Lemma 6 of [26]. Consider the sequenceof stopping time ( τ n ) n ∈ N , where τ n ( ω ) ∶ = inf { t ≥ ∣ S t ( ω ) ≥ n } ∧ T. Note that ( τ n ) n ∈ N defines a localizing sequence for all λ -consistent local price systems asfor ( Q , ˜ S Q ) ∈ CPS ( , T, λ ) we have˜ S Q t ≤ ( + λ ) S t ≤ ( + λ ) n, (3.10)for all 0 ≤ t < τ n and hence Proposition 6.1 of [50] can be applied. Further, it holds that τ n ↑ T a.s. Fix ¯ λ > λ . First we consider the interval ⟦ , σ ⟧ . Choose δ ≤ λ such that δ + ( + δ )( λ + δ )/( − δ ) < ¯ λ . By Assumption 1 there exists ( Q ( δ ) , ˜ S ( δ )) ∈ CPS loc ( , σ, δ ) a δ -consistent local price system on the interval ⟦ , σ ⟧ . We have1 − δ ≤ ˜ S τ n ∧ σ ( δ ) S τ n ∧ σ ≤ + δ. (3.11)For n ∈ N define f n ∶ = ⎧⎪⎪⎨⎪⎪⎩ f on { τ n ≥ σ } , ˜ S ( δ ) τ n on { τ n < σ } . Hence by (3.10) we get f n ∈ L ( F τ n ∧ σ , P ) , and for h n ∶ = f n / S τ n ∧ σ we have1 − λ < h n < + λ, and ∣ ˜ S τ n ∧ σ ( δ ) − f n ∣ < ( λ + δ ) S τ n ∧ σ ≤ λ + δ − δ ˜ S τ n ∧ σ ( δ ) . This implies that f n ∈ L ( F τ n ∧ σ , Q ( δ )) as well as f ∈ L ( F σ , Q ( δ )) by (3.9) and the factthat f ≤ ( + λ ) S σ ≤ + λ − λ ˜ S σ ( δ ) . Consequently, for ¯ S nρ ∶ = E Q ( δ ) [ f n ∣ F ρ ] and a stopping time ρ with 0 ≤ ρ ≤ ( τ n ∧ σ ) , ∣ E Q ( δ ) [ f n ∣ F ρ ] − E Q ( δ ) [ ˜ S τ n ∧ σ ( δ ) ∣ F ρ ]∣ < ˜ S ρ ( δ ) λ + δ − δ ≤ S ρ ( λ + δ )( + δ ) − δ , thus using (3.11) we get ( − ¯ λ ) S ρ < ¯ S nρ < ( + ¯ λ ) S ρ . (3.12)We show that ¯ S nρ converges almost surely to a random variable ¯ S Q ( δ ) ρ for all 0 ≤ ρ ≤ σ . Wehave that E Q ( δ ) [ f n ∣ F ρ ] = E Q ( δ ) [ { τ n ≥ σ } f ∣ F ρ ] + E Q ( δ ) [ { τ n < σ } ˜ S τ n ( δ ) ∣ F ρ ] . The main difference with respect to the proof of Lemma 6 of [26] is that we cannot use the martingaleproperty of consistent price systems as in [26], because we are now in the local setting. Hence we needsome further technicalities.
11y the Theorem of Monotone Convergence it follows that E Q ( δ ) [ { τ n ≥ σ } f ∣ F ρ ] a.s. Ð → E Q ( δ ) [ f ∣ F ρ ] . (3.13)For the second term we have E Q ( δ ) [ { τ n < σ } ˜ S τ n ( δ ) ∣ F ρ ] = E Q ( δ ) [ ˜ S τ n ∧ σ ( δ ) ∣ F ρ ] − E Q ( δ ) [ { τ n ≥ σ } ˜ S τ n ∧ σ ( δ ) ∣ F ρ ] = ˜ S τ n ∧ ρ ( δ ) − E Q ( δ ) [ { τ n ≥ σ } ˜ S σ ( δ ) ∣ F ρ ] . Since { τ n ≥ σ } ≤ { τ n + ≥ σ } for all n ∈ N , we can apply the Theorem of Monotone Convergenceto conclude ¯ S nρ a.s. Ð→ E Q ( δ ) [ f ∣ F ρ ] + ˜ S ρ ( δ ) − E Q ( δ ) [ ˜ S σ ( δ ) ∣ F ρ ] = ∶ ¯ S Q ( δ ) ρ . (3.14)We define the process ¯ S Q ( δ ) = ( ¯ S Q ( δ ) t ) ≤ t ≤ σ by (3.14). Since ( E Q ( δ ) [ f ∣ F t ] − E Q ( δ ) [ ˜ S σ ( δ ) ∣ F t ]) ≤ t ≤ σ is a Q ( δ ) -martingale, it admits a unique c`adl`ag modification. Further, ˜ S ( δ ) has a uniquec`adl`ag modification. Therefore, ¯ S Q ( δ ) admits a unique c`adl`ag modification as a local Q ( δ ) -martingale. By (3.12) ¯ S Q ( δ ) lies in the bid-ask spread for ¯ λ by construction. Thus ( Q ( δ ) , ¯ S Q ( δ ) ) is a ¯ λ -consistent local price system on ⟦ , σ ⟧ satisfying ¯ S Q ( δ ) σ = f . With thesame construction as in the proof of Lemma 6 of [26] we can now show the existence of aconsistent local price system ( ̂ Q , ̂ S ) ∈ CPS loc ( σ, T, ¯ λ ) such that ̂ S σ = f . We refer to [48]for further details. We use this result to extend ( Q ( δ ) , ¯ S Q ( δ ) ) to a consistent local pricesystem ( ˇ Q , ˇ S ) ∈ CPS loc ( , T, ¯ λ ) on the entire interval [ , T ] .We now define ( ˇ Q , ˇ S ) ∈ CPS loc ( , T, ¯ λ )) which satisfies ˇ S σ = f . Set ˇ Q byd ˇ Q d P ∶ = d Q ( δ ) d P E P [ d ̂ Q d P ∣ F σ ] d ̂ Q d P , and ˇ S t ∶ = ⎧⎪⎪⎨⎪⎪⎩ ¯ S Q ( δ ) t , for 0 ≤ t ≤ σ ̂ S t , for σ ≤ t ≤ T. Then ( ˇ Q , ˇ S ) ∈ CPS loc ( , T, ¯ λ ) and ˇ S σ = ̂ S σ = ¯ S Q ( δ ) σ = f . Remark 3.7.
Note that in the case of a consistent price system in the non-local sense,Lemma 3.6 coincides with Lemma 6 of [26].
The following Corollary 3.8 can be proved in the same way as Corollary 3 in [26] becausethe construction does not use the martingale property of the consistent price systems.
Corollary 3.8.
Let Assumption 1 hold. For any stopping time ≤ σ ≤ T , probabilitymeasure Q ∼ P ∣ F σ on F σ and random variable f ∈ L ( F σ , P ) with ( − λ ) S σ ≤ f ≤ ( + λ ) S σ , there exists a λ -consistent local price system ( Q , ˜ S Q ) ∈ CPS loc ( σ, T, λ ) such that ˜ S Q σ = f and Q ∣ F σ = Q .
12e can now show time independence for the essential supremum in the definition of thefundamental value.
Theorem 3.9.
Under Assumption 1 the following identity holds: ess sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T ) E Q [ ˜ S Q T ∣ F σ ] = ess sup ( Q , ˜ S Q ) ∈ CPS loc ( ,T ) E Q [ ˜ S Q T ∣ F σ ] . Proof. ) If ( Q , ˜ S Q ) ∈ CPS loc ( , T ) , then ( Q , ˜ S Q ∣ ⟦ σ,T ⟧ ) ∈ CPS loc ( σ, T ) . Thus, CPS loc ( , T ) ⊆ CPS loc ( σ, T ) we immediately get thatess sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T ) E Q [ ˜ S Q T ∣ F σ ] ≥ ess sup ( Q , ˜ S Q ) ∈ CPS loc ( ,T ) E Q [ ˜ S Q T ∣ F σ ] . ) Let λ n ∶ = λ + n for n ∈ N . Note that there is n ∈ N such that λ n <
1. Fix ( Q , ˜ S Q ) ∈ CPS loc ( σ, T, λ ) with ˜ Z , ˜ Z be the associated (local) P -martingales as in Proposition 2.4.Let n ≥ n . By Lemma 3.6 and Corollary 3.8 there exists ( Q , ¯ S ¯ Q ) ∈ CPS loc ( , T, λ n ) suchthat ¯ S nσ = ˜ S Q σ . Let ¯ Z , ¯ Z be the associated (local) P -martingales as in Proposition 2.4.We define a λ n -consistent local price system ( ̂ Q , ̂ S ̂ Q ) ∈ CPS loc ( , T, λ n ) by ̂ Z it ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ¯ Z it , ≤ t ≤ σ, ˜ Z it ¯ Z iσ ˜ Z iσ , σ ≤ t ≤ T, for i = ,
2. By construction it then holds that ̂ S t = ˜ S Q t for all t ≥ σ and E Q [ ˜ S Q T ∣ F σ ] = E P [ ˜ Z T ∣ F σ ] ( ˜ Z σ ) − = E P [ ˜ Z T ¯ Z σ ˜ Z σ ∣ F σ ] ( ˜ Z σ ) − ̂ Z σ ̂ Z σ ˜ Z σ ¯ Z σ = E P [ ̂ Z T ∣ F σ ] Z σ ¯ Z σ ¯ Z σ ˜ Z σ ¯ Z σ = E P [ ̂ Z T ∣ F σ ] ( ¯ Z σ ) − ˜ S Q σ ¯ S ¯ Q σ = E P [ ̂ Z T ∣ F σ ] ( ¯ Z σ ) − = E ̂ Q [ ̂ S ̂ Q T ∣ F σ ] . Since ( Q , ˜ S Q ) ∈ CPS loc ( σ, T, λ ) and n ∈ N are arbitrary, we can conclude that for all n ∈ N F n ∶ = ess sup ( ̂ Q , ̂ S ̂ Q ) ∈ CPS loc ( ,T,λ n ) E ̂ Q [ ̂ S ̂ Q T ∣ F σ ] ≥ ess sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T,λ ) E Q [ ˜ S Q T ∣ F σ ] . (3.15)In particular, as the right hand side does not depend on n ∈ N we getlim n →∞ F n ≥ ess sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T,λ ) E Q [ ˜ S Q T ∣ F σ ] . (3.16)The limit is well defined because the sequence is monotonically decreasing and is lowerdominated by (3.15), as CPS loc ( σ, T, λ n ) ⊇ CPS loc ( σ, T, λ n + ) ⊇ CPS loc ( σ, T, λ ) . Then F ∶ = lim n →∞ F n is of the form F = ess sup ( ̂ Q , ̂ S ̂ Q ) ∈⋂ n ∈ N CPS loc ( ,T,λ n ) E ̂ Q [ ̂ S ̂ Q T ∣ F σ ] . (3.17)13t is left to show that ⋂ n ∈ N CPS loc ( , T, λ n ) = CPS loc ( , T, λ ) . The first implication fol-lows directly because CPS loc ( , T, λ ) ⊆ CPS loc ( , T, λ n ) for all n ∈ N . Hence we getCPS loc ( , T, λ ) ⊆ ⋂ n ∈ N CPS loc ( , T, λ n ) . For the reverse implication let ( ̂ Q , ̂ S ̂ Q ) ∈ ⋂ n ∈ N CPS loc ( , T, λ n ) . By definition we have for each t ∈ [ , T ]( − ( λ + n )) S t ≤ ̂ S ̂ Q t ≤ ( + λ + n ) S t , ∀ n ∈ N . However, this can only be true if ( ̂ Q , ̂ S ̂ Q ) ∈ CPS ( , T, λ ) which yields the reverse implica-tion. Putting (3.16) and (3.17) together we obtainess sup ( ̂ Q , ̂ S ̂ Q ) ∈ CPS loc ( ,T,λ ) E ̂ Q [ ̂ S ̂ Q T ∣ F σ ] = F ≥ ess sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T,λ ) E Q [ ˜ S Q T ∣ F σ ] . (3.18) In this section we study some basic properties of the fundamental value and of asset pricebubbles in our setting.
Lemma 3.10.
The process ˜ F = ( ˜ F t ) t ∈ [ ,T ] defined by ˜ F t ∶ = ess sup ( Q , ˜ S Q ) ∈ CPS ( ,T,λ ) E Q [ ˜ S Q T ∣ F t ] , t ∈ [ , T ] , (3.19) is unique to within evanescent processes.Proof. Since Proposition 3.4 and Theorem 3.9 hold for all stopping times 0 ≤ σ ≤ T thisfollows directly from the Optional Cross-Section Theorem, see Theorem 86 in Chapter IVof [17]. Proposition 3.11.
Let Assumption 2 hold. Then we have for all stopping times ≤ σ ≤ T that ess sup ( Q , ˜ S Q ) ∈ CPS ( ,T,λ ) E Q [ ˜ S Q T ∣ F σ ] = ( + λ ) S σ . In particular, there is no asset price bubble in the market model.Proof.
Let n ∈ N such that n ≤ λ . By Assumption 2 there exists a ( Q n , ˜ S n ) ∈ CPS ( , T, n ) for all n ∈ N /{ } . Define µ n ∶ = + λ + n for n ≥ n . Then ( Q n , µ n ˜ S n ) ∈ CPS ( , T, λ ) for all n ≥ n , since for t ∈ [ , T ] we have ( − λ ) S t ≤ ( − n ) S t ≤ ˜ S nt ≤ µ n ˜ S nt ≤ µ n ( + n ) S t = ( + λ ) S t . ( + λ ) S σ ≥ ess sup ( Q , ˜ S Q ) ∈ CPS ( ,T,λ ) E Q [ ˜ S Q T ∣ F σ ] ≥ ess sup n ≥ n E Q n [ µ n ˜ S nT ∣ F σ ] = ess sup n ≥ n µ n ˜ S nσ , where we have used the martingale property of ( Q n , µ n ˜ S n ) ∈ CPS ( , T, λ ) . For the essen-tial supremum we get ∣( + λ ) S σ − ess sup n ≥ n µ n ˜ S nσ ∣ ≤ ∣( + λ ) S σ − ess sup n ≥ n µ n ( − n ) S σ ∣ = ∣( + λ ) S σ ( − ess sup n ≥ n − n + n )∣ = . Hence we can conclude that ess sup ( Q , ˜ S Q ) ∈ CPS ( ,T,λ ) E Q [ ˜ S Q T ∣ F σ ] = ( + λ ) S σ . Lemma 3.12.
If the asset price S = ( S t ) t ∈ [ ,T ] is a semimartingale and the set M loc ( S ) of equivalent local martingale measures for S is not empty, then ( Q , µS ) ∈ CPS loc ( , T ) for Q ∈ M loc ( S ) and µ ∈ [ − λ, + λ ] , and F σ = ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ∣ F σ ] ≥ ess sup Q ∈M loc ( S ) E Q [ µS T ∣ F σ ] . (3.20) Proof.
Equation (3.20) immediately follows by the observation that {( Q , µS ) ∶ Q ∈ M loc ( S ) , µ ∈ [ − λ, + λ ]} ⊂ CPS loc ( , T ) . (3.21) Theorem 3.13.
Suppose that Assumption 1 holds. Then ˜ F = ( ˜ F t ) t ∈ [ ,T ] defined in (3.19) admits a c`adl`ag modification with respect to P .Proof. We first show that ˜ F admits a c`adl`ag modification with respect to some Q ∈Q loc ( , T ) . Since P and Q are equivalent we can conclude that ˜ F has also a c`adl`agmodification with respect to P .By Theorem 48 in [18], ˜ F admits a c`adl`ag modification with respect to Q if and only if forevery uniformly bounded increasing sequence ( α n ) n ∈ N of stopping times lim n →∞ E Q [ ˜ F α n ] exists and if lim n →∞ E Q [ ˜ F β n ] = E Q [ ˜ F lim n →∞ β n ] for every decreasing sequence ( β n ) n ∈ N of bounded stopping times. For convenience, we write CPS loc = CPS loc ( , T, λ ) and Z loc =Z loc ( , T, λ ) in the sequel. Note that we use the representation of ( Q , ˜ S Q ) ∈ CPS loc fromProposition 2.4. Fix some Q ∈ Q loc . As in Proposition 4.3 of [43] we first show theidentity E Q ⎡⎢⎢⎢⎢ ⎣ ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ∣ F σ ]⎤⎥⎥⎥⎥⎦ = sup ( Q , ˜ S Q ) ∈ CPS E Q [ E Q [ ˜ S Q T ∣ F σ ]] , (3.22)15or all stopping times σ with values in [ , T ] . For the first direction we use monotonicityto obtain E Q ⎡⎢⎢⎢⎢ ⎣ ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ∣ F σ ]⎤⎥⎥⎥⎥⎦ ≥ sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ ]] . (3.23)For the reverse direction we use Theorem 3.9 to show thatΦ ∶ = { E Q [ ˜ S Q T ∣ F σ ] ∶ ( Q , ˜ S Q ) ∈ CPS loc ( σ, T )} is directed upwards, i.e. for E Q [ ˜ S Q T ∣ F σ ] , E ¯ Q [ ¯ S ¯ Q T ∣ F σ ] ∈ Φ there exists E ̂ Q [ ̂ S ̂ Q ∣ F σ ] ∈ Φsuch that E ̂ Q [ ̂ S ̂ Q T ∣ F σ ] ≥ E Q [ ˜ S Q T ∣ F σ ] ∨ E ¯ Q [ ¯ S ¯ Q T ∣ F σ ] . We define A σ ∶ = { E Q [ ˜ S Q T ∣ F σ ] ≥ E ¯ Q [ ¯ S ¯ Q T ∣ F σ ]} ∈ F σ . Let Z = ( Z , Z ) and ¯ Z = ( ¯ Z , ¯ Z ) be the processes associated to ( Q , ˜ S Q ) and ( Q , ¯ S ¯ Q ) respectively, as in Proposition 2.4. Then we define d ̂ Q d P = ̂ Z T E P [ ̂ Z T ] ∶ = A σ Z T + A cσ ¯ Z T E P [ A σ Z T + A cσ ¯ Z T ] , (3.24)and for σ ≤ t ≤ T , ̂ Z t ∶ = A σ Z t + A cσ ¯ Z t (3.25)with corresponding ̂ S ̂ Q t = ̂ Z t ̂ Z t . (3.26)Obviously, ̂ Z satisfies all requirements from Definition 2.3, i.e., ̂ Z ∈ Z loc . Clearly, ( − λ ) S t ≤ ̂ S ̂ Q t ≤ ( + λ ) S t for all t ∈ [ σ, T ] . For the local martingale property let ( τ n ) n ∈ N be a localizingsequence for ˜ S Q and ¯ S ¯ Q . For σ ≤ s ≤ t ≤ T we get E ̂ Q [( ̂ S ̂ Q t ) τ n ∣ F s ] = E P ⎡⎢⎢⎢⎢ ⎣( ̂ Z t ̂ Z t ) τ n ̂ Z T E P [ ̂ Z T ] ∣ F s ⎤⎥⎥⎥⎥⎦ E P [ ̂ Z T ]̂ Z s ∧ τ n = E P [( A σ Z t + A cσ ¯ Z t ) τ n ∣ F s ] ̂ Z s ∧ τ n = ( A σ E P [( Z t ) τ n ∣ F s ] + A cσ E P [( ¯ Z t ) τ n ∣ F s ]) ̂ Z s ∧ τ n = ( A σ Z s ∧ τ n + A cσ ¯ Z s ∧ τ n ) ̂ Z s ∧ τ n = ( ̂ S s ) τ n , where we used that A σ , A cσ are F σ ⊂ F s measurable. In particular, by Theorem A.33 of[21], there exists an increasing sequence ( E Q n [ ˜ S nT ∣ F σ ]) n ∈ N ⊂ Φess sup ( Q , ˜ S Q ) ∈ CPS loc ( ,T ) E Q [ ˜ S Q T ∣ F σ ] = ess sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T ) E Q [ ˜ S Q T ∣ F σ ] = lim n →∞ E Q n [ ˜ S nT ∣ F σ ] . (3.27)16hus, we obtain by the Theorem of monotone convergence E Q ⎡⎢⎢⎢⎢⎣ ess sup ( Q , ˜ S Q ) ∈ CPS loc ( ,T,λ ) E Q [ ˜ S Q T ∣ F σ ]⎤⎥⎥⎥⎥⎦ = lim n →∞ E Q [ E Q n [ ˜ S nT ∣ F σ ]] ≤ sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T,λ ) E Q [ E Q [ ˜ S Q T ∣ F σ ]] = sup ( Q , ˜ S Q ) ∈ CPS loc ( ,T,λ ) E Q [ E Q [ ˜ S Q T ∣ F σ ]] . (3.28)The last equality in (3.28) holds due to similar arguments as in the proof of Theorem 3.9.This concludes the proof of (3.22).Let now ( σ n ) n ∈ N be a sequence of stopping times with values in [ , T ] such that σ n ↓ σ as n tends to infinity. We now prove thatlim n →∞ E Q [ ˜ F σ n ] = E Q [ lim n →∞ ˜ F σ n ] = E Q [ ˜ F σ ] . If the limit exists, then lim n →∞ E Q [ ˜ F σ n ] < ∞ for all stopping times 0 ≤ σ ≤ T and Q ∈ Q loc ( , T ) . Namely,lim n →∞ E Q [ ˜ F σ n ] = lim n →∞ E Q ⎡⎢⎢⎢ ⎣ ess sup ( Q ,SQ ) ∈ CPS loc E Q [ ˜ S Q T ∣ F σ n ]⎤⎥⎥⎥⎦ ≤ lim n →∞ E Q ⎡⎢⎢⎢⎣ ess sup ( Q ,SQ ) ∈ CPS loc ˜ S Q σ n ⎤⎥⎥⎥⎦ ≤ lim n →∞ E Q [( + λ ) S σ n ] ≤ lim n →∞ E Q [ + λ − λ ˜ S Q σ n ] ≤ + λ − λ ˜ S Q . Analogously, we can show that E Q [ ˜ F σ ] < ∞ . Using (3.22), the Fatou’s lemma, and thefact that ( E Q [ ˜ S Q T ∣ F t ]) σ ≤ t ≤ T is right-continuous we obtainlim n →∞ sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ n ]] ≥ lim n →∞ E Q [ E Q [ ˜ S Q T ∣ F σ n ]] ≥ E Q [ lim inf n →∞ E Q [ ˜ S Q T ∣ F σ n ]] = E Q [ E Q [ ˜ S Q T ∣ F σ ]] . (3.29)Since (3.29) holds for all ( Q , ˜ S Q ) ∈ CPS loc we get thatlim n →∞ E Q [ ˜ F σ n ] = lim n →∞ sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ n ]] ≥ sup ( Q , ˜ S Q ) CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ ]] = E Q [ ˜ F σ ] , (3.30)where the last equality follows by (3.22). It is left to show the second direction, namely,lim n →∞ sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ n ]] ≤ sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ ]] . (3.31)To this propose, we first show that the family { ˜ F σ n ∶ n ∈ N } is uniformly Q -integrable.For n ∈ N we have E Q [ ˜ F σ n ] = E Q ⎡⎢⎢⎢⎢⎣ ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ∣ F σ n ]⎤⎥⎥⎥⎥⎦ ≤ E Q ⎡⎢⎢⎢⎢⎣ ess sup ( Q , ˜ S Q ) ∈ CPS loc ˜ S Q σ n ⎤⎥⎥⎥⎥⎦ ≤ E Q [( + λ ) S σ n ] ≤ E Q [ + λ − λ ˜ S Q σ n ] ≤ + λ − λ ˜ S Q < ∞ . ǫ > δ = δ ( ǫ ) > A ∈ F T with Q ( A ) < δ we have E Q ⎡⎢⎢⎢⎢ ⎣ ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ∣ F σ n ] A ⎤⎥⎥⎥⎥⎦ < ǫ, (3.32)for all n ∈ N . Define the stopping times ( τ m ) m ∈ N by τ m ∶ = inf { t ≥ ∶ S t ≥ m } ∧ T. (3.33)Then we have τ m ↑ T , Q -a.s. as m tends to ∞ . In particular, lim m →∞ Q ( τ m = T ) = ǫ > δ ∶ = δ ( ǫ ) > A ∈ F T with Q ( A ) < δ then (3.32) holds.There exists M ( δ ) ∈ N such that Q ( τ m < T ) < δ for all m ≥ M ( δ ) . Therefore, we obtainfor m ≥ M ( δ ) sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ n ]] = E Q ⎡⎢⎢⎢⎢⎣ ess sup ( Q , ˜ S Q ) ∈ CPS E Q [ ˜ S Q T ∣ F σ n ]⎤⎥⎥⎥⎥⎦ = E Q [ ˜ F σ n { τ m = T } + ˜ F σ n { τ m < T } ]] ≤ E Q [ ˜ F σ n { τ m = T } ] + ǫ. (3.34)On { τ m = T } we have that S τ m t ≤ m for all 0 ≤ t < τ m and S t = S τ m t , hence ˜ F σ n = ˜ F σ n ∧ τ m = ( + λ ) S σ n ∧ τ m = ( + λ ) S σ n for all n ∈ N , by Proposition 3.11. Since the process S is c`adl`agwe get lim n →∞ S σ n = S σ . To conclude the proof, we apply the dominated convergencetheorem by using that ( ˜ F σ n ) n ∈ N is uniformly Q -integrable. Hence by (3.34) we get thatlim n →∞ sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ E Q [ ˜ S Q T ∣ F σ n ]] ≤ lim n →∞ E Q [ ˜ F σ n { τ m = T } ] + ǫ = E Q [ lim n →∞ ˜ F σ n ∧ τ m { τ m = T } ] + ǫ = E Q [ lim n →∞ ( + λ ) S σ n ∧ τ m { τ m = T } ] + ǫ = E Q [( + λ ) S σ ∧ τ m { τ m = T } ] + ǫ = E Q [ ˜ F σ { τ m = T } ] + ǫ ≤ E Q [ ˜ F σ ] + ǫ. (3.35)Since ǫ > F has left finite limits if given a uniformly bounded increasingsequence ( σ n ) n ∈ N of stopping times we have that lim n →∞ E Q [ ˜ F σ n ] exists and is finite.The fact that the limits is finite if it exists is due to the uniform Q -integrability. Fix ǫ > δ ∶ = δ ( ǫ ) > n →∞ ( + λ ) S σ n exists and is finite since S is a c`adl`ag process and ( σ n ) n ∈ N is a uniformly bounded increasing sequence of stopping18imes. Then we havelim sup n →∞ E Q [ ˜ F σ n ] ≤ lim sup n →∞ E Q [ ˜ F σ n { τ m = T } ] + ǫ = E Q [ lim sup n →∞ ˜ F σ n { τ m = T } ] + ǫ ≤ E Q [ lim sup n →∞ ( + λ ) S σ n { τ m = T } ] + ǫ = E Q [ lim inf n →∞ ( + λ ) S σ n { τ m = T } ] + ǫ = E Q [ lim inf n →∞ ˜ F σ n { τ m = T } ] + ǫ ≤ lim inf n →∞ E Q [ ˜ F σ n ] + ǫ. (3.36)Since it holds that lim inf n →∞ E Q [ ˜ F σ n ] ≤ lim sup n →∞ E Q [ ˜ F σ n ] and ǫ > In this section we provide several examples which illustrate the general results of Section2 and 3. In Example 4.1 we start by showing a market model under transaction costwhere the asset price, driven by a fractional Brownian motion, has a bubble in the senseof Definition 3.1. Then we study how the presence of an asset price bubble in a marketmodel without transaction costs may influence the appearance of a bubble in the analogousmarket model with transaction costs, and vice versa. To this purpose, we introduce theframework of [28] for frictionless market models and consider examples where the assetprice is a semimartingale. In Example 4.5 we illustrate a standard market model such thatthere is no bubble, neither with nor without transaction cost. In Example 4.6, the marketmodel has no bubble under transaction cost but there is a bubble without transaction costin the sense of [28]. In Example 4.7 we illustrate how bubble’s birth is already includedin our model.
Example 4.1.
This example is based on Example 7.1 of [24]. Let W H be a fractionalBrownian motion with Hurst index 0 < H <
1. We define X t ∶ = exp ( W Ht + µt ) , t ≥ , for µ ≥
0. Let F X ∶ = ( F Xt ) t ≥ be the (completed) natural filtration of the process X . Notethat X admits a consistent price system in the non-local sense on the interval [ , T ] forall T > τ ∶ = inf { t ∈ R ∶ X t = } , S t ∶ = X τ ∧ tan t , ≤ t < π , S t = , t ≥ π . Define G t ∶ = F tan t , 0 ≤ t < π /
2, and G π / ∶ = F ∞ . Consider T ≥ π /
2. We now show that thereexists no consistent price system in the non-local sense for any λ ∈ ( , ) . By contradictionassume that there exists a consistent price system ( Q , ˜ S Q ) for S in the non-local sense fora λ ∈ ( , ) . Then we have1 − λ ≤ ˜ S Q t = E Q [ ˜ S Q T ∣ F t ] ≤ + λ a.s. (4.1)for all 0 ≤ t ≤ T , and hence also 1 − λ ( + λ ) ≤ S t ≤ + λ ( − λ ) , (4.2)which is not possible because S is not bounded from above for 0 < t < π /
2. Thus, wecan conclude that there is no consistent price system in the non-local sense. However, S satisfies Assumption 1, i.e., for every λ > S . Since X admits for all λ > ( Q , ˜ S Q ) on [ , T ] for all T > ( Q , ( ˜ S Q ) τ ) is also a consistent local price system for S on [ , T ] . We now showthat there is a bubble in this market model with transaction costs for λ < /
3. For anyconsistent local price system ( Q , ˜ S Q ) for S we have ( − λ ) S ≤ ˜ S Q ≤ ( + λ ) S , and because S = − λ ≤ ˜ S Q T ≤ + λ . This implies for λ < / S Q ≥ − λ > + λ ≥ ˜ S Q T (4.3)for all consistent local price systems. Thus, we have ( + λ ) S ≥ ess sup ( Q , ˜ S Q ) ∈ CPS loc ˜ S Q > ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ] . (4.4)Therefore, we can conclude by equation (4.4) that the the asset S has a bubble undertransaction cost at time t = Remark 4.2.
Due to the well-known arbitrage arguments, see [16], the process X inExample 4.1 cannot be considered to describe asset price dynamics in a market modelwithout transaction costs. Hence in the case a comparison with an analogous frictionlessmarket model makes no-sense. Note also that the process X can be replaced by any c`adl`agprocess which is not bounded and admits a consistent local price system on [ , T ] for all T > and for all λ > .
20e now investigate the relation between asset price bubbles in market models with andwithout transaction costs. To this purpose we use [28] as reference for the frictionlessmarket case. We now briefly recall and re-adapt the framework of [28] to be coherentwith our setting outlined in Section 2. In particular, we assume that the asset price S is given by a c`adl`ag non-negative semimartingale such that M loc ( S ) ≠ ∅ . Under theseassumptions, NFLVR holds, see [16]. Put S ∶ = ( B, S ) . We denote by σ L ( S ) the set of all R -valued processes ν = ( ν t , ν t ) σ ≤ t ≤ T which are predictable on ⟦ σ, T ⟧ and for which thestochastic integral process ∫ tσ ν s d S s , σ ≤ t ≤ T , is defined in the sense of 2-dimensionalstochastic integration, see [46, Section III.6]. Definition 4.3 (Definition 2.5, [28]) . Fix a stopping time 0 ≤ σ ≤ T . The space σ L sf ( S ) of self-financing strategies (for S ) on ⟦ σ, T ⟧ consists of all 2-dimensional processes ν whichare predictable on ⟦ σ, T ⟧ , belong to σ L ( S ) , and such that the value process V ( ν )( S ) of ν satisfies the self-financing condition V ( ν )( S ) ∶ = ν ⋅ S = ν σ ⋅ S σ + ∫ σ ν u d S u on ⟦ σ, T ⟧ . Definition 4.4 (Definition 3.1, [28]) . The fundamental value of the asset S at time t ∈ [ , T ] is defined by S ∗ t ∶ = ess inf { v ∈ L + ( F t , P ) ∶ ∃ ν ∈ ∗ L sf + ( S ) with V T ( ν )( S ) ≥ S T and V t ( ν )( S ) ≤ v, P -a.s. } . (4.5)We say that the market model has a strong bubble if S ∗ and S are not indistinguishable, i.e.,if P ( S ∗ σ < S σ ) > ≤ σ ≤ T and define the process β NoTC = ( β NoTC t ) ≤ t ≤ T by β NoTC t ∶ = S t − S ∗ t , t ∈ [ , T ] .Note that Definition 4.4 differs from Definition 3.1 of [28] since we require v ∈ L + ( F t , P ) in (4.5) to be consistent with Definition 3.1. In this setting the duality from Theorem 3.2of [43] holds and we get S ∗ σ = ess sup Q ∈ M loc ( S ) E Q [ S T ∣ F σ ] . (4.6)In the more general framework of [28] it is possible that the duality does not hold, seeRemark 3.11 of [28] and the comment before for more information. Example 4.5.
Let S be a true Q -martingale for some probability measure Q ∼ P . Then,˜ S Q ∶ = (( + λ ) S t ) ≤ t ≤ T is a true Q -martingale and ( Q , ˜ S Q ) is a consistent price systemin the non-local sense for S . For any stopping time 0 ≤ τ ≤ T we obtain by Proposition3.3 that ( + λ ) S τ ≥ ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ∣ F τ ] ≥ E Q [( + λ ) S T ∣ F τ ] = ( + λ ) S τ = ( + λ ) S τ . (4.7)21ence there is no bubble in the market model with transaction costs. Alternatively, wecan observe that Assumption 2 is satisfied and thus we can apply Proposition 3.11.From Definition 4.4 we have for any stopping time 0 ≤ τ ≤ TS τ ≥ ess sup Q ∈ M loc ( S ) E Q [ S T ∣ F τ ] ≥ E Q [ S T ∣ F τ ] = S τ . So there is also no bubble in the market model without transaction cost in the sense of[28].
Example 4.6.
In this example we assume that S is given by a three-dimensional inverseBessel process, i.e., S t ∶ = ∥ B t ∥ − , t ∈ [ , T ] , (4.8)where ( B t ) t ∈ [ ,T ] = ( B t , B t , B t ) t ∈ [ ,T ] is a three-dimensional Brownian motion with B = ( , , ) and consider the filtration F S defined by F St ∶ = σ ( S s ∶ s ≤ t ) . Example 5.2 in [28],shows that there is a bubble in the market model without transaction cost in the senseof Definition 4.4. Note that there is also a P -bubble in the sense of [47] as in the case ofa complete market model these definitions coincide. However, by Theorem 5.2 of [24] wehave that for all λ > ( Q , ˜ S Q ) ∈ CPS, where ˜ S Q is a true Q -martingale suchthat ( − λ ) S t ≤ ˜ S Q t ≤ ( + λ ) S t , for all t ∈ [ , T ] . (4.9)In the notation of [24], we say that ˜ S Q is λ -close to S . In particular, Assumption 2 issatisfied and thus we obtain by Proposition 3.11 that there is no bubble in any marketmodel with proportional transaction costs λ >
0. This example shows that proportionaltransaction costs can prevent bubbles’ formation.
Example 4.7.
This example is based on Example 5.4 of [28]. It illustrates that bubblebirth (see [47], [7]) is naturally included in our model.Let γ be a random variable with values in ( , ] , 0 < P ( γ = ) < P ( γ ≥ t > ) = t ∈ ( , T ) and consider the filtration F γ generated by H t = { γ ≤ t } , t ∈ [ , ] . Then γ is a F γ stopping time, which represents the time when the bubble is born. Further, let W be a Brownian motion independent of γ . Denote by F W the natural filtration generatedby W and define the filtration F = ( F t ) t ∈ [ , ] by F t ∶ = F Wt ∨ F γt ∨ N , t ∈ [ , ] , where N denotes the P -nullsets of F W ∨ F γ . Then γ is also an F -stopping time. Let S = ( S t ) ≤ t ≤ be the unique strong solution to the SDE dS t = S t ( µ d t + v ( t, γ ) d W t ) , S = , (4.10)with µ ∈ R and v ∶ [ , ] → [ v , ∞ ) given by v ( t, u ) = v ( + − t { u ≤ t < } ) , (4.11)22or v >
0. Then S is a geometric Brownian motion up to γ . At time γ the term 1 /( − t ) starts to influence the volatility which explodes until time 1. This implies that S convergesto 0 as t tends to 1. We determine the fundamental value F σ of S at time σ <
1. Inparticular, we see that there is no bubble before time γ but the bubble starts at γ . Thefundamental value F σ at time σ < F σ = ( + λ ) S σ { γ > σ } . (4.12)Note that S ( ω ) = ω ∈ { ω ∈ Ω ∶ γ ( ω ) ≤ σ ( ω )} . We define the strategy ϕ = ( ϕ t , ϕ t ) t ∈ ⟦ σ,T ⟧ on ⟦ σ, T ⟧ by ( ϕ t , ϕ t ) = ⎧⎪⎪⎪⎪⎪ ⎨⎪⎪⎪⎪⎪⎩(( + λ ) S σ { γ > σ } , ) , for t = σ, ( , { γ > σ } ) , for σ < t < , ( , ) , for t = . That is, using the initial capital ( + λ ) S σ { γ > σ } we trade in such a way that we hold theasset at time 1. If at time σ , γ has already happened, we know that the volatility blowsup and we can buy the asset at time 1 at price 0. However, if γ happens strictly after σ we do not know if the volatility will blow up and thus we buy the asset at time σ atprice ( + λ ) S σ in order to hold the asset at time 1. As this strategy ϕ super-replicatesthe position ( , ) , we conclude that F σ ≤ ( + λ ) S σ { γ > σ } .For the reverse direction, “ ≥ ” we use the duality from Proposition 3.4. By Example 5.4 of[28] we get ess sup Q ∈ M loc ( S ) E Q [ S ∣ F σ ] = S σ { γ > σ } . From this we obtain F σ = ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ S ∣ F σ ] ≥ ess sup Q ∈ M loc ( S ) E Q [( + λ ) S ∣ F σ ] = ( + λ ) S σ { γ > σ } . Indeed, we have F σ = ( + λ ) S σ { γ > σ } . This implies that F σ = ( + λ ) S σ on { σ < γ } and F σ = { σ ≥ γ } . In particular, we canconclude that γ is then the time at which the bubble is born. In this section we study whether transaction costs can prevent bubbles’ formation andtheir impact on bubbles’ size.In the setting of [28] outlined in Section 4, we assume that the asset price S is a semi-martingale and that M loc ( S ) ≠ ∅ . At time t ∈ [ , T ] we obtain by Lemma 3.12 that ( + λ ) S t ≥ F t = ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S QT ∣ F t ] ≥ ess sup Q ∈ M loc ( S ) E Q [( + λ ) S T ∣ F t ] = ( + λ ) S ∗ t .
23n particular, we have β t = ( + λ ) S t − F t ≤ ( + λ ) ⎛⎝ S t − ess sup Q ∈ M loc ( S ) E Q [ S T ∣ F t ]⎞⎠ = ( + λ ) β NoTC t . (5.1)From (5.1) we immediately obtain for t ∈ [ , T ] that if β NoTC t =
0, then β t = β NoTC t > β t > t ∈ [ , T ] . Let β NoTC t ≠
0. Then β t β NoTC t ≤ + λ, (5.2)which means that the quotient of the bubbles is bounded by the factor ( + λ ) . Furthermore,we have − λβ NoTC ≤ β NoTC t − β t ≤ β NoTC t . (5.3)It is easy to see that both bounds in (5.3) can be obtained. In Example 4.7 we get β t − β NoTC t = ( + λ ) S t { γ ≤ t } − S t { γ ≤ t } = λS t { γ ≤ t } = λβ NoTC t . Furthermore, we have in Example 4.6 that β t ≡ ( + λ ) β NoTC t − β t = ( + λ ) ⎛⎝ S t − ess sup Q ∈ M loc ( S ) E Q [ S T ∣ F t ]⎞⎠ − ( + λ ) S t + ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S QT ∣ F t ] = ess sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S QT ∣ F t ] − ( + λ ) ess sup Q ∈ M loc ( S ) E Q [ S T ∣ F t ] = ∶ ∆ t,T ( λ ) . (5.4)By rearranging equation (5.4) we obtain then β t = ( + λ ) β NoTC t − ∆ t,T ( λ ) . (5.5)Clearly, it holds ∆ t,T ( λ ) ∈ ⟦ , ( + λ ) β NoTC t ⟧ . Consider Example 4.6, where S is a 3-dimensional inverse Bessel process with respect to P and set t =
0. Then β = t,T ( λ ) = sup ( Q , ˜ S Q ) ∈ CPS loc E Q [ ˜ S Q T ] − ( + λ ) sup Q ∈ M loc ( S ) E Q [ S T ] = ( + λ ) S − ( + λ ) E P [ S T ] = ( + λ ) ( − Φ ( √ T )) , (5.6)where Φ denotes the cumulative distribution function of the standard normal distribution,see [20]. For T tending to infinity, then ∆ t,T ( λ ) tends to ( + λ ) .24 emark 5.1. From equation (5.1) we can see that if a market model without transactioncosts has no asset price bubble, then the analogue market model with transaction has noasset price bubble either. In other words, the introduction of transaction costs into a marketcannot generate asset price bubbles. Conversely, by (5.1) it follows that, if a market modelwith transaction costs has an asset price bubble, the corresponding frictionless marketmodel has an asset price bubble as well.In our model the introduction of transaction costs can possibly prevent the occurrence ofan asset price bubble. This can be seen in Example 4.6 where we have an asset pricebubble in the sense of Definition 4.4 but no bubble in presence of transaction costs withrespect to Definition 3.1. However, Example 4.7 shows that it is possible to have an assetprice bubble in both market models, with and without transaction costs. In particular, thepresence of transaction costs does not guarantee the absence of asset price bubbles.
The lack of symmetry between buying, holding and selling a share of an asset suggeststo study separately bubbles for the bid price and the ask price. In Section 3 we haveintroduced an asset price bubble for the ask price. If we wish to similarly define an assetprice bubble for the bid price, a possible approach would be to define the fundamentalvalue by the sub-replication price of holding one share of the asset at time T . However,in our setting it is not possible to use the common definition of sub-replication pricesbecause our definition of trading strategies allows to throw money away, see [49]. Thiscan be illustrated by considering the classical definition of the sub-replication price, see[33]. Denote by A ( x ) the the set of self-financing admissible trading strategies accordingto Definition 2.2 of [33]. For a derivative Z T ∈ L + ( F , P ) we have that the sub-replicationprice c at time 0 is given by c = sup { x ∈ R ∶ ∃ ( α , α ) ∈ A ( x ) , x + ∫ T α t ⋅ dS t ≤ Z T } , (6.1)If Z T ∈ L + ( F , Q ) for all Q ∈ M loc ( S ) is bounded from above we have by Theorem 8.2 of[33] that c = inf Q ∈ M loc ( S ) E Q [ Z T ] . (6.2)As we can throw money away in our setting, the supremum in (6.1) will always be infinite.Alternatively, we can consider the claim X T = ( , − ) and define the fundamental valueby the super-replication price for X T . By applying Theorem 1.5 of [50] (see Theorem A.1in the Appendix) we obtain that in this case the fundamental price would coincide withsup ( Q , ˜ S Q ) ∈ CPS E Q [ − ˜ S Q T ] . However, the super-replication price is well-defined in this caseonly if Assumption 2 is in force, which implies that we only allow consistent price systemsin the non-local sense and thus we get by Proposition 3.11 thatsup ( Q , ˜ S Q ) ∈ CPS E Q [ − ˜ S Q T ] = − inf ( Q , ˜ S Q ) ∈ CPS E Q [ ˜ S Q T ] = − ( − λ ) S . (6.3)25f we consider the case of consistent local price systems, we need strategies which areadmissible in the the num´eraire-based sense. In particular, a claim X T = ( X T , X T ) canonly be super-replicated by a strategy ϕ = ( ϕ t , ϕ t ) ≤ t ≤ T admissible in the num´eraire-basedsense, i.e. ϕ T = X T , if X T + ( X T ) + ( − λ ) S T − ( X T ) − ( + λ ) S T ≥ − M, (6.4)for a constant M >
0. But for X T = ( , − ) equation (6.4) can only be fulfilled if S T isbounded. This is a strong restriction and in general not fulfilled.Let us consider another approach. The fundamental value in the sense of Definition 3.1coincides with the super-replication price of holding one share of the asset at time T . For abid-bubble we want to compare the fundamental value with the bid-price ( − λ ) S . As thebid-price is the selling price, it seems reasonable to introduce the definition of fundamentalprice by considering the liquidation value of one share of the asset, i.e., to super-replicatethe cash position X T = (( − λ ) S T , ) . Definition 6.1.
The fundamental value for the bid-price F bid = ( F bidt ) t ∈ [ ,T ] of an asset S at time t ∈ [ , T ] in a market model with proportional transaction costs 0 < λ < F bidt ∶ = ess inf { X t ∈ L + ( F t , Q loc ) ∶ ∃ ϕ ∈ V loc t,T ( X t , λ ) with ϕ t = ( X t , ) and ϕ T = (( − λ ) S T , )} . By Proposition 3.5 we obtain that F bidt = ess sup ( Q , ˜ S Q ) ∈ CPS loc ( σ,T ) E Q [( − λ ) S T ∣ F t ] . (6.5)It is easy to see that F bidt ≤ F t for all t ∈ [ , T ] . However, it is hard to determinesup ( Q , ˜ S Q ) ∈ CPS loc ( ,T ) E Q [( − λ ) S T ] = sup Q ∈ Q loc E Q [( − λ ) S T ] explicitly, because we donot know the behavior of S T under a measure Q ∈ Q loc . A Super-replication Theorems
For sake of completeness we provide the super-replication theorems (Theorem 1.4, Theo-rem 1.5) of [50]. Note that Theorem 1.5 of [50] coincides with Theorem 4.1 of [11]. Wedenote by A M ( X σ , σ, T, λ ) (resp. A loc M ( X σ , σ, T, λ ) the set of all terminal values of self-financing trading strategies ϕ , ϕ ∈ V M ( X σ , σ, T, λ ) (resp. ϕ ∈ V loc M ( X σ , σ, T, λ ) ). We use thenotation A σ,T ( X σ , λ ) = ⋃ M A M ( X σ , σ, T, λ ) (resp. A loc σ,T ( X σ , λ ) = ⋃ M A loc M ( X σ , σ, T, λ ) ). Theorem A.1 (Theorem 1.5, [50]) . Let Assumption 2 hold. We consider a contingentclaim X T = ( X T , X T ) which pays X T many units of the bond and X T many units of therisky asset at time T . The random variable X T is assumed to satisfy X T + ( X T ) + ( − λ ) S T − ( X T ) − ( + λ ) S T ≥ − M ( + S T ) , for some M > . Then the following assertions are equivalent(i) There is a self-financing trading strategy ϕ with ϕ ≡ on ⟦ , σ ⟧ and ϕ T = ( X T , X T ) which is admissible in a num´eraire-free sense. ii) For every ( Q , ˜ S Q ) ∈ CPS ( σ, T ) we have E Q [ X T + X T ˜ S Q T ] ≤ . (A.1) Theorem A.2 (Theorem 1.4, [50]) . Let Assumption 1 hold. We consider a contingentclaim X T = ( X T , X T ) which pays X T many units of the bond and X T many units of therisky asset at time T . The random variable X T is assumed to satisfy X T + ( X T ) + ( − λ ) S T − ( X T ) − ( + λ ) S T ≥ − M, (A.2) for some M > . Then the following assertions are equivalent:(i) There is a self-financing trading strategy ϕ on ⟦ σ, T ⟧ with ϕ ≡ on ⟦ , σ ⟧ and ϕ T = ( X T , X T ) which is admissible in a num´eraire-based sense.(ii) For every ( Q , ˜ S Q ) ∈ CPS loc ( σ, T ) we have E Q [ X T + X T ˜ S Q T ] ≤ . (A.3)Note that in Theorem A.2 we consider the claim X T = ( X T , X T ) instead of X T = ( X T , ) as in Theorem 1.4 of [50]. However, the proof is similar. For details on the proof, we referto [48]. Acknowledgement
We would like to thank Martin Schweizer for helpful remarks and insightful discussions.
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