A constraint-based notion of illiquidity
aa r X i v : . [ q -f i n . M F ] A p r A CONSTRAINT-BASED NOTION OF ILLIQUIDITY
THOMAS KRABICHLER AND JOSEF TEICHMANN
Abstract.
This article introduces a new mathematical concept of illiquiditythat goes hand in hand with credit risk. The concept is not volume- butconstraint-based, i.e., certain assets cannot be shorted and are ineligible asnuméraire. If those assets are still chosen as numéraire, we arrive at a two-price economy. We utilise Jarrow & Turnbull’s foreign exchange analogy thatinterprets defaultable zero-coupon bonds as a conversion of non-defaultableforeign counterparts. In the language of structured derivatives, the impact ofcredit risk is disabled through quanto-ing. In a similar fashion, we look atbond prices as if perfect liquidity was given. This corresponds to asset pricingwith respect to an ineligible numéraire and necessitates Föllmer measures. Introduction
Let us assume that a financial agent sells buckets and promises to fill each ofthem with one litre of water in three months. If one buys such a claim, then,amongst others, one is exposed to the following somewhat interlinked risks. Firstand foremost, there is the risk that the supplier cannot honour their obligations atmaturity for whatever reason. For instance, there may not be enough water aroundor the buckets may not prove to be watertight.Furthermore, let us assume that the only water supply is from a single lakeand that the lake is frozen in its entirety due to severe weather conditions. Asa consequence, the water cannot be delivered in due time and only after havinginvested energy in order to liquefy the desired amount. During the lifetime ofthe contract, there are several reasons why such a bucket may change hands. Forexample, one is fortunate, and the extra amount of water is redundant. In any case,a bucket holder runs the risk that the recoverable value is adversely affected, andsignificant discounts have to be accepted when reselling the claim. Even thoughone is dealing with a bona fide counterparty and the buckets are of high quality,water might not be demanded for in the market at all. What is more, sound adviceor rumours may circulate that the water supplier is not trustworthy and not asreliable in delivering the promised amount as initially expected. Therefore, onewants to get rid of the claim as quickly as possible in terms of a fire sale. In bothscenarios, the recoverable value of the bucket is typically reduced substantially.These considerations are symptomatic for every financial contract. Broadlyspeaking, one suffers a loss because either something is not backed sufficiently orsomething is not readily available due to a lack of liquidity. Aspects of liquidity arediverse and relate to both the market and the commodity itself. Regarding interestrate modelling, the commodity of interest is usually money. Despite the fact thatliquidity is an intuitive concept, which we all know from our day-to-day experience,
Mathematics Subject Classification. it appears to be tricky to translate it into a sound mathematical framework. Acentrepiece of this article is an attempt to clarify this notion.
Credit risk refers to possible financial losses that a holder of a claim may sufferover the lifetime of the contract. Indisputably, there is an intricate connectionbetween credit risk and aspects of liquidity . The IMF Working Paper WP/02/232defines four different but intertwined terms of liquidity; see [22] for further details.Predominantly relevant for quantitative risk modelling are two of them, namely the asset liquidity , describing the immediacy and the transaction cost with which theasset in scope can be converted into legal tender, and the institutional liquidity ,standing for the issuer’s ability to meet their settlement obligations. The promiseof a risky loan can be understood, amongst others, as the lender’s compensation forbeing exposed to inflation risk , institutional liquidity risk of the issuer and assetliquidity risk. • The inflation risk is caused by the time value of money. The notional’spurchasing power might weaken over the loan’s lifetime and a lender wantsto be compensated for that risk. • The institutional liquidity risk, more commonly known as default risk, isthe event of a liquidity squeeze at expiry or even a premature insolvencyof the debtor party implicating only a partial or zero recovery of the loan’sface value. • The asset liquidity risk comprises the fact that the lenders forgo theirown institutional liquidity over the lifetime of the product. If they facean intermediary liquidity squeeze and a fire sale is their last resort, theymight have to accept significant discounts on the fair value of the loan.Noteworthy, the notion of asset liquidity solely relates to the loan itself butnot to the liquidity of the issuer’s assets.Filipović and Trolle corroborate in [6] that, subsequent to the credit crunch in2007/2008, asset liquidity constituted a significant fraction of the risk compensationin the money market. From the phenomenological viewpoint, asset illiquidityinvolves two prices for a certain good. One price is the fundamental value , whichis the intrinsic economic value or the minimal cost to replicate this product as ifthere was perfect liquidity. The other price is a market value which is derived fromtransactions. One typically has to accept a certain discount when converting anilliquid good into cash. The resulting difference between the fundamental valueand the market value cannot be exploitet. Below, we translate this observationinto a rigorous mathematical statement; illiquidity causes the alleged arbitrageopportunity to be inadmissible. We analyse interest rate modelling in the presenceof illiquidity by exploiting a neat foreign exchange (FX) modelling framework.The
FX-analogy was originally introduced in a working paper by Jarrow &Turnbull. A comprehensive exposition can be found in [17], in particular also ina two-filtration setting. This framework enables a joint modelling framework forinstitutional liquidity and asset liquidity that goes hand in hand with credit risk.Here we consider the Jarrow & Turnbull setting with only one filtration. However,the exchange rate together with the foreign bank account is only a local martingalein general (if discounted by the domestic bank account). This situation can becategorised in four different ways depending on the properties of the discountedforeign bank account. It can constitute different sorts of liquidity crises in themarket, since the foreign bank account can possibly not be taken as a numéraire
CONSTRAINT-BASED NOTION OF ILLIQUIDITY 3 without changing price structures. We therefore arrive at a two-price economydepending on whether we discount in domestic or foreign terms. We do introducearbitrage opportunities in the market if one is allowed to short the foreign bankaccount.Throughout the article, we use the following notion of liquidity.
Definition 1.1 (Liquidity and Liquidity Risk) . Liquidity is an entity’s ability toincur debts immediately. A possible lack of liquidity, affecting issued loans and theentity’s solvency likewise, is referred to as liquidity risk .As illustrated by [8], asset liquidity in modern financial markets is a key butelusive concept. The above notion of liquidity was proposed by the authors in [17]and pursues the idea of [16]. In their introductory section of [21], Lehalle andLaruelle define illiquidity for a specific demanded quantity as round trip cost. Thisconcept can be linked to the proposed theory below by extending the idea to roll-over strategies; see Section 5.2 in [17]. Ruf and others generalised in [1] and [24]the change of numéraire technique to dominating Föllmer measures in order modelhyperinflation in multi-currency settings. This was further substantiated in [7] inthe context of defaultable numéraires. Chau and Tankov study the same setting in[3] for optimal arbitrage. In a different direction goes the liquidity risk approachby Çetin, Jarrow and Protter; see [2]. They model prices for a single default-freeasset, e.g., for a zero-coupon bond with a fixed maturity, both time- and order-size-dependent. This leads to the concept of a supply curve which characterises thecomposition of the order book at a given time instance. Similarly, Madan studiestwo-price economies in [23] in order to account for other risks such as liquidity.The article is structured as follows. In Section 2, we recall the famous FX-analogy by Jarrow & Turnbull. In the Sections 3 and 4, we present the key ideaswithout going into any technical details. The necessary technical toolkit is derivedin the Sections 5 and 6. The centrepiece of this article is Section 7, where we presentfour distinct market scenarios for aspects of liquidity.2.
The Jarrow & Turnbull Setting
Let [0 , ∞ ) be the considered timeline. We denote by (cid:0) P ( t, T ) (cid:1) ≤ t ≤ T the càdlàgprice process of a non-defaultable zero-coupon bond with maturity T ≥ andpayoff P ( T, T ) = 1 . Furthermore, we denote by (cid:0) e P ( t, T ) (cid:1) ≤ t ≤ T the càdlàg priceprocess of a defaultable zero-coupon bond with the same maturity and a randompayoff < e P ( T, T ) ≤ . We assume that P ( T, T ) and e P ( T, T ) are written in thesame currency. The distribution of the final recovery e P ( T, T ) is strongly linked tothe riskiness of the issuer’s business model. It needs to be noted that we do notallow the final payoff to become zero and we make this assumption for any maturity T ≥ . We shall see straightaway why this assumption is of crucial importance forour modelling approach. Consequently, we may introduce another term structure (cid:8) Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ via Q ( t, T ) := e P ( t,T ) e P ( t,t ) . Note that we have Q ( T, T ) = 1 and,hence, that this synthetic series is default-free. By setting S t := e P ( t, t ) , weget e P ( t, T ) = S t Q ( t, T ) . Although this rewriting is very elementary, it opensan extremely nice modelling opportunity for defaultable zero-coupon bonds. Werecognise that credit risk can be analysed in an FX-like setting. THOMAS KRABICHLER AND JOSEF TEICHMANN
Paradigm 2.1 (Jarrow & Turnbull 1991) . The series P ( t, T ) and Q ( t, T ) areconsidered as non-defaultable zero-coupon bonds in different currencies. e P ( t, T ) may be interpreted as conversion of foreign default-free counterparts. S t = e P ( t, t ) is referred to as recovery rate or spot FX rate .In order to utilise the FX-analogy from a mathematical perspective, we fix thefollowing setup. Assumption 2.2 (The General FX-like Setting) . Let (Ω , F , F , Q ) with F = ( F t ) t ≥ be a filtered probability space satisfying the usual conditions. We consider Q asrisk-neutral pricing measure. By B = ( B t ) t ≥ we describe the accumulation ofthe domestic risk-free bank account with initial value of one monetary unit andby q B = ( q B t ) t ≥ its foreign counterpart. Furthermore, let (cid:8) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ , (cid:8) e P ( t, T ) (cid:9) ≤ t ≤ T < ∞ and (cid:8) Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ be three F -adapted families capturingthe stochastic evolution of the term structure of zero-coupon bond prices. Moreprecisely, ω P (cid:0) t, T (cid:1) ( ω ) , ω e P (cid:0) t, T (cid:1) ( ω ) and ω Q (cid:0) t, T (cid:1) ( ω ) are supposedto be positive and F t -measurable a.s. for all ≤ t ≤ T < ∞ . Additionally, themappings t P ( t, T ) , t e P ( t, T ) and t Q ( t, T ) are supposed to havecàdlàg paths a.s. for all ≤ T < ∞ . The corresponding payoffs satisfy(1) P ( T, T ) = 1 , < S T := e P ( T, T ) ≤ in the domestic currency, and the relation(2) Q ( t, T ) = e P ( t, T ) S t in some synthetic foreign currency. We assume the three properties in (1) and (2)to hold a.s. for all ≤ t ≤ T < ∞ . Finally, we assume absence of arbitrage in thesense that the discounted price processes P ( t, T ) B t , S t Q ( t, T ) B t = e P ( t, T ) B t , S t q B t B t for ≤ t ≤ T are Q -local martingales for each T ≥ .The FX-like setting provides a powerful machinery to study credit and liquidityrisk; see [17] or [18] for further details and examples.3. A First Step Towards Modelling Illiquidity
In many articles on mathematical finance, it is assumed inherently that enteringan arbitrarily large short position in the numéraire is admissible. After the existenceof a bank account, this is yet another very strong assumption. Despite beingcontroversial, this finding may serve as the basic idea to model consequences ofasset liquidity constraints. All we have to do is to reverse the rationale.
A financialasset subject to asset liquidity constraints cannot be shorted arbitrarily. Hence, itcannot serve as numéraire either.
See also Lemma 4.12 and Proposition 4.13 in [17].We are going to illustrate the basic idea by the following example, which is lentfrom the introductory section of [16]. This article links the concept of illiquidity to ineligible numéraires and bubbles ; see Theorem 2.8 in [16]. We will formalise thisidea in the subsequent Section 7. Meanwhile, we prepare the necessary technicalbuilding blocks. CONSTRAINT-BASED NOTION OF ILLIQUIDITY 5
Example 3.1 (Klein, Schmidt & Teichmann 2013) . We consider the general FX-like setting of Assumption 2.2 and fix some maturity
T > . We assume that Q isa true martingale measure for (cid:0) e P ( t, T ) (cid:1) ≤ t ≤ T . Additionally, we assume full initialrecovery S = 1 and that F contains only trivial information. We define the Q -local martingale Z = ( Z t ) ≤ t ≤ T as Z t := S t q B t S B t . If Z is a true Q -martingale, then Z represents a density process for some equivalent measure q Q ≈ Q , and the classicalchange of numéraire technique says Q (0 , T ) = E q Q (cid:2) q B T (cid:3) . If Z is a strict Q -localmartingale, hence also a strict Q -supermartingale with E Q (cid:2) Z T (cid:3) < , then we maystill define a locally equivalent measure q Q (cid:12)(cid:12) F T ≈ Q (cid:12)(cid:12) F T via d q Q d Q (cid:12)(cid:12)(cid:12)(cid:12) F T := Z T E Q (cid:2) Z T (cid:3) . Consistently, we get q Q (0 , T ) := E q Q (cid:20) q B T (cid:21) = 1 E Q (cid:2) Z T (cid:3) E Q (cid:20) Z T q B T (cid:21) = 1 E Q (cid:2) Z T (cid:3) Q (0 , T ) > Q (0 , T ) , i.e., we end up in a two-price economy. The price of a defaultable zero-coupon bond S q Q (0 , T ) with respect to the numéraire q B and the pricing measure q Q is higher than e P (0 , T ) = S Q (0 , T ) in the initial market. Nonetheless, we proclaim that the pricedifference cannot be exploited due to asset liquidity constraints. In this context, Z isnot considered as being an eligible numéraire. While being a strict local martingale, Z exceeds any barrier with positive probability. In order to emulate a replicatingstrategy and build the synthetic asset consisting of a long position in Q (0 , T ) and,more crucially, a short position in q Q (0 , T ) , an arbitrarily negatively valued shortposition in either finite or infinitesimal roll-overs of defaultable zero-coupon bondswould be required; and partial recovery must not hold over the replication periodeither. As indicated by Definition 6.1 in [16], the replication involves the reciprocalvalue of e P ( t, t + dt ) . Analogously, the strict local martingale property of Z featuresthe market phenomenon that no market participant is willing to lend capital basedon entering a repurchase agreement (repo) with this synthetic asset. The bubblemay burst any time and devalue the collateral strongly. This is synonymous with acontingent or qualified interest in holding defaultable zero-coupon bonds; see alsoExample 7.14 below for further clarification. (cid:3) Note that the above argument only works for the time instance t = 0 and issomehow maturity-dependent. In contrast to Example 3.1, we do not charge thepayoff with the defect of Z T . Instead, we put mass into a hidden default, whichcan only be seen under the new measure q Q . Correspondingly, the enhanced settingfounds on the existence of a foreign bank account together with an associateddominating pricing measure.The above concept of liquidity raises several discussion points. In mathematicalterms, it can hardly be analysed on a stand-alone basis. Once credit risk has beendisabled and S t ≡ prevails, one will end up in a pathological model, in which e P ( t, T ) = P ( t, T ) holds for all ≤ t ≤ T < ∞ . Seen from time t and with respectto a selected numéraire, there must be a unique value for the payoff of one monetaryunit at time T . Unless shorting either of the zero-coupon bonds is not admissible,any discrepancy between P ( t, T ) and e P ( t, T ) could be exploited as a free lunch THOMAS KRABICHLER AND JOSEF TEICHMANN with a simple buy-and-hold strategy. To this extent, asset liquidity can hardly beisolated. In a financial model, one cannot contrast two financial assets equippedwith the identical payoff but two different levels of liquidity. Contrasting is thenatural approach for studying credit risk; see Section 2. It needs to be noted thatthis statement is only affecting financial modelling. In the real world, there mightbe coequal bonds within the same discrete rating class but deviating yields. Fromthe modelling viewpoint, the impact of liquidity can only be uncovered by a changeof numéraire together with an associated change of measure. The marketable price e P ( t, T ) is with respect to some objective numéraire. In our case, this is normallythe bank account B = ( B t ) t ≥ . The fundamental value or intrinsic economic value S t q Q ( t, T ) is with respect to some synthetic numéraire q B = ( q B t ) t ≥ capturing theassumed accumulation of returns, if roll-overs of the considered financial asset werepossible. The resulting price difference is the premium solely attributed to assetliquidity; see also Remark 7.4 below.4. The Long Story Short
Conceptually, in the centre of the enhanced FX-like setting lies a process Z thatis either exogenously given or the result of a roll-over strategy in defaultable zero-coupon bonds. Due to asset liquidity constraints, this portfolio value process isin a bubble state under a risk-neutral measure Q . The bubble state is embodiedby the strict local martingale property of Z , whose characteristic sample pathsshow a hump with a far end being below its long-term mean. Consequently, nocounterparty is willing to accept it as collateral in the context of lending money.The bubble could burst any time and devalue the collateral. If one believes in thebubble state, one is tempted to short the asset and take advantage of the anticipatedprice collapse. However, in mathematical terms, shorting Z is not admissible under Q , since − Z is not a Q -local martingale. The presence of the bubble goes hand inhand with assigning zero probability to a liquidity event .A dominating Föllmer measure q Q for Z , as introduced in the next section, enablesto generalise the change of numéraire technique. Under q Q , the discounted value ofthe numéraire evolves flat on the level one. Thus, the bubble is not visible under q Q and Z = 1 seems to be priced correctly. Allowing Z as numéraire implies a newpricing regime. The change in credit lines gives reason for new superreplicationprices. Generally, as pinpointed by the model of the th kind below, this happenswithout order relation, i.e., prices can either rise or fall compared to the initialsetting. Z is the natural numéraire in the economic pricing of the defaultable zero-coupon bonds and determines the so-called fundamental value. It treats them asif perfect asset liquidity was given. Since there is no order relation in the prices,the illiquidity premium can attain both signs. Though, illiquidity appears morenatural than hyperliquidity. The latter is a pathological phenomenon of imperfectmarkets. An entirely positive illiquidity premium is easily achievable by meansof the model of the nd kind. The scenario in which no one is willing to holdthe defaultable zero-coupon bonds under any circumstances whatsoever translatesinto the promised yield exceeding any rational level. The abrupt devaluation leadsto a hyperinflation in the foreign market and Z explodes. As seen under Q , theliquidity event is featured in terms of a hidden (i.e., improbable) default of foreignzero-coupon bonds occurring at the stopping time τ . It is as if one took the equity CONSTRAINT-BASED NOTION OF ILLIQUIDITY 7 of the considered issuer as numéraire. The singularity occurring at time τ refers toequity on the edge of becoming negative.Under mild technical assumptions, the financial market under q Q is arbitrage-free;see Lemma 6.5. In the original market with respect to Q , the difference betweenmarket prices and fundamental prices cannot be exploited due to admissibilityconstraints. If these were ignored, one could materialise the discrepancy. Anoptimal arbitrage profit would lurk in the self-financing, yet inadmissible, replicationof { τ>T } with B and Z .We utilise the quanto-ing technique from Jarrow & Turnbull in order to analyseinstitutional liquidity risk; see Paradigm 2.1. To this end, we somehow disabledefault risk through introducing the foreign term structure T Q ( t, T ) . For assetliquidity risk, we go the other way round. By changing the numéraire from ( Q , B ) to ( q Q , Z ) , we discover a previously unlikely default feature Q ( T, T ) = { τ>T } . Froma risk management perspective, new scenarios are added in order for absence ofarbitrage to prevail. Essentially, institutional liquidity risk and asset liquidity riskare modelled jointly in a similar way and interact with each other. This can beperceived as a duality of credit and liquidity risk.5. Föllmer Measures
The following probabilistic exposition is inspired by [1], [11], [19] and [24]. Undersuitable conditions, supermartingales may be seen as generalised density processes.The importance of Föllmer measures in the context of mathematical finance can berecognised by Theorem 4.14 in [11]. Let (Ω , F , F , Q ) with F = ( F t ) t ≥ be a filteredprobability space. Definition 5.1 (Standard System) . The filtration F is called a standard system ,if (Ω , F t ) is isomorphic to some separable complete metric space with its Borel σ -algebra for each t ≥ , and if it holds T n ∈ N A n = ∅ for all non-decreasing sequences ( t n ) n ∈ N and for all non-increasing sequences of atoms ( A n ) n ∈ N with A n ∈ F t n foreach n ∈ N .We assume that F is the right-continuous modification of a standard system.Since we are going to work with dominating local martingale measures, we do notaugment F with the Q -nullsets. See Lemma 6.4 below, or [11], [19] and [24] forfurther details.Let Z = ( Z t ) t ≥ be a non-negative Q -local martingale with Z = 1 and càdlàgpaths. We define the stopping times(3) τ n := n ∧ inf (cid:8) t ≥ (cid:12)(cid:12) Z t > n (cid:9) , τ := lim n →∞ τ n . Each τ n for n ∈ N is the capped hitting time of an open set and, hence, an F -stoppingtime according to Lemma 6.6 (iii) in [13]. τ also is an F -stopping time since { τ ≤ t } = \ ε ∈ Q ∩ (0 , ∞ ) [ m ∈ N ∞ \ n = m (cid:8) τ n ≤ t + ε (cid:9) and F is assumed to be right-continuous. If ( σ n ) n ∈ N denotes a localising sequenceof Z , then it holds by Fatou’s Lemma for all ≤ u ≤ t ≤ ∞ E Q (cid:2) Z t (cid:12)(cid:12) F u (cid:3) = E Q h lim inf n →∞ Z t ∧ σ n (cid:12)(cid:12)(cid:12) F u i ≤ lim inf n →∞ E Q (cid:2) Z t ∧ σ n (cid:12)(cid:12) F u (cid:3) = Z u . THOMAS KRABICHLER AND JOSEF TEICHMANN
Therefore, Z is a Q -supermartingale. Setting u = 0 and taking expectations onboth sides of the above argument yields E Q (cid:2) Z t (cid:3) ≤ for all t ≥ . This togetherwith the càdlàg property of Z guarantees Q (cid:2) τ < ∞ (cid:3) = 0 , i.e., Z does not explode infinite time under Q . According to Section 2 in [19], there exists a unique probabilitymeasure q Q , the so-called Föllmer measure of Z , on the sub- σ -field F τ – := σ (cid:16)(cid:8) A ∩ { τ > t } (cid:12)(cid:12) A ∈ F t for some t ≥ (cid:9)(cid:17) , such that it holds(4) q Q (cid:2) A ∩ { τ > t } (cid:3) = E Q (cid:2) Z t A (cid:3) for all A ∈ F t and all t ≥ . Consistently, this may be extended to E q Q (cid:2) H t { τ>t } (cid:3) = E Q (cid:2) Z t H t (cid:3) for all q Q -integrable F t -measurable random variables H t . Particularly, itholds q Q (cid:2) τ = ∞ (cid:3) = lim k →∞ q Q (cid:2) τ > k (cid:3) = lim k →∞ E Q (cid:2) Z k (cid:3) . If we set(5) q Z t := ( Z t { τ>t } , on { Z t > } , , otherwise,then we will get for A ∈ F t and H t = q Z t A the inverse transformation formula(6) Q (cid:2) A ∩ { Z t > } (cid:3) = Q (cid:2) A ∩ { Z t > } ∩ { τ > t } (cid:3) = E q Q (cid:2) q Z t A (cid:3) . Analogously, we get E Q (cid:2) H t { Z t > } (cid:3) = E q Q (cid:2) q Z t H t (cid:3) for all Q -integrable F t -measurablerandom variables H t . In the sequel, the processes Z = ( Z t ) t ≥ and q Z = ( q Z t ) t ≥ are referred to as generalised density processes .Even though Equation (4) characterises q Q only on F τ – , we want q Q to be definedon the whole σ -field F . Consistent with Definition 2.1 and Proposition 2.3 in [24],where Föllmer measures are considered from a formal perspective, we make thefollowing definition. Definition 5.2 (Föllmer Pair) . Let (Ω , F , F , Q ) with F = ( F t ) t ≥ be a filteredprobability space, where F is the right-continuous modification of a standard system,and Z = ( Z t ) t ≥ be a non-negative Q -supermartingale with càdlàg paths and Z = 1 . Furthermore, let q Q be another probability measure on F and τ be astopping time. Then, ( q Q , τ ) is called a Föllmer pair for Z , if Q (cid:2) τ = ∞ (cid:3) = 1 andEquation (4) holds for all A ∈ F t and all t ≥ .Theorem 3.1 in [24] provides an existence and (non-)uniqueness result for Föllmermeasures on state spaces; see Definition C.3 in [24] for a reference. Beyond the timeinstance τ , q Q can be extended arbitrarily without breaking Equation (4); see alsoitem (iii) in the Appendix B of [1].Given two probability measures Q and q Q , where q Q is a Föllmer measure withrespect to Z , then τ is uniquely determined up to a q Q -nullset and Z is uniquelydetermined up to a Q -evanescent set; see Proposition 2 in [26]. If Z is a true Q -martingale, then we also have q Q (cid:2) τ < ∞ (cid:3) = 0 . In this case, Z becomes the classical Radon-Nikodym density process for the locally absolutely continuous measure q Q ≪ Q . More precisely, it holds q Q (cid:12)(cid:12) F t ≪ Q (cid:12)(cid:12) F t for all t ≥ . If, in addition, Z is strictlypositive Q -a.s., then Q and q Q are locally equivalent. If Z is strictly positive Q -a.s.but not necessarily a true Q -martingale, then only the local relation Q (cid:12)(cid:12) F t ≪ q Q (cid:12)(cid:12) F t is assured for each t ≥ . Generally, there is no order relation in the sense of ≪ CONSTRAINT-BASED NOTION OF ILLIQUIDITY 9 between Q and q Q . As highlighted in [24], Q (cid:12)(cid:12) F t and q Q (cid:2) · (cid:12)(cid:12) τ ≤ t (cid:3)(cid:12)(cid:12)(cid:12) F t are even singular,given that q Q (cid:2) τ ≤ t (cid:3) > . The first one has full mass on the event { τ = ∞} , whilethe other assigns zero mass to it. Lemma 5.3 (Generalised Bayes Formula) . Consider the setting of Definition 5.2.Then the following Bayes formula for conditional expectations holds q Q -a.s.(7) { Z t > } E q Q (cid:2) H T { τ>T } (cid:12)(cid:12) F t (cid:3) = q Z t E Q (cid:2) Z T H T (cid:12)(cid:12) F t (cid:3) for all ≤ t ≤ T < ∞ and all F T -measurable random variables H T , which are both Q - and q Q -integrable. Proof . Let A ∈ F T and B ∈ F t . Then, by applying the above transformationformulae (4) and (6) forth and back, we may write E q Q h A B { τ>T } { Z t > } i = E Q h Z T A B { Z t > } i = E Q h E Q (cid:2) Z T A (cid:12)(cid:12) F t (cid:3) B { Z t > } i = E q Q h q Z t E Q (cid:2) Z T A (cid:12)(cid:12) F t (cid:3) B i . The standard machine from measure theory yields the assertion. (cid:3) If Z is strictly positive Q -a.s. but not necessarily a true Q -martingale, thenFormula (7) simplifies to(8) E q Q (cid:2) H T { τ>T } (cid:12)(cid:12) F t (cid:3) = 1 Z t { τ>t } E Q (cid:2) Z T H T (cid:12)(cid:12) F t (cid:3) , which holds a.s. under Q and q Q alike. The choice t = 0 and H T ≡ in (7) results inthe relation q Q (cid:2) τ > T (cid:3) = E Q (cid:2) Z T (cid:3) . Hence, T E Q (cid:2) Z T (cid:3) describes the distributionof the explosion time under q Q . Equation (8) reminds of a popular intensity-basedpricing formula when default risk is modelled via filtration enlargement; e.g., seeCorollary 7.3.4.2 in [12].The next lemma describes how to realise the setting of Definition 5.2. The lemmais based on Proposition 2.5 in [24] and can be seen as a generalisation of Theorem 1in [5]. It is also presented as Theorem 1.1 in [15]. Lemma 5.4 (Inverse Construction Scheme) . For a start, let (Ω , F , F , q Q ) be afiltered probability space satisfying the usual conditions. Moreover, let q Z = ( q Z t ) t ≥ with q Z = 1 be a non-negative uniformly integrable ( F , q Q ) -martingale. Define thelocally absolutely continuous measure Q on F ∞ := W t ≥ F t via d Q d q Q (cid:12)(cid:12)(cid:12)(cid:12) F t := q Z t . Converse to (3), define for each n ∈ N the F -stopping times τ n := inf (cid:8) t ≥ (cid:12)(cid:12) q Z t < /n (cid:9) , τ := lim n →∞ τ n . Then ( q Q , τ ) forms a Föllmer pair for the Q -supermartingale Z := q Z − . In addition,the following two equivalence statements hold: • Z is a Q -local martingale if and only if(9) q Q (cid:2) τ < ∞ , q Z τ – = 0 (cid:3) = 0 , i.e., q Z does not jump to zero q Q -a.s. • Z is a true Q -martingale if and only if q Z is strictly positive q Q -a.s.Noteworthy, the standard bottom-up approach as in [19] or [24], with a possiblynon-unique q Q , is consistent with the proposed top-down construction. Accordingto Lemma 3 in [19], Condition (9) is satisfied naturally. As is well-known, q Z willstay in zero after τ q Q -a.s.; e.g., see Proposition II.3.4 in [25]. Proof . It needs to be shown that the following four items hold:(1) Q (cid:2) τ < ∞ (cid:3) = 0 .(2) Equation (4) is satisfied for all A ∈ F t and all t ≥ .(3) Z really is a Q -supermartingale.(4) The martingale properties of Z under Q are equivalent to the stated pathproperties of q Z under q Q .We proceed in successive steps: Proof of 1 . By the right-continuity of q Z , we havethe upper bound q Z τ n ≤ n on the event where τ n is finite. Hence, it holds for all t ≥ and for all n ∈ NQ (cid:2) τ ≤ t (cid:3) ≤ Q (cid:2) τ n ≤ t (cid:3) = E q Q (cid:2) q Z t { τ n ≤ t } (cid:3) = E q Q (cid:2) q Z τ n { τ n ≤ t } (cid:3) ≤ n . We required the uniform integrability of Z for the optional stopping theorem inthe penultimate step; see Theorem II.3.2 in [25] and the integral counterexamplethereafter. Proof of 2 . By construction, it holds(10) E Q (cid:2) H t (cid:3) = E q Q (cid:2) q Z t H t (cid:3) for all Q -integrable F t -measurable random variables H t . Let A ∈ F t . If we set H t := Z t A , then we can write q Q (cid:2) A ∩{ τ > t } (cid:3) = E q Q (cid:2) q Z t Z t A ∩{ τ>t } (cid:3) = E Q (cid:2) Z t A ∩{ τ>t } (cid:3) = E Q (cid:2) Z t A (cid:3) . In the first equation, we exploited that q Z t > holds q Q -a.s. on { τ > t } .Then, we used Formula (10). Eventually, we could omit the restriction to the event { τ > t } , because it has full Q -mass anyway. Proof of 3 . According to the first step of the proof, Z as the inverse of q Z iswell-defined Q -a.s. Furthermore, Z is a Q -supermartingale since it holds for all ≤ u ≤ t < ∞ and for all A ∈ F u E Q (cid:2) Z t A (cid:3) = q Q (cid:2) A ∩ { τ > t } (cid:3) ≤ q Q (cid:2) A ∩ { τ > u } (cid:3) = E Q (cid:2) Z u A (cid:3) . Proof of 4 . The argument is motivated by Example 4.1 in [24]. If Condition (9)is not satisfied, then Z cannot form a Q -local martingale. In fact, any localisingsequence that preserves the expectation at the stopping time under Q must remainfinite with positive Q -probability. This certainly contravenes the local martingaleproperty of Z ; see Example 4.1 in [24] for the exact details. If Condition (9)is met, then ( τ n ∧ n ) n ∈ N constitutes a localising sequence. Indeed, let ρ be anarbitrary bounded stopping time. All we need to show is that Z τ n ∧ nρ = Z ρ ∧ τ n ∧ n is Q -integrable and that E Q (cid:2) Z τ n ∧ nρ (cid:3) = E Q (cid:2) Z τ n ∧ n (cid:3) holds for all n ∈ N ; e.g., see CONSTRAINT-BASED NOTION OF ILLIQUIDITY 11
Theorem II.3.5 in [25]. What we already know is the validity of the generalisedFöllmer property(11) q Q (cid:2) A ∩ { τ > ρ } (cid:3) = E Q (cid:2) Z ρ A (cid:3) for all A ∈ F ρ := σ (cid:16)(cid:8) A ∈ F (cid:12)(cid:12) A ∩{ ρ ≤ t } ∈ F t for all t ≥ (cid:9)(cid:17) and all finite stoppingtimes ρ ; see Proposition 2.3 together with Definition 2.1 in [24]. Alternatively, seethe first part in the proof of Lemma 6.5 below. On the one hand, Z is a Q -supermartingale. Thus, the optional stopping theorem yields(12) E Q (cid:2) Z τ n ∧ nρ (cid:3) ≤ E Q (cid:2) Z (cid:3) = 1 . On the other hand, Condition (9) is equivalent to saying that q Q (cid:2) ( τ n ∧ n ) < τ (cid:3) = 1 for all n ∈ N . Consequently, (11) gives(13) E Q (cid:2) Z τ n ∧ nρ (cid:3) = q Q (cid:2) τ > ( ρ ∧ τ n ∧ n ) (cid:3) ≥ q Q (cid:2) τ > ( τ n ∧ τ ) (cid:3) = 1 . Combining (12) and (13) yields E Q (cid:2) Z τ n ∧ nρ (cid:3) ≡ .The last equivalence does not require separate attention. It follows straightforwardlywith the same arguments as in the proof of Lemma 6.3 below. This concludes ourproof. (cid:3) Remark 5.5 (Uniform Integrability of q Z ) . It needs to be noted that the uniformintegrability of q Z in Lemma 5.4 is not a necessary condition. We only required it inthe first step of the proof and in (12). Generally, after having chosen a particularmodel, one may verify Q (cid:2) τ < ∞ (cid:3) = 0 and the local martingale property of Z under Q alternatively; for instance, see Example 5.26 in [17] for an illustration. Ina non-pathological setting for which q Z is uniformly integrable, it generally holds q Q (cid:2) τ < ∞ (cid:3) < . Indeed, if it held q Q (cid:2) τ < ∞ (cid:3) = 1 , then we would end up with therequisite q Z t = E q Q (cid:2) q Z ∞ (cid:12)(cid:12) F t (cid:3) ≡ due to q Z ∞ = lim t →∞ q Z t = 0 ; see Theorem II.3.1 in[25]. (cid:3) Stochastic Basis of the Enhanced FX-like Setting
Let (Ω , F , F ) with F = ( F t ) t ≥ denote a filtered space that is equipped with twoexogenously given probability measures Q and q Q , where the local relation Q (cid:12)(cid:12) F t ≪ q Q (cid:12)(cid:12) F t holds for every t ≥ . The stochastic basis carries three F -adapted familiesof zero-coupon bond price processes (cid:8) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ , (cid:8) e P ( t, T ) (cid:9) ≤ t ≤ T < ∞ and (cid:8) Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ , such that the familiar payoff relations (1) and (2) from the FX-like approach are satisfied. The recovery rate process S = ( S t ) t ≥ is still definedvia S t := e P ( t, t ) . Furthermore, the stochastic basis carries the two genuine bankaccount numéraires B = ( B t ) t ≥ and q B = ( q B ) t ≥ from the domestic and theforeign market respectively. We restrict ourselves to a particular setting. The firsttwo items are of a technical nature. The last one is motivated by the observationsfrom Example 3.1. Assumption 6.1 (The Enhanced FX-like Setting) . (1) F satisfies the usual conditions under q Q .(2) All involved processes (cid:0) P ( t, T ) (cid:1) ≤ t ≤ T , (cid:0) e P ( t, T ) (cid:1) ≤ t ≤ T , (cid:0) Q ( t, T ) (cid:1) ≤ t ≤ T forany T ≥ as well as S , B and q B are non-negative q Q -a.s. Moreover, theyall admit q Q -indistinguishable càdlàg versions. (3) Concerning absence of arbitrage, the discounted price processes admit thefollowing properties:(a) (cid:0) B t − P ( t, T ) (cid:1) ≤ t ≤ T defines a Q -local martingale for each maturity T ≥ .(b) (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T defines a Q -local martingale for each maturity T ≥ .(c) Z = ( Z t ) t ≥ with Z t := S t q B t S B t defines a Q -local martingale and q B remains finite perpetually as seen under Q . q B may nonetheless divergeand reach the cemetery state { + ∞} in finite time as seen under q Q .(d) Let the explosion time τ be defined as before in (3). Then, with anabuse of notation, the process q Z = ( q Z t ) t ≥ with q Z t := Z t { τ>t } , inthe sense of (5) is a true q Q -martingale and coincide with the densityprocess of Q (cid:12)(cid:12) F t with respect to its locally dominating counterpart q Q (cid:12)(cid:12) F t .The following four lemmata illustrate some implications of Assumption 6.1. Lemma 6.2 (Föllmer Pair) . Let Assumption 6.1 be met. Then, ( q Q , τ ) is a Föllmerpair for Z in the sense of Definition 5.2. Proof . Since Z is a Q -local martingale, it does not explode in finite time. Thus,we have Q (cid:2) τ < ∞ (cid:3) = 0 . Moreover, by item 3. d) of Assumption 6.1, we have forall t ≥ and for all Q -integrable F t -measurable functions H t (14) E Q (cid:2) H t (cid:3) = E q Q (cid:2) q Z t H t (cid:3) . Let A ∈ F t . If we choose H t := Z t A , then we get q Q (cid:2) A ∩ { τ > t } (cid:3) = E q Q (cid:2) q Z t H t (cid:3) = E Q (cid:2) Z t A (cid:3) , where we used Formula (14) in the second equation. This yields theassertion. (cid:3) Lemma 6.2 will help us in the next section to extend the idea of Example 3.1 ina measurable way to arbitrary time instances t ≥ . Lemma 6.3 (Perfect Liquidity) . Let Assumption 6.1 be met. Then, the followingstatements are equivalent:(1) Q (cid:12)(cid:12) F t ≈ q Q (cid:12)(cid:12) F t for each t ≥ .(2) q Q (cid:2) τ < ∞ (cid:3) = 0 .(3) Z is a true Q -martingale. Proof . "1. = ⇒ { τ ≤ t } is a Q -nullset for all t ≥ . Due to the assumed local equivalence, { τ ≤ t } is also a q Q -nullset for all t ≥ . Consequently, as { τ ≤ n } ∞ n =1 is an increasing sequence and q Q is σ -additive, we have q Q (cid:2) τ < ∞ (cid:3) = q Q (cid:20) ∞ [ n =1 { τ ≤ n } (cid:21) = lim n →∞ q Q (cid:2) τ ≤ n (cid:3) = 0 . "2. = ⇒ { τ < ∞} is a q Q -nullset, then Z q Z becomes q Q -indistinguishablefrom a constant process at the level . This obviously forms a martingale under q Q .According to the Bayes formula for conditional expectations, a process X = ( X t ) t ≥ CONSTRAINT-BASED NOTION OF ILLIQUIDITY 13 is a Q -martingale if and only if q ZX is a q Q -martingale; see Formula (8)."3. = ⇒ q Q (cid:12)(cid:12) F t ≪ Q (cid:12)(cid:12) F t . Let A ∈ F t be an arbitrary Q -nullset. Firstly, if Z is a true Q -martingale, then we have t E Q (cid:2) Z t (cid:3) ≡ .Secondly, as shown in the previous lemma, q Q is a Föllmer measure of Z . Thus, bydefinition, we have q Q (cid:2) τ ≤ t (cid:3) = 1 − q Q (cid:2) Ω ∩{ τ > t } (cid:3) = 1 − E Q (cid:2) Z t Ω (cid:3) = 1 − E Q (cid:2) Z t (cid:3) = 0 for any t ≥ . Now we can proceed similarly as in the first step of the proof in orderto verify that { τ < ∞} is also a q Q -nullset. Therefore, we may easily conclude with q Q (cid:2) A (cid:3) = q Q (cid:2) A ∩ { τ > t } (cid:3) = E Q (cid:2) Z t A (cid:3) = 0 . (cid:3) As indicated by Example 3.1, the strict local martingale property of Z featuresaspects of illiquidity. Thus, Lemma 6.3 provides equivalent characterisations of amarket that is equipped with perfect asset liquidity. In this case, we could just aswell consider the general FX-like setting of Assumption 2.2 instead. Lemma 6.4 (Incomplete Filtration) . Let Assumption 6.1 be met and let Z be astrict Q -local martingale. Then, F cannot be complete under Q . Proof . We proceed by contradiction. If F was Q -complete, then we would have { τ ≤ T } ∈ F for all T ≥ . This is because { τ ≤ T } is contained in the Q -nullset { τ < ∞} . On the contrary, it exists a T > such that q Q (cid:2) τ ≤ T (cid:3) > .Otherwise, if no such T existed, then Z would be a true Q -martingale accordingto the previous lemma. However, this would be a contradiction to the premisethat Z is a strict Q -local martingale. Thus, under the assumption that F was Q -complete, it would hold q Q -a.s. Z { τ> } = q Z = E q Q (cid:2) q Z T (cid:12)(cid:12) F (cid:3) , and, as { τ>T } = { τ>T } { τ>T } and { τ>T } is F -measurable, E q Q (cid:2) q Z T (cid:12)(cid:12) F (cid:3) { τ>T } .Combining these two representations of yields q Q (cid:2) τ > T (cid:3) = 1 , which is obviouslya contradiction to our choice of T . (cid:3) The argument in the proof of Lemma 6.4 is very intuitive. If the explosiontime τ of the bubble Z is already known beforehand, then Q and q Q essentiallyhave to be equivalent. Lemma 6.4 highlights that modelling asset liquidity in theproposed way involves certain technical obstacles. We can no longer assume that F fulfils the usual conditions under Q ; see also Example 2.8 in [24]. Nonetheless,if F were not complete under q Q , it could be augmented to an (cid:0) F , q Q (cid:1) -complete F straightforwardly. τ would remain an F -stopping time and Z an (cid:0) F , Q (cid:1) -localmartingale according to Lemma 1 in [19]. Additionally, Equation (4) would easilyextend its area of validity to all A ∈ F t . Thus, the first item of Assumption 6.1does not pose any problems.The last lemma in this section says under what circumstances the two-priceeconomy in the foreign market does not involve arbitrage opportunities, at leastup to the default time τ . If it also holds Q ( T, T ) = { τ>T } q Q -a.s., which is notfar-fetched in the light of Lemma 7.11 below, then absence of arbitrage can evenbe guaranteed for all times. Lemma 6.5 (Absence of Arbitrage) . Let Assumption 6.1 be satisfied. Moreover,let ( σ n ) n ∈ N be a localising sequence for (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T under Q , which satisfiesthe monotonous convergence property lim n →∞ σ n = T also under q Q . Then (cid:0) σ n ) n ∈ N is also a q Q -localising sequence for the q Q -local martingale (cid:0) q B − t Q ( t, T ) { τ>t } (cid:1) ≤ t ≤ T . Remark 6.6 (Regularity Assumption) . The additional assumption about ( σ n ) n ∈ N is necessary and, unfortunately, cannot be relaxed. It is just uncertain how thecharacteristic properties of ( σ n ) n ∈ N carry over when changing to the dominatingmeasure q Q . They still prevail on { τ = ∞} , but not necessarily on the Q -nullset { τ < ∞} . As a matter of fact, one cannot modify ( σ n ) n ∈ N on { τ < ∞} to ( e σ n ) n ∈ N and still comply with the requirements lim n →∞ e σ n = T q Q -a.s. and { e σ n ≤ t } ∈ F t for all t ≥ and all n ∈ N . Lemma 6.5 holds naturally when one equips theenhanced FX-like setting with HJM-dynamics; see Section 6.5 in [17]. (cid:3) Proof . Similarly as in the proof of Theorem 2.1 in [1], one extends Formula (8)to(15) E q Q (cid:2) H σ ∧ T { τ>σ ∧ T } (cid:12)(cid:12) F t (cid:3) = 1 Z σ ∧ t { τ>σ ∧ t } E Q (cid:2) Z σ ∧ T H σ ∧ T (cid:12)(cid:12) F t (cid:3) for all stopping times σ . Indeed, by construction of the Föllmer measure, itholds d q Q (cid:12)(cid:12) F τn – = Z τ n d Q (cid:12)(cid:12) F τn – , where Z τ n is well-defined since the stopped process (cid:0) Z τ n t (cid:1) t ≥ forms a uniformly integrable martingale; see also Lemma A.3 in [1]. Let A ∈ F σ ∧ T , where F σ ∧ T := σ (cid:16)(cid:8) A ∈ F (cid:12)(cid:12) A ∩ { σ ∧ T ≤ t } ∈ F t for all t ≥ (cid:9)(cid:17) . Then, we can write q Q (cid:2) A ∩ { τ > σ ∧ T } (cid:3) = lim n →∞ q Q (cid:2) A ∩ { τ n > σ ∧ T } (cid:3) = lim n →∞ E Q (cid:2) Z τ n A ∩{ τ n >σ ∧ T } (cid:3) = lim n →∞ E Q (cid:2) Z σ ∧ T A ∩{ τ n >σ ∧ T } (cid:3) = E Q (cid:2) Z σ ∧ T A (cid:3) , where we utilised Q (cid:2) τ = ∞ (cid:3) = 1 and dominated convergence in the last equation.Having Formula (4) generalised to capped stopping times, we can proceed exactlyas in the proof of Lemma 5.3 in order to derive (15). Let ( σ n ) n ∈ N be a localisingsequence for (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T under Q . Thereby, ( σ n ) n ∈ N is increasing with lim n →∞ σ n = T Q -a.s. and it holds(16) E Q (cid:20) e P ( σ n ∧ t, T ) B σ n ∧ t (cid:12)(cid:12)(cid:12)(cid:12) F u (cid:21) = e P ( σ n ∧ u, T ) B σ n ∧ u Q -a.s. for all ≤ u ≤ t ≤ T and all n ∈ N . Consequently, combining (15) and (16)yields E q Q (cid:20) Q ( σ n ∧ t, T ) B σ n ∧ t { τ>σ n ∧ t } (cid:12)(cid:12)(cid:12)(cid:12) F u (cid:21) = B σ n ∧ u S σ n ∧ u q B σ n ∧ u { τ>σ n ∧ u } E Q (cid:20) S σ n ∧ t q B σ n ∧ t B σ n ∧ t Q ( σ n ∧ t, T ) q B σ n ∧ t (cid:12)(cid:12)(cid:12)(cid:12) F u (cid:21) = B σ n ∧ u S σ n ∧ u q B σ n ∧ u { τ>σ n ∧ u } E Q (cid:20) e P ( σ n ∧ t, T ) B σ n ∧ t (cid:12)(cid:12)(cid:12)(cid:12) F u (cid:21) = B σ n ∧ u S σ n ∧ u q B σ n ∧ u { τ>σ n ∧ u } e P ( σ n ∧ u, T ) B σ n ∧ u CONSTRAINT-BASED NOTION OF ILLIQUIDITY 15 = Q ( σ n ∧ u, T ) B σ n ∧ u { τ>σ n ∧ u } q Q -a.s. for all ≤ u ≤ t ≤ T and all n ∈ N . (cid:3) Remark 6.7 (Weak Ineligibility) . In order to end up in a non-trivial case of theenhanced FX-like setting as under Assumption 6.1, Z does not necessarily have tobe ineligible in the sense that Z is a strict local martingale under any equivalentseparating measure; e.g., see Definition 4.11 in [17]. We only require that Z is atleast a strict local martingale under the exogenously chosen risk-neutral referencemeasure Q . As a consequence, the local absolute continuity of Q with respect to q Q will also be strict. However, there may well exist a locally equivalent measure b Q ≈ Q under which Z forms a true martingale; e.g., see Example 5.5 in [10].Therefore, we slightly weaken the notion of ineligibility. (cid:3) The Illiquidity Premium in the Enhanced FX-like Setting
Motivated by Example 3.1, we make the following definition.
Definition 7.1 (Illiquidity Deflator) . Let the enhanced FX-like setting as specifiedin Assumption 6.1 be given. We define the illiquidity deflator as the Q -localmartingale Z = ( Z t ) t ≥ with Z t := S t q B t S B t . The illiquidity deflator Z is vulnerable to hyperinflation as q B tends to explodeonce the recovery rate has depreciated. As long as full recovery is given, Z may beinterpreted as the limiting value process of roll-over strategies in defaultable zero-coupon bonds with declining holding periods; see the exposition in [16]. Z deflatesilliquidity since the resulting pricing machinery is conducted as if there was perfectliquidity in the market.Let the explosion time τ be defined as in Equation (3). A priori, the randomvariable q B t q B T for ≤ t ≤ T < ∞ is only well-defined on the event { τ > T } .Nonetheless, we can extend its domain beyond τ almost arbitrarily. Once the foreignbank account process is about to reach the cemetery state { + ∞} , we suspend theprevious regime and replace q B by a suitable (0 , ∞ ) -valued q B ◦ . More precisely, aswe cannot override the value of q B t in hindsight if τ happens to lie within the range ( t, T ] , we consider for ≤ t ≤ T < ∞ the stochastic discount factors(17) q B t q B T { τ>T } + q B ◦ t q B ◦ T { τ ≤ T } . Still, we will stick to the notation q B t q B T . Under Q , the changes are concentratedon a nullset and raise no issues. Under q Q , the density q Z t = Z t { τ>t } for t ≥ will not be affected by the modification either. For instance, we may want to set q B t q B T ≡ whenever T ≥ τ . Since this may be too restrictive in some applications,we simply proclaim a general integrability condition. As we shall see below, thevalues of q B t q B T beyond τ have a crucial impact on the term structure of illiquidity;see also the Examples 7.7 and 7.8 below. τ naturally describes the default time. Ifno counterparty is willing to lend capital under any circumstances, refinancing cost inevitably explode and the business model ceases to be viable. After this regimeswitch, q B ◦ accounts for the risky interest rate term structure of the post-bankruptcyera. Assumption 7.2 (Integrability of the Stochastic Discount Factors) . Let the settingof Assumption 6.1 be given. In the sense of Ansatz (17), we assume that thestochastic discount factor q B t q B T is (0 , ∞ ) -valued q Q -a.s. and integrable with respect to q Q for any ≤ t ≤ T < ∞ . Definition 7.3 (Liquidity Adjusted Price and Illiquidity Premium) . Let the settingof Assumption 6.1 be given, Assumption 7.2 be fulfilled and ≤ t ≤ T < ∞ . Wedefine the t - liquidity adjusted price of a foreign zero-coupon bond maturing at time T as q Q ( t, T ) := E q Q (cid:20) q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . The price difference L ( t, T ) := S t q Q ( t, T ) − e P ( t, T ) = S t (cid:0) q Q ( t, T ) − Q ( t, T ) (cid:1) is referredto as the illiquidity premium of e P ( t, T ) . S t q Q ( t, T ) may be interpreted as a domestic fair t -value of a defaultable zero-coupon bond seen from a foreign investor. Thus, its deviance from e P ( t, T ) is anatural candidate in order to quantify asset liquidity. A positive illiquidity premium L ( t, T ) infers an illiquid zero-coupon bond e P ( t, T ) , whereas a negative illiquiditypremium relates to hyperliquidity; see below for further clarification. It needs tobe noted that the foreign market may only be free of arbitrage under q Q withinthe stochastic interval [0 , τ ) . L ( t, T ) is mainly of interest on { τ > t } . After τ ,all obligations are unwinded and the hitherto existing liquidity framework becomesredundant. Remark 7.4 (Mathematical Concept of Illiquidity) . Limited institutional liquidityat time t goes along with a low asset liquidity of e P ( t, t + dt ) , that inevitably affectsthe recovery rate adversely. As the classical FX-like approach covers both defaultand migration risk, the enhanced FX-like setting unites the two aspects of asset andinstitutional liquidity. What is more, since the recovery rate S enters the illiquiditydeflator Z , the enhanced FX-like setting features an interdependence between creditand liquidity risk. Therefore, we deem the ineligibility of Z the right mathematicalconcept to describe illiquidity. If required, aspects of liquidity can be analysedon a stand-alone basis simply by setting the recovery rate S ≡ ; however theseconsiderations are delicate due to inherent arbitrage. One rather interprets theimpact of illiquidity as a deviance from the intrinsic economic value. Consequently,it suffices to consider one term structure together with an associated numéraire. (cid:3) Definition 7.5 (Illiquidity Factor) . The term-dependent ratio q Ξ( t, T ) = Q ( t, T ) q Q ( t, T ) for ≤ t ≤ T < ∞ is referred to as illiquidity factor . q Ξ( t, T ) < features illiquidity,whereas q Ξ( t, T ) = 1 describes an equilibrium between supply and demand.We tacitly assume that the term structure (cid:8) q Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ is strictly positive q Q -a.s. Thus, the illiquidity factors in Definition 7.5 are well-defined. This subtlety CONSTRAINT-BASED NOTION OF ILLIQUIDITY 17 is not pedantic in the light of Lemma 7.11 below. One may consider four differentcases of the enhanced FX-like setting; see Table 1. Each case describes a distinctmarket situation. Z = ( Z t ) t ≥ is a Z = ( Z t ) t ≥ is a stricttrue Q -martingale Q -local martingale (cid:16) e P ( t,T ) B t (cid:17) ≤ t ≤ T is a true model of the st kind, model of the nd kind,efficient market, illiquid market, Q -martingale L ( t, T ) ≡ L ( t, T ) ≥ (cid:16) e P ( t,T ) B t (cid:17) ≤ t ≤ T is a strict model of the rd kind, model of the th kind,hyperliquid market, general market, Q -local martingale L ( t, T ) ≤ L ( t, T ) state-dependent Table 1.
This table provides an overview of the four distinctmarket situations in which the enhanced FX-like setting may beconsidered. In the models of the st and the rd kind, we have thelocal equivalence Q (cid:12)(cid:12) F t ≈ q Q (cid:12)(cid:12) F t . For those of the nd and th kind,it only holds Q (cid:12)(cid:12) F t ≪ q Q (cid:12)(cid:12) F t .7.1. Model of the 1st Kind.
Let the setting of Assumption 6.1 be given and letboth (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T and Z = ( Z t ) t ≥ be true Q -martingales. In this case, theclassical change of numéraire technique yields q Q ( t, T ) = E q Q (cid:20) q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = Q ( t, T ) . Hence, this model captures perfect liquidity with L ( t, T ) ≡ and q Ξ( t, T ) ≡ . Thisis a special case of the general FX-like setting.7.2. Model of the 2nd Kind.
Likewise, let the setting of Assumption 6.1 begiven. Moreover, let (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T be a true Q -martingale, whereas Z =( Z t ) t ≥ is a strict Q -local martingale. If τ denotes the explosion time as defined inEquation (3), then it holds by the Bayes formula (8) both Q -a.s. and q Q -a.s. S t q Q ( t, T ) = S t E q Q (cid:20) q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ≥ S t E q Q (cid:20) q B t q B T { τ>T } (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) (18) = { τ>t } S t B t S t q B t E Q (cid:20) S T q B T B T q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = { τ>t } e P ( t, T ) . The liquidity adjusted prices also infer a non-defaultable term structure, since q Q ( T, T ) = 1 holds q Q -a.s. for all T . Due to Q (cid:2) τ = ∞ (cid:3) = 1 , the illiquidity premia arenon-negative Q -a.s. This is, however, not necessarily the case q Q -a.s., since e P ( t, T ) still might exceed S t q Q ( t, T ) on { τ ≤ t } . Remark 7.6 (Model of the nd Kind) . The presented concept of the illiquiditypremium heavily relies on the premise that Q is a true martingale measure for thedefaultable zero-coupon bonds. It cannot be relaxed without destroying the Q -a.s.order relation Q ( t, T ) ≤ q Q ( t, T ) . If (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T formed a strict Q -localmartingale and, hence, a Q -supermartingale, then we would have both Q -a.s. and q Q -a.s. S t E q Q (cid:20) q B t q B T { τ>T } (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = B t q B t { τ>t } E Q (cid:20) S T q B T B T q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ≤ { τ>t } e P ( t, T ) , where we used the Bayes formula (8) in the first equation. Thus, the argument(18) would not withstand any longer. Illiquidity as an asymmetry between supplyand demand can attain two states. We deem the risk that you hardly find a buyerfor certain corporate loans higher and more often present in the real world thanthe opposite situation in which an abundance of market participants is craving fora hardly available asset. Thus, we typically have L ( t, T ) ≥ , i.e., the corporateloans are traded below their intrinsic economic value. Equivalently, the issuer of theloans have to bear a higher interest rate burden such that market participants arewilling to invest. All in all, the model of the nd kind may be somehow consideredas standard case in the presence of illiquidity. For instance, this can be utilised forthe modelling of the interbank market; see Chapter 7 in [17]. (cid:3) Example 7.7 (Flat Post-Default Curve) . Let the enhanced FX-like setting ofthe nd kind be given. If it holds q B t q B T ≡ for any T ≥ τ , then a straightforwardcalculation yields q Q ( t, T ) = E q Q (cid:20) q B t q B T { τ ≤ T } (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) + E q Q (cid:20) q B t q B T { τ>T } (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = q Q (cid:2) τ ≤ T (cid:12)(cid:12) F t (cid:3) + Q ( t, T ) { τ>t } . Therefore, L ( t, T ) { τ>t } = S t q Q (cid:2) t < τ ≤ T (cid:12)(cid:12) F t (cid:3) , whereas the post-default illiquiditypremium reduces to L ( t, T ) { τ ≤ t } = (cid:0) S t − e P ( t, T ) (cid:1) { τ ≤ t } . The model features aliquidity premium that is, up to time τ , proportional to the conditional defaultprobability. After τ , the liquidity premium is the deviance of the time value ofthe defaultable zero-coupon bond from the most current recovery rate. Thus, if T e P ( t, T ) does not become flat after τ , bond holders must bear an illiquiditydiscount during the unwinding process. A pertinent model choice for q Q ( t, T ) mayconsist of an HJM-framework for Q ( t, T ) and an intensity-based approach for τ under q Q ; see Chapter 6 and Section A.2 in [17] for further details. (cid:3) Example 7.8 (Non-trivial Post-Default Curve) . Let the enhanced FX-like settingof the nd kind be given. If the stochastic discount factors q B t q B T are replaced by q B ◦ t q B ◦ T for all ≤ τ ≤ T < ∞ and t ≤ T , where q B ◦ itself induces the termstructure T q Q ◦ ( t, T ) for ≤ t ≤ T < ∞ and is conditionally independent CONSTRAINT-BASED NOTION OF ILLIQUIDITY 19 from τ , then similar calculations as in the previous example yield L ( t, T ) { τ>t } = S t q Q ◦ ( t, T ) q Q (cid:2) t < τ ≤ T (cid:12)(cid:12) F t (cid:3) and L ( t, T ) { τ ≤ t } = S t (cid:0) q Q ◦ ( t, T ) − Q ( t, T ) (cid:1) { τ ≤ t } . Inthis case, the corresponding pre-default illiquidity premium is given by a productof the conditional default probability times the conversion of the term structure T q Q ◦ ( t, T ) into the domestic market. After τ , the illiquidity premia becomemere spreads. (cid:3) Remark 7.9 (Analytical Tractability of the Illiquidity Premium) . In order tocalculate an illiquidity premium as modelled in the previous two examples, oneshould be able to derive the cumulative distribution function of the explosion time τ under q Q . Exemplarily, the corresponding Laplace transform can be characterisedif Z is a one-dimensional diffusion; e.g., see [14] or [20]. Other examples with semi-explicit formulae for the distribution of the explosion time can be constructed basedon Section 6 of [14]. Unfortunately, only little is known in this regard if Z followsgeneral Itô-dynamics. This is in contrast to ordinary differential equations forwhich the explosion time is known explicitly; e.g., see [9]. Facing that difficulty, theauthors chose an indirect approach to model the illiquidity premium in Section 6.5of [17]. (cid:3) Proposition 7.10 (Forward Measures) . Let the setting of Assumption 6.1 begiven. Moreover, let (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T be a true Q -martingale. Then we havethe absolute continuity e Q T ≪ q Q T and d e Q T d q Q T (cid:12)(cid:12)(cid:12)(cid:12) F t = q Q (0 , T ) Q (0 , T ) Q ( t, T ) q Q ( t, T ) = q Ξ( t, T ) q Ξ(0 , T ) . Particularly, (cid:0) q Ξ( t, T ) (cid:1) ≤ t ≤ T is a q Q T -martingale. Proof . It holds for any ≤ t ≤ Td e Q T d q Q T (cid:12)(cid:12)(cid:12)(cid:12) F t = d e Q T d Q (cid:12)(cid:12)(cid:12)(cid:12) F t × d Q d q Q (cid:12)(cid:12)(cid:12)(cid:12) F t × d q Q d q Q T (cid:12)(cid:12)(cid:12)(cid:12) F t = e P ( t, T ) e P (0 , T ) B t × S B t S t q B t × q Q (0 , T ) q B t q Q ( t, T ) , which yields the assertion. (cid:3) As it turns out, the term structure T Q ( t, T ) is no longer default-free whenit is considered under the dominating measure q Q . In fact, the corresponding payoffs Q ( T, T ) = { τ>T } are of the all-or-nothing type. Full recovery is given throughoutuntil but excluding τ , and zero recovery prevails thereafter. Even though fullconsistency is ensured due to Q (cid:2) τ < ∞ (cid:3) = 0 , this observation is counter-intuitivein the light of Paradigm 2.1. Lemma 7.11 (Illiquidity as an Invisible Default Event) . Let Assumption 6.1 besatisfied. Moreover, let (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T be a true Q -martingale. Then it holds Q ( t, T ) { τ ≤ t } = q Ξ( t, T ) { τ ≤ t } = 0 q Q -a.s. for all ≤ t ≤ T < ∞ . Remark 7.12 (Explicit Arbitrage) . Let us assume that it holds Q ( t, T ) < q Q ( t, T ) and hypothetically (despite asset liquidity constraints) that both term structuresare marketable. This opens the way for an explicit arbitrage under Q , but notnecessarily under q Q . Provided that the strategy is admissible, shorting q Q ( t, T ) and entering a long position in Q ( t, T ) leaves you with a free lunch. As seen under Q , thepayoffs q Q ( T, T ) = Q ( T, T ) = 1 a.s. offset each other. Hence, the initial discrepancycan fully be consumed by the financial agent. In contrast, as seen under q Q , thesmaller price Q ( t, T ) accounts for the possible shortfall in Q ( T, T ) = { τ>T } thatmay occur with strictly positive q Q -likelihood. Thus, the arbitrage opportunitypresumably vanishes under q Q . Proposition 4.20 in [17] tells us that exploiting this Q -arbitrage is even optimal in some sense. It is certainly scalable arbitrarily; seealso Example 7.14 below. (cid:3) Proof . By construction of the enhanced FX-like approach, it holds under thestated premises for all ≤ t ≤ T < ∞ e Q T (cid:2) τ ≤ t (cid:3) = E q Q T " d e Q T d q Q T (cid:12)(cid:12)(cid:12)(cid:12) F t { τ ≤ t } = q Ξ(0 , T ) − E q Q T h q Ξ( t, T ) { τ ≤ t } i , where we used Proposition 7.10 in the last step. Therefore, it must hold q Q T -a.s. q Ξ( t, T ) { τ ≤ t } = 0 . Utilising the equivalence q Q T ≈ q Q yields the assertion. (cid:3) Lemma 7.13 (Absence of Arbitrage) . Let the setting of Assumption 6.1 be given.Moreover, let (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T be a true Q -martingale. In that case, the process (cid:0) q B − t Q ( t, T ) (cid:1) ≤ t ≤ T is a q Q -martingale. Proof . Using the previous lemma back and forth as well as the Bayes formula (8),we get q Q -a.s. for all ≤ u ≤ t ≤ T < ∞ E q Q (cid:20) Q ( t, T ) q B t (cid:12)(cid:12)(cid:12)(cid:12) F u (cid:21) = E q Q (cid:20) Q ( t, T ) q B t { τ>t } (cid:12)(cid:12)(cid:12)(cid:12) F u (cid:21) = B u S u q B u { τ>u } E Q (cid:20) S t q B t B t Q ( t, T ) q B t (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = B u S u q B u { τ>u } e P ( u, T ) B u = Q ( u, T ) q B u . This proves the assertion. (cid:3)
Example 7.14 (Explicit Arbitrage) . The following construction is based on thearticle [5] that studies arbitrage opportunities in FX markets. The relevance of thisarticle for our credit and liquidity risk setting is apparent.Let | W = ( | W ) t ≥ be a Brownian motion on (Ω , F , F , q Q ) with its completed naturalfiltration F = ( F t ) t ≥ . The density process q Z = ( q Z t ) t ≥ with q Z t := 1 + | W τ ∧ t and τ := inf (cid:8) t > (cid:12)(cid:12) | W t = − (cid:9) is a Brownian motion started at the level and stoppedonce it has hit the origin. We define the locally absolutely continuous measure Q ≪ q Q on F ∞ := W t ≥ F t via d Q d q Q (cid:12)(cid:12) F t := q Z t . Then it holds Q (cid:2) τ ≤ T (cid:3) = 0 for each T ≥ and f W t := | W t − Z t q Z u du defines a Q -Brownian motion. As seen under Q , q Z is a Bessel process of dimensionthree (
Bes ) started at . Its inverse Z t = q Z − t satisfies dZ t = − Z t d f W t and isthe stereotypical example of a strict Q -local martingale. As exposed in [5], the Bes process satisfies the no-arbitrage property with respect to simple integrands.However, it permits arbitrage with respect to general admissible integrands. Thus,while the submarket (1 , Z ) with Z t = S t q B t S B t and the numéraire B = ( B t ) t ≥ fulfils CONSTRAINT-BASED NOTION OF ILLIQUIDITY 21 (NFLVR) under Q , a riskless profit can be made after having conducted an unduechange of numéraire to Z resulting in (1 , q Z ) . This approach allows to specify thearbitrage strategy explicitly.Let us fix a maturity T > for which q Q (cid:2) τ ≤ T (cid:3) > holds. The reflection principlefor Brownian motion and the Markov property yield as derived in Section 5 of [3] q Q (cid:2) τ ≤ T (cid:12)(cid:12) F t (cid:3) = ( , on { τ ≤ t } , (cid:16) − q Z t √ T − t (cid:17) , on { τ > t } . Applying Itô’s formula results on { τ > t } in the replication strategy q Q (cid:2) τ ≤ T (cid:12)(cid:12) F t (cid:3) = q Q (cid:2) τ ≤ T (cid:3)| {z } =2Φ (cid:0) − √ T (cid:1) < − r π Z τ ∧ t √ T − u e − | Z uT − u d q Z u .a T := q Q (cid:2) τ > T (cid:3) / q Q (cid:2) τ ≤ T (cid:3) = 1 / (cid:0) − √ T (cid:1) − is a strictly positive constant.The payoff f := { τ>T } − a T { τ ≤ T } can be perfectly replicated started from zeroinitial wealth. Indeed, the corresponding self-financing delta hedging strategy H =( H t ) ≤ t ≤ T is given by H t = (1 + a T ) r π √ T − t e − | Z tT − t . The strategy certainly is a T -admissible with respect to the numéraire Z both under Q and q Q ; otherwise, there would be an arbitrage opportunity. As seen under Q , itresults a.s. in the riskless payoff { τ>T } = 1 . In the traditional perspective withrespect to the numéraire B , the strategy H is not admissible. Indeed, unless Z is a true Q -martingale, which is prevented by the well-posedness of a T , shorting Z t q Q (cid:2) τ ≤ T (cid:12)(cid:12) F t (cid:3) cannot be bounded from below. Indeed, the Bayes formula (8) says Z t q Q (cid:2) τ ≤ T (cid:12)(cid:12) F t (cid:3) = Z t − { τ>t } E Q (cid:2) Z T (cid:12)(cid:12) F t (cid:3) . The subtrahend is a.s. bounded fromabove by one, whereas Z exceeds any level with positive Q -probability. (cid:3) Remark 7.15 (Exogeneity of the Foreign Numéraire) . In fact, the foreign bankaccount process q B = ( q B t ) t ≥ in Example 7.14 is somehow exogenously given and notthe infinitesimal roll-over of (cid:8) Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ . As such, it would often imply finitevariation sample paths, where as the mentioned q B is generated beyond the shortrate paradigm. If both B and q B were inherited from short rate processes r = ( r t ) t ≥ and q r = ( q r t ) t ≥ respectively, then the resulting Q -dynamics of S = ( S t ) t ≥ wouldbe dS t = S t ( r t − q r t ) dt − S t Z t d f W t . In order to keep the recovery rate S within thetarget zone (0 , , it must hold B t Z t ≤ q B t Q -a.s. for all t ≥ . Maintaining both thiscondition and full analytical tractability appears hardly achievable. For instance,one might postulate q B t = X t B t Z t for some auxiliary Itô-process X = ( X t ) t ≥ with state space [1 , ∞ ) . The diffusion part of X must be X t Z t d f W t and needsto be compensated in the drift accordingly in order to ensure X t ≥ . Keepingthis stochastic volatility process in the upper half-space is not trivial. In orderto circumvent this perplexity, the authors propose an indirect HJM-approach inSection 6.5 of [17]. To this end, the model features are inspired by the enhanced FX-like approach, but the characteristic of the foreign bank account is only secondary.In contrast, the next example presents a tractable case, where the foreign bankaccount coincides with the infinitesimal roll-over. (cid:3) Example 7.16 (Pure Illiquidity) . This model is a modification of Example 8.1in [16]. We consider the limiting case S ≡ , i.e., default risk is disabled. Thus,the term structures (cid:8) P ( t, T ) } ≤ t ≤ T < ∞ , (cid:8) e P ( t, T ) } ≤ t ≤ T < ∞ and (cid:8) Q ( t, T ) } ≤ t ≤ T < ∞ all coincide. Let x ∈ R \ { } , f : [0 , ∞ ) −→ (0 , ∞ ) denote a strictly positive,deterministic, càdlàg function and f W = ( f W t ) t ≥ be a four-dimensional Brownianmotion on some filtered probability (Ω , F , F , Q ) with F = ( F t ) t ≥ satisfying theusual conditions. We interpret the auxiliary process X = ( X t ) t ≥ with X t := (cid:13)(cid:13) x + f W t (cid:13)(cid:13) f ( t ) as the evolution of the market’s growth optimal portfolio. Its inverse is a strict Q -local martingale. Utilising X as natural numéraire, the marketable zero-couponbond prices fulfil P ( t, T ) = E Q (cid:20) X t X T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = f ( T ) f ( t ) (cid:18) − e − k x + f Wt k− T − t ) (cid:19) ; see (8.2) in [16]. Provided that the mesh size of the discretisations can be controlledglobally (see Example 8.1 in [16] for the exact details), the infinitesimal roll-overis Q -a.s. given by q B = ( q B t ) t ≥ with q B t = f (0) f ( t ) . If the discount factors remaindeterministic as seen under a dominating Föllmer measure q Q for the strict Q -localmartingale Z = ( Z t ) t ≥ with Z t = q B t X t , then we easily get q P ( t, T ) = E q Q (cid:20) q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = f ( T ) f ( t ) . Thus, the fundamental values for all future times are already fixed at time t = 0 .The only driver for random fluctuations is the level of liquidity. The correspondingilliquidity premium L ( t, T ) = q P ( t, T ) − P ( t, T ) is positive for all ≤ t < T < ∞ and vanishes at maturity. (cid:3) A general construction scheme and further examples can be found in Section 5.4of [17].7.3.
Model of the 3rd Kind.
Let the setting of Assumption 6.1 be given. Thistime, let (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T be a strict Q -local martingale, whereas Z = ( Z t ) t ≥ is a true Q martingale. By Fatou’s Lemma, the discounted prices of the defaultablezero-coupon bonds also form Q -supermartingales. Consequently, we have E Q (cid:20) B t S T B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ≤ e P ( t, T ) . Therefore, by the classical change of numéraire technique, we have q Q ( t, T ) = E q Q (cid:20) q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = 1 S t E Q (cid:20) B t S T B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ≤ Q ( t, T ) . We observe that the defaultable zero-coupon bonds are overpriced and traded abovetheir intrinsic economic value. For instance, this market inefficiency may be drivenby emotions or prestige, or the simple fact that the intrinsic economic value is notreadily observable. More commonly, this situation may occur in a regime with anabundance of foreign investors who bear the additional cost due to (more than)offsetting convenience effects of the currency conversion. Hereby, the domestic
CONSTRAINT-BASED NOTION OF ILLIQUIDITY 23 currency of the model acts as a safe haven. This is why even negative nominalinterest rates could be enforced in Switzerland over the last couple of years.
Example 7.17 (Hyperliquidity) . We consider the enhanced FX-like setting fromAssumption 6.1 in the limiting case S ≡ and q B = B Q -a.s. Furthermore, weset B t := X t and B t − P ( t, T ) := X t − for a Bes -process X = ( X t ) t ≥ starting at X = 1 . In this case, we have Z ≡ , Q = q Q and q P ( t, T ) = E q Q (cid:20) q B t q B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = E Q (cid:20) X t X T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = 1 − (cid:18) − X t √ T − t (cid:19) < P ( t, T ) for all ≤ t < T < ∞ ; see page 69 in [4] (cid:3) Model of the 4th Kind.
Let the setting of Assumption 6.1 be given. Lastly,let both (cid:0) B t − e P ( t, T ) (cid:1) ≤ t ≤ T and Z = ( Z t ) t ≥ be a strict Q -local martingales. Onthe one hand, we have similarly as in the model of the nd kind q Q ( t, T ) ≥ { τ>t } S t E Q (cid:20) B t S T B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . Particularly, the statement still holds Q -a.s. if we drop the restriction to the event { τ > t } . On the other hand, exactly the same argument as in the model of the rd kind carries over S t E Q (cid:20) B t S T B T (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ≤ Q ( t, T ) . The model structure only maintains a common lower bound for Q ( t, T ) and q Q ( t, T ) under Q . Without any further model assumptions, nothing can be said about thedirection of the illiquidity premium. A priori, we are in a general market situationin which the sign of L ( t, T ) is state-dependent. The F t -event (cid:8) L ( t, T ) < (cid:9) relatesto an exorbitant demand for loans maturing at time T . Remark 7.18 (Analytical Intractability of the Illiquidity Premium) . It is worthmentioning that neither Example 7.7 nor Example 7.8 can be considered in thisgeneral market situation; more precisely, their premises cannot ever be satisfied.As L ( t, T ) may attain negative values on { τ > t } , then q Q (cid:2) t < τ ≤ T (cid:12)(cid:12) F t (cid:3) wouldhave to become negative as well. Obviously, this would be absurd. (cid:3) Conclusion
This article introduces a new mathematical concept of illiquidity that goes handin hand with credit risk. Utilising the FX-analogy, the recovery rate stands for bothinstitutional liquidity and that of the lending market. Asset liquidity constraintsare nothing else than a hidden default; one sees two prices for a certain good, butone cannot exploit the price difference. At the explosion time, nobody is willing tohold the considered asset regardless of the promised yield being beyond any rationallevel. This is the occurrence of total illiquidity and coincides with the default time.In this sense, credit and liquidity risk can be modelled essentially in the same way.
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