Admissible Trading Strategies under Transaction Costs
aa r X i v : . [ q -f i n . P R ] M a y Admissible Trading Strategies underTransaction Costs
Walter Schachermayer ∗ June 13, 2018
Abstract
A well known result in stochastic analysis reads as follows: foran R -valued super-martingale X = ( X t ) ≤ t ≤ T such that the terminalvalue X T is non-negative, we have that the entire process X is non-negative. An analogous result holds true in the no arbitrage theoryof mathematical finance: under the assumption of no arbitrage, anadmissible portfolio process x + ( H · S ) verifying x + ( H · S ) T ≥ x + ( H · S ) t ≥ , for all 0 ≤ t ≤ T .In the present paper we derive an analogous result in the pres-ence of transaction costs. In fact, we give two versions: one witha num´eraire-based, and one with a num´eraire-free notion of admissi-bility. It turns out that this distinction on the primal side perfectlycorresponds to the difference between local martingales and true mar-tingales on the dual side.A counter-example reveals that the consideration of transactioncosts makes things more delicate than in the frictionless setting. We consider a stock price process S = ( S t ) ≤ t ≤ T in continuous time witha fixed horizon T. This stochastic process is assumed to be based on a fil-tered probability space (Ω , F , ( F t ) ≤ t ≤ T , P ) , satisfying the usual conditionsof completeness and right continuity. We assume that S is adapted and has ∗ Fakult¨at f¨ur Mathematik, Universit¨at Wien, Nordbergstrasse 15, A-1090 Wien, [email protected] . Partially supported by the Austrian ScienceFund (FWF) under grant P25815, the European Research Council (ERC) under grantFA506041 and by the Vienna Science and Technology Fund (WWTF) under grant MA09-003. `adl`ag (right continuous, left limits), and strictly positive trajectories, i.e. thefunction t → S t ( ω ) is c`adl`ag and strictly positive, for almost each ω ∈ Ω . In mathematical finance a key assumption is that the process S is freeof arbitrage . The Fundamental Theorem of Asset Pricing states that thisproperty is essentially equivalent to the property that S admits an equivalentlocal martingale measure (see, [10], [4], or the books [5],[14]). Definition 1.1.
The process S admits an equivalent local martingale mea-sure, if there is a probability measure Q ∼ P such that S is a local martingaleunder Q . Fix a process S satisfying the above assumption and note that Def.1.1implies in particular that S is a semi-martingale as this property is invariantunder equivalent changes of measure. Turning to the theme of the paper,we now consider trading strategies , i.e. S -integrable predictable processes H = ( H t ) ≤ t ≤ T . We call H admissible if there is M > H · S ) t ≥ − M, P − a.s. for 0 ≤ t ≤ T. (1)The stochastic integral( H · S ) t = Z t H u dS u , ≤ t ≤ T, (2)then is a local Q -martingale by a result of Ansel-Stricker under each equiva-lent local martingale measure Q (see [1] and [17]). Assumption (1) also im-plies that the local martingale H · S is a super-martingale (see [5], Prop.7.2.7)under each equivalent local martingale measure Q . We thus infer from theeasy result mentioned in the first line of the abstract that ( H · S ) T ≥ − x al-most surely implies that ( H · S ) t ≥ − x almost surely under Q (and thereforealso under P ), for all 0 ≤ t ≤ T . In fact, we may replace the deterministictime t by a [0 , T ]-valued stopping time τ. We resume our findings in the subsequent well-known Proposition (com-pare [15], Prop.4.1).
Proposition 1.2.
Let the process S admit an equivalent local martingalemeasure, let H be admissible, and suppose that there is x ∈ R + such that x + ( H · S ) T ≥ , P − a.s. (3) Then x + ( H · S ) τ ≥ , P − a.s. (4) for every [0 , T ] -valued stopping time τ .
2e now introduce transaction costs: fix 0 ≤ λ <
1. We define the bid-askspread as the interval [(1 − λ ) S, S ]. The interpretation is that an agent canbuy the stock at price S , but sell it only at price (1 − λ ) S . Of course, thecase λ = 0 corresponds to the usual frictionless theory.In the setting of transaction costs the notion of consistent price systems ,which goes back to [11] and [3], plays a role analogous to the notion ofequivalent martingale measures in the frictionless theory (Definition 1.1). Definition 1.3.
Fix > λ ≥ . A process S = ( S t ) ≤ t ≤ T satisfies the condi-tion ( CP S λ ) of having a consistent price system under transaction costs λ if there is a process e S = ( e S t ) ≤ t ≤ T , such that (1 − λ ) S t ≤ e S t ≤ S t , ≤ t ≤ T, (5) as well as a probability measure Q on F , equivalent to P , such that ( e S t ) ≤ t ≤ T is a local martingale under Q .We say that S admits consistent price systems for arbitrarily small trans-action costs if ( CP S λ ) is satisfied, for all > λ > . For continuous process S , in [9] the condition of admitting consistent pricesystems for arbitrarily small transaction costs has been related to the condi-tion of no arbitrage under arbitrarily small transaction costs, thus proving aversion of the Fundamental Theorem of Asset Pricing under small transactioncosts (compare [13] for a large amount of related material).It is important to note that we do not assume that S is a semi-martingaleas one is forced to do in the frictionless theory [4, Theorem 7.2]. Only theprocess e S appearing in Definition 1.3 has to be a semi-martingale, as itbecomes a local martingale after passing to an equivalent measure Q. To formulate a result analogous to Proposition 1.2 in the setting of trans-action costs we have to define the notion of R -valued self-financing tradingstrategies . Definition 1.4.
Fix a strictly positive stock price process S = ( S t ) ≤ t ≤ T withc`adl`ag paths, as well as transaction costs > λ > . A self-financing trading strategy starting with zero endowment is a pairof predictable, finite variation processes ( ϕ t , ϕ t ) ≤ t ≤ T such that ( i ) ϕ = ϕ = 0 , ii ) denoting by ϕ t = ϕ , ↑ t − ϕ , ↓ t and ϕ t = ϕ , ↑ t − ϕ , ↓ t , the canonicaldecompositions of ϕ and ϕ into the difference of two increasing processes,starting at ϕ , ↑ = ϕ , ↓ = ϕ , ↑ = ϕ , ↓ = 0 , these processes satisfy dϕ , ↑ t ≤ (1 − λ ) S t dϕ , ↓ t , dϕ , ↓ t ≥ S t dϕ , ↑ t , ≤ t ≤ T. (6) The trading strategy ϕ = ( ϕ , ϕ ) is called admissible if there is M > such that the liquidation value V liqt satisfies V liqτ ( ϕ , ϕ ) := ϕ τ + ( ϕ τ ) + (1 − λ ) S τ − ( ϕ τ ) − S τ ≥ − M, (7) a.s., for all [0 , T ] -valued stopping times τ . The processes ϕ t and ϕ t model the holdings at time t in units of bond andstock respectively. We normalize the bond price by B t ≡
1. The differentialnotation in (6) needs some explanation. If ϕ is continuous, then (6) has tobe understood as the integral requirement. Z τσ ((1 − λ ) S t dϕ , ↓ t − dϕ , ↑ t ) ≥ , a.s. (8)for all stopping times 0 ≤ σ ≤ τ ≤ T , and analogously for the seconddifferential inequality in (6). The above integral makes pathwise sense asRiemann-Stieltjes intregral, as ϕ is continuous and of finite variation and S is c`adl`ag. Things become more delicate when we also consider jumps of ϕ :note that, for every stopping time τ the left and right limits ϕ τ − and ϕ τ + exist as ϕ is of bounded variation. But the three values ϕ τ − , ϕ τ and ϕ τ + mayvery well be different. As in [2] we denote the increments by∆ ϕ τ = ϕ τ − ϕ τ − , ∆ + ϕ τ = ϕ τ + − ϕ τ . (9)For totally inaccessible stopping times τ , the predictability of ϕ implies that∆ ϕ τ = 0 almost surely, while for accessible stopping times τ it may happenthat ∆ ϕ τ = 0 as well as ∆ + ϕ τ = 0 . To the assumption that (8) has to hold true for the continuous part of ϕ the following requirements therefore have to be added to take care of thejumps of ϕ. ∆ ϕ , ↑ τ ≤ (1 − λ ) S τ − ∆ ϕ , ↓ τ , ∆ ϕ , ↓ τ ≥ S τ − ∆ ϕ , ↑ τ (10)and in the case of right jumps∆ + ϕ , ↑ τ ≤ (1 − λ ) S τ ∆ + ϕ , ↓ τ , ∆ + ϕ , ↓ τ ≥ S τ ∆ + ϕ , ↑ τ , (11)4olding true a.s. for all [0 , T ]-valued stopping times τ. Let us give an economicinterpretation of the significance of (10) and (11). For simplicity we let λ = 0 . Think of a predictable time τ , say the time τ of a speech of the chairman ofthe Fed. The speech does not come as a surprise. It was announced some timebefore which - mathematically speaking - corresponds to the predictability of τ . It is to be expected that this speech will have a sudden effect on the priceof a stock S , say a possible jump from S τ − ( ω ) = 100 to S τ ( ω ) = 110 (recallthat S is assumed to be c`adl`ag). A trader may want to follow the followingstrategy: she holds a position of ϕ τ − ( ω ) stocks until “immediately before thespeech”. Then, one second before the speech starts, she changes the positionfrom ϕ τ − ( ω ) to ϕ τ ( ω ) causing an increment of ∆ ϕ τ ( ω ). Of course, the price S τ − ( ω ) still applies, corresponding to (10). Subsequently, the speech startsand the jump ∆ S τ ( ω ) = S τ ( ω ) − S τ − ( ω ) is revealed. The agent may nowdecide “immediately after learning the size of ∆ S τ ( ω )” to change her positionfrom ϕ τ ( ω ) to ϕ τ + ( ω ) on the base of the price S τ ( ω ) which corresponds to(11).We have chosen to define the trading strategy ϕ by explicitly specifyingboth accounts, the holdings in bond ϕ as well as the holdings in stock ϕ . It would be sufficient to only specify ϕ similarly as in the frictionlesstheory where we usually only specify the process H in (1) which correspondsto ϕ in the present notation. Given a predictable finite variation process ϕ = ( ϕ t ) ≤ t ≤ T starting at ϕ = 0 , which we canonically decompose into thedifference ϕ = ϕ , ↑ − ϕ , ↓ , we may define the process ϕ by dϕ t = (1 − λ ) S t dϕ , ↓ t − S t dϕ , ↑ t . The resulting pair ( ϕ , ϕ ) obviously satisfies (6) with equality holding truerather than inequality. Not withstanding, it is convenient in (6) to considertrading strategies ( ϕ , ϕ ) which allow for an inequality in (6), i.e. for “throw-ing away money”. But it is clear from the preceding argument that we mayalways pass to a dominating pair ( ϕ , ϕ ) where equality holds true in (6).In the theory of financial markets under transaction costs the super-martingale property of the value process is formulated in Proposition 1.6below. First we have to recall a definition from [7] which extends the notionof a super-martingale beyond the framework of c`adl`ag processes. Definition 1.5.
An optional process X = ( X t ) ≤ t ≤ T is called an optionalstrong super-martingale if, for all stopping times ≤ σ ≤ τ ≤ T we have E [ X τ | F σ ] ≤ X σ , (12) where we impose that X τ is integrable.
5n optional strong super-martingale can be decomposed in the style ofDoob-Meyer which is known under the name of Mertens decomposition (see[7]). X is an optional strong super-martingale if and only if it can be decom-posed into X = M − A, (13)where M is a local martingale (and therefore c`adl`ag) as well as a super-martingale, and A an increasing predictable process (which is l`adl`ag but hasno reason to be c`agl`ad or c`adl`ag). This decomposition then is unique.One may also define the notion of a local optional strong supermartin-gale in an obvious way. In this case the process M in (15) only is requiredto be a local martingale and not necessarily a super-martingale, while therequirements on A remain unchanged. Proposition 1.6.
Fix S , transaction costs > λ > , and an admissibleself-financing trading strategy ϕ = ( ϕ , ϕ ) as above. Suppose that ( e S, Q ) isa consistent price system under transaction costs λ . Then the process e V t := ϕ t + ϕ t e S t , ≤ t ≤ T, satisfies e V ≥ V liq almost surely and is an optional strong super-martingaleunder Q .Proof. The assertion e V ≥ V liq is an obvious consequence of e S ∈ [(1 − λ ) S, S ].We have to show that e V decomposes as in (13). Arguing formally, wemay apply the product rule to obtain d e V t = ( dϕ t + e S t dϕ t ) + ϕ t d e S t (14)so that e V t = Z t ( dϕ u + e S u dϕ u ) + Z t ϕ u d e S u . (15)The first term in (15) is decreasing by (6) and the fact that e S ∈ [(1 − λ ) S, S ].The second term defines, at least formally speaking, a local Q-martingale as e S is so. Hence the sum of the two integrals should be an (optional strong)super-martingale.The justification of the above formal reasoning deserves some care (com-pare the proof of Lemma 8, in [2]). Suppose first that ϕ is continuous. Inthis case ϕ is a semi-martingale so that we are allowed to apply Itˆo calculusto e V . Formula (15) therefore makes perfect sense as an Itˆo integral, bearingin mind that ϕ has finite variation, which coincides with the pointwise inter-pretation of the integral via partial integration. The first integral in (15) is a6ell-defined decreasing predictable process. As regards the second integral,note that by the admissibility of ϕ it is uniformly bounded from below. Henceby a result of Ansel-Stricker ([1], see also [17]) it is a local Q-martingale aswell as a super-martingale. Hence e V is indeed a super-martingale under Q (in the classical c`adl`ag sense).Passing to the case when ϕ is allowed to have jumps, the process e V neednot be c`adl`ag anymore. It still is an optional process and we have to verifythat it decomposes as in (13). Assume first that ϕ is of the form ϕ t = ( f , f ) K τ,T K ( t ) , (16)where ( f , f ) = ∆ + ( ϕ τ , ϕ τ ) are F τ -measurable bounded random variablesverifying (11) and τ is a [0 , T ]-stopping time. We obtain e V t = [∆ + ϕ τ + (∆ + ϕ τ ) e S t ] K τ,T K ( t )= [∆ + ϕ τ + (∆ + ϕ τ ) e S τ ] K τ,T K ( t ) + (∆ + ϕ τ )( e S t − e S τ ) K τ,T K ( t ) . (17)Again, the first term is a decreasing predictable process and the second termis a local martingale under Q .Next assume that ϕ is of the form ϕ t = ( f , f ) J τ,T K ( t ) , (18)where τ is a predictable stopping time, and ( f , f ) = ∆( ϕ τ , ϕ τ ) are bounded F τ − -measurable random variables verifying (10). Similarly as in (17) weobtain e V t = [∆ ϕ τ + (∆ ϕ τ ) e S t ] J τ,T K ( t )= [∆ ϕ τ + (∆ ϕ τ ) e S τ − ] J τ,T K ( t ) + (∆ ϕ τ )( e S t − e S τ − ) J τ,T K ( t ) . (19)Once more, the first term is a decreasing predictable process (this time itis even c`adl`ag) and the second term is a local martingale under Q .Finally we have to deal with a general admissible self-financing tradingstrategy ϕ . To show that e V is of the form (13) we first assume that the totalvariation of ϕ is uniformly bounded. We decompose ϕ into its continuous andpurely discontinuous part ϕ = ϕ c + ϕ pd . We also may find a sequence ( τ n ) ∞ n =1 of [0 , T ] ∪ {∞} -valued stopping times such that the supports ( J τ n K ) ∞ n =1 are7utually disjoint and ∞ S n =1 J τ n K exhausts the right jumps of ϕ . Similarly, wemay find a sequence ( τ pn ) ∞ n =1 of predictable stopping times such that theirsupports ( J τ pn K ) ∞ n =1 are mutually disjoint and ∞ S n =1 J τ pn K exhausts the left jumpsof ϕ . We apply the above argument to ϕ c , and to each ( τ n , ∆ + ϕ τ n ) and( τ pn , ∆ ϕ τ pn ), and sum up the corresponding terms in (15), (17) and (19). Thissum converges to e V = M − A , where M is a local Q-martingale and A an increasing process, as we have assumed that the total variation of ϕ isbounded (compare [12] and the proof of Lemma 8 in [2]). By the boundednessfrom below we conclude that M is also a super-martingale.Passing to the case where ϕ has only finite instead of uniformly boundedvariation, we use the predictability of ϕ to find a localizing sequence ( σ k ) ∞ k =1 such that each stopped process ϕ σ k has uniformly bounded variation. Applythe above argument to each ϕ σ k to obtain the same conclusion for ϕ .Summing up, we have shown that e V admits a Mertens decomposition(13) and therefore is an optional strong super-martingale.We can now state the analogous result to Proposition 1.2 in the presenceof transaction costs. Theorem 1.7.
Fix the c`adl`ag, adapted process S and > λ > as above,and suppose that S satisfies ( CP S λ ′ ) , for each 1 > λ ′ > . Let ϕ = ( ϕ t , ϕ t ) ≤ t ≤ T be an admissible, self-financing trading strategyunder transaction costs λ, starting with zero endowment, and suppose thatthere is x > s.t. for the terminal liquidation value V liqT we have a.s. V liqT ( ϕ , ϕ ) = ϕ T + ( ϕ T ) + (1 − λ ) S T − ( ϕ T ) − S T ≥ − x. (20) We then also have that V liqτ ( ϕ , ϕ ) = ϕ τ + ( ϕ τ ) + (1 − λ ) S τ − ( ϕ τ ) − S τ ≥ − x, (21) a.s., for every stopping time ≤ τ ≤ T. Proof.
Supposing that (21) fails, we may find λ > α > , and a stoppingtime 0 ≤ τ ≤ T, such that either A = A + or A = A − satisfies P [ A ] > , where A + = { ϕ τ ≥ , ϕ τ + ϕ τ − λ − α S τ < − x } , (22) A − = { ϕ τ ≤ , ϕ τ + ϕ τ (1 − α ) S τ < − x } . (23)8ndeed, focusing on (22) and denoting by A + ( α ) the set in (22) we have ∪ α> A + ( α ) = { ϕ τ ≥ , ϕ τ + ϕ τ (1 − λ ) S τ < − x } , showing that the failure of(21) implies the existence of α > P [ A ] > . Choose 0 < λ ′ < α and a λ ′ -consistent price system ( e S, Q ) . As e S takesvalues in [(1 − λ ′ ) S, S ], we have that (1 − α ) e S as well as − λ − α e S take values in[(1 − λ ) S, S ] as (1 − λ ′ )(1 − λ ) > (1 − λ ) and (1 − λ ′ ) − λ − α > − λ. It followsthat ((1 − α ) e S, Q ) as well as ( − λ − α e S, Q ) are consistent price systems undertransaction costs λ. By Proposition 1.6 we obtain that (cid:16) ϕ t + ϕ t (1 − α ) e S t (cid:17) ≤ t ≤ T and (cid:16) ϕ t + ϕ t − λ − α e S t (cid:17) ≤ t ≤ T are optional strong Q -super-martingales. Arguing with the second processusing e S ≤ S, we obtain from (22) the inequality E Q [ V liqT | A + ] ≤ E Q (cid:20) ϕ T + ϕ T − λ − α e S T (cid:12)(cid:12)(cid:12) A + (cid:21) ≤ E Q (cid:20) ϕ τ + ϕ τ − λ − α e S τ (cid:12)(cid:12)(cid:12) A + (cid:21) ≤ E Q (cid:20) ϕ τ + ϕ τ − λ − α S τ (cid:12)(cid:12) A + (cid:21) < − x. Arguing with the first process and using that e S ≥ (1 − λ ′ ) S ≥ (1 − α ) S (which implies that ϕ τ (1 − α ) e S τ ≤ ϕ τ (1 − α ) S τ on A − ) we obtain from (23)the inequality E Q [ V liqT | A − ] ≤ E Q h ϕ T + ϕ T (1 − α ) e S T | A − i ≤ E Q h ϕ τ + ϕ τ (1 − α ) e S τ | A − i ≤ E Q (cid:2) ϕ τ + ϕ τ (1 − α ) S τ | A − (cid:3) < − x. Either A + or A − has strictly positive probability; hence we arrive at a con-tradiction to V liq T ≥ − x almost surely. In this section we derive results analoguous to Proposition 1.6 and Theorem1.7 in a num´eraire-free setting. This is inspired by the discussion of thenum´eraire-based versus num´eraire-free setting in [9] and [16] (compare also[8], [13], [18], [19]).We complement the above notions of admissibility and consistent pricesystems by the following num´eraire-free variants.9 efinition 2.1.
In the setting of Definition 1.4 we call a self-financing strat-egy ϕ admissible in a num´eraire-free sense if there is M > such that V liqτ ( ϕ , ϕ ) := ϕ τ + ( ϕ τ ) + (1 − λ ) S τ − ( ϕ τ ) − S τ ≥ − M (1 + S τ ) , a.s. , (24) for each [0 , T ] -valued stopping time τ . While the control of the portfolio process ϕ in (7) is in terms of M units ofbond (which is considered as num´eraire), the present condition (24) stipulatesthat the risk involved by the trading strategy ϕ can be super-hedged byholding M units of bond plus M − λ units of stock. Definition 2.2.
Fix > λ ≥ . In the setting of Definition 1.3 we calla pair ( e S, Q ) = (( e S t ) ≤ t ≤ T,Q ) satisfying (5) a consistent price process in thenon-local sense if e S is a true martingale under Q , not only a local martingale. The passage from the num´eraire-based to num´eraire-free admissibility forthe primal objects, i.e. the trading strategies ϕ , perfectly corresponds to thepassage from local martingales to martingales in Definition 2.2 for the dualobjects, i.e. the consistent price systems. This is the message of the twosubsequent results (compare also [16]). Proposition 2.3.
In the setting of Proposition 1.6 fix a self-financing tradingstrategy ϕ = ( ϕ , ϕ ) which we now assume to be admissible in the num´eraire-free sense. Also fix ( e S, Q ) which we now assume to be a λ -consistent pricesystem in the non-local sense, i.e. e S is a true Q -martingale. We again mayconclude that the process e V t := ϕ t + ϕ t e S t , ≤ t ≤ T, satisfies e V ≥ V liq almost surely and is an optional strong super-martingaleunder Q .Proof. We closely follow the proof of Proposition 1.6 which carries over ver-batim, also under the present weaker assumption of num´eraire-free admis-sibility. Again, we conclude that the second integral in (15) is a local Q -martingale from the fact that e S is a local Q -marginale and ϕ is predictableand of finite variation. The only subtlety is the following: contrary to thesetting of Proposition 1.6 we now may only deduce the obvious implicationthat e V = ( e V t ) ≤ t ≤ T is a local optional strong super-martingale under Q .What needs extra work is an additional argument which finally showsthat the word local may be dropped, i.e. that e V again is an optional strongsuper-martingale under Q . 10y the num´eraire-free admissibility condition we know that there is some M > , T ]-valued stopping times τ , e V τ ≥ V liqτ ≥ − M (1 + S τ ) , a.s. (25)We also know that e S is a uniformly integrable martingale under Q . Hencethe family of random variables e S τ as well as that of S τ (note that S τ ≤ e S τ − λ ), where τ ranges through the [0 , T ]-valued stopping times, is uniformlyintegrable.We have to show that, for all stopping times 0 ≤ ρ ≤ σ ≤ T we have E Q [ e V σ |F ρ ] ≤ e V ρ . (26)We know that e V is a local optional strong super-martingale under Q , so thatthere is a localizing sequence ( τ n ) ∞ n =1 of stopping times such that E Q [ e V σ ∧ τ n |F ρ ∧ τ n ] ≤ e V ρ ∧ τ n , n ≥ . (27)Using (25) we may deduce (26) from (27) by the (conditional version of the)following well-known variant of Fatou’s lemma: Let ( f n ) ∞ n =1 be a sequence ofrandom variables on (Ω , F , R ) converging almost surely to f and such thatthe negative parts ( f − n ) ∞ n =1 are uniformly Q -integrable. Then E Q [ f ] ≤ lim inf n →∞ E Q [ f n ] . Remark 2.4.
We have assumed in Proposition 1.6 as well as in the aboveProposition 2.3 that Q is equivalent to P . In fact, we may also assume that Z T vanishes on a non-trivial set so that Q is only absolutely continuous w.r. to P . The assertions of the two propositions still remain valid for P -absolutelycontinuous Q , provided that we replace the requirements almost surely by Q -almost surely .We now state and prove the num´eraire-free version of Theorem 1.7. Theorem 2.5.
In the setting of Theorem 1.7 suppose now that S satisfies ( CP S λ ′ ) in the non-local sense, for each > λ ′ > . As in Theorem 1.7, let ϕ be admissible, but now in the num´eraire-free sense, and let x > such that V liqT ( ϕ , ϕ ) ≥ − x. (28) We then also have V liqτ ( ϕ , ϕ ) ≥ − x, (29) a.s., for every stopping time ≤ τ ≤ T. Proof.
The proof of Theorem 1.7 carries over verbatim to the present set-ting, replacing the application of Proposition 1.6 by an application of itsnum´eraire-free version Proposition 2.3.11
A Counter-Example
The assumption (
CP S λ ′ ), for each λ ′ > , cannot be dropped in Proposition1.7 as shown by the example presented in the next lemma. Lemma 3.1.
Fix > λ ≥ λ ′ > and C > . There is a continuous process S = ( S t ) ≤ t ≤ satisfying ( CP S λ ′ ) , and a λ -self-financing, admissible tradingstrategy ( ϕ , ϕ ) = ( ϕ t , ϕ t ) ≤ t ≤ such that V liq ( ϕ , ϕ ) ≥ − , a.s. (30) while P (cid:20) V liq ( ϕ , ϕ ) ≤ − C (cid:21) > . (31) Proof.
In order to focus on the central (and easy) idea of the construction wefirst show the assertion for the constant C = 2 − λ and under the assumption λ = λ ′ . In this case we can give a deterministic example, i.e. S, ϕ and ϕ will not depend on the random element ω ∈ Ω . Define S = S = 1 , and S = 1 − λ where we fix T = 1.To make S = ( S t ) ≤ t ≤ T continuous, we interpolate linearly, i.e. S t = 1 − tλ, ≤ t ≤ , (32) S t = 1 − − t ) λ, ≤ t ≤ . (33)Note that condition ( CP S λ ) is satisfied, as the constant process e S t ≡ (1 − λ ) defines a λ -consistent price system: it trivially is a martingale (underany probability measure) and takes values in [(1 − λ ) S, S ] . Starting from the initial endowment ( ϕ , ϕ ) = (0 , , we might invest,at time t = 0 , the maximal amount into the stock so that at time t = 1condition (30) holds true. In other words, we let ϕ + = − ϕ + be the biggestnumber such that (1 − λ ) ϕ + + ϕ + ≥ − , which clearly gives ϕ + = λ . Hence ( ϕ t , ϕ t ) = ( − λ , λ ) , for all 0 < t ≤ T, is aself-financing strategy, starting at ( ϕ , ϕ ) = (0 ,
0) for which (30) is satisfied.Looking at (31) we calculate V ( ϕ , ϕ ) = (1 − λ ) · (1 − λ ) · λ − λ = − λ.
12n order to replace λ ′ = λ by an arbitrarily small constant λ ′ >
0, and C = 2 − λ by an arbitrarily large constant C > , we make the follow-ing observation: if the initial endowment ( ϕ , ϕ ) = (0 ,
0) were replaced by( ϕ , ϕ ) = ( M, , for some large M , the agent could play the above game ona larger scale: she could choose ( ϕ t , ϕ t ) = ( M − M +1 λ , M +1 λ ) , for 0 < t ≤ V ( ϕ , ϕ ) = M − M +1 λ + (1 − λ ) M +1 λ = − . As regards the liquidation value V liq , we now assume S = 1 − λ ′ (insteadof S = 1 − λ in (32) and (33)) to make sure that ( CP S λ ′ ) holds true. Theliquidation value at time t = then becomes V liq ( ϕ , ϕ ) = M − M +1 λ + (1 − λ )(1 − λ ′ ) M +1 λ = M − ( M + 1)[1 + λ ′ ( 1 λ − −∞ , as M → ∞ in view of 0 < λ ′ ≤ λ < ϕ , ϕ ) = (0 , , the idea is that,during the time interval [0 , ] , the price process S provides the agent with theopportunity to become rich with positive probability, i.e. P [( ϕ , ϕ ) = ( M, > . We then play the above game, conditionally onthe event { ( ϕ , ϕ ) = ( M, } and with [0 ,
1] replaced by [ , . The subsequent construction makes this idea concrete. Let ( F t ) ≤ t ≤ begenerated by a Brownian motion ( W t ) ≤ t ≤ . Fix disjoint sets A + and A − in F such that P [ A + ] = f M − and P [ A − ] = 1 − P [ A + ] , where f M > M = − f M (1 − λ ′ ) . The set A + is split into two sets A ++ and A + − suchthat A ++ and A + − are in F and P (cid:20) A ++ (cid:12)(cid:12)(cid:12)(cid:12) F (cid:21) = P (cid:20) A + − (cid:12)(cid:12)(cid:12)(cid:12) F (cid:21) = 12 A + . We define S by S = f M − A ++ A + − on A − S t = E (cid:20) S |F t (cid:21) , ≤ t ≤ , (34)so that ( S t ) ≤ t ≤ is a continuous P -martingale. The numbers above weredesigned in such a way that S = 1 , and S = ( f M on A +12 on A − To define S t also for < t ≤ S t = S on A ++ ∪ A − while, conditionally on A + − , we repeat the above deterministic constructionon [ ,
1] : S t = 1 − t − ) λ ′ , ≤ t ≤ ,S t = 1 − − t ) λ ′ , ≤ t ≤ . This defines the process S. Condition (
CP S λ ′ ) is satisfied as ( e S t ) ≤ t ≤ :=((1 − λ ′ ) S t ∧ ) ≤ t ≤ is a P -martingale taking values in the bid-ask spread[(1 − λ ′ ) S t , S t ] ≤ t ≤ . Let us now define the strategy ( ϕ , ϕ ) : starting with ( ϕ , ϕ ) = (0 , ϕ t , ϕ t ) = ( − , , for 0 < t ≤ . In prose: the agent buys onestock at time t = 0 and holds it until time t = . At time t = she sells thestock again, so that ( ϕ , ϕ ) = ( − (1 − λ )2 ,
0) on A − , while ( ϕ + , ϕ + ) =( − f M (1 − λ ′ ) ,
0) = ( M,
0) on A + . On A − we simply define ( ϕ t , ϕ t ) = ( − − λ , , for all < t ≤ A − .On A + we define ( ϕ t , ϕ t ) = ( M, , for < t ≤ . In prose: during ] , ]the agent does not invest into the stock and is happy about the M bonds inher portfolio. At time t = we distinguish two cases: on A ++ we continueto define ( ϕ t , ϕ t ) = ( M, , also for < t ≤ . On A + − we let ( ϕ t , ϕ t ) =( M − M +1 λ , M +1 λ ) , for < t ≤ . As discussed above, inequality (30) then holdstrue almost surely, while V ( ϕ , ϕ ) attains the value M − ( M +1)[1+ λ ′ ( λ − −∞ as M tends to ∞ . This happens with positive probability P [ A + − ] > . The construction of the example now is complete.14 cknowledgement.
I warmly thank Irene Klein without whose encourage-ment this note would not have been written and who strongly contributedto its shaping. Thanks go also to Christoph Czichowsky for his advice onsome of the subtle technicalities of this note. I thank an anonymous refereefor careful reading and for pointing out a number of inaccuracies.
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