5 The local duality theory

Transcription

1 5 The local duality theory In this section we extend the duality theory to the setting where the corresponding concepts such as no arbitrage, existence of consistent price systems etc. only hold locally. For example, this situation arises naturally in the stochastic portfolio theory as promoted by R. Fernholz and I. Karatzas. We refer to the paper [6] by I. Karatzas and C. Kardaras (compare also [63]) where the local duality theory is developed in the classical frictionless setting. Recall that a property pp q of a stochastic process S ps t q ďtďt holds locally if there is a sequence of stopping times pτ n q 8 n increasing to infinity such that each of the stopped processes S τn ps t^τn q ďtďt has property pp q. We say that pp q is a local property if the fact that S has property pp q locally implies that S has property pp q. In the subsequent definition we formulate the notion of a super-martingale deflator in the frictionless setting. The tilde super-scripts indicate that we are in the frictionless setting. Definition 5.. (see [6]) Let S p S t q ďtďt be a semi-martingale based on and adapted to pω, F, pf t q ďtďt, P.q The set of equivalent super-martingale deflators Z e are defined as the s, 8r-valued processes p Z t q ďtďt, starting at Z, such that, for every S-integrable predictable process H p H t q ďtďt verifying ` p H Sq t ě, ď t ď T, (3) the process Z t p ` p H Sq t q, ď t ď T (4) is a super-martingale under P. Dropping the super-script e we obtain the corresponding class Z of r, 8r-valued super-martingale deflators. We call Z P Z a local martingale deflator if, in addition, Z is a local martingale. We say that S satisfies the property pesdq (resp. peldq) of existence of an equivalent super-martingale (resp. local martingale) deflator if Z e H (resp. there is a local martingale Z in Z e ). We remark that, for a probability measure Q equivalent to P under which S is a local martingale, we have that the density process Z t Er dq dp F ts defines a local martingale deflator. We first give an easy example of a process S, for which (NFLVR) fails while there does exist a super-martingale deflator (see [6, Ex. 4.6] for a more sophisticated example, involving the three-dimensional Bessel process). In 9

2 fact, we formulate this example in such a way that it also highlights the persistence of this phenomenon under transaction costs. Proposition 5.. There is a continuous semi-martingale S ps t q ďtď, based on a Brownian filtration pf t q ďtď, such that there is an equivalent super-martingale deflator pz t q ďtď for S. On the other hand, for ď λ ă, there does not exist a λ-consistent price system p S, Qq associated to S. Proof: Let W pw t q tě be an pf t q tě -Brownian motion, where pf t q tě is the natural (right-continuous, saturated) filtration generated by W. Define the martingale Z Ep W q and let N Z, i.e. so that N satisfies the SDE Z t expp W t t q, t ě, N t exppw t ` t q, t ě, dn t N t dw t ` dt. Define the stopping time τ as τ inftt : Z t u inftt : N t u, and note that τ is a.s. finite. Define the stock price process S as the timechanged restriction of N to the stochastic interval, τ, i.e. S t N tanp π pt^τqq, ď t ď. By Girsanov there is only one candidate for the density process of an equivalent martingale measure, namely Z tanp π pt^τqq. But the example ďtď is cooked up in such a way that only is a local martingale. Of course, Z tanp π pt^τqq ďtď Z tanp π pt^τqq ďtď is an equivalent local martingale deflator. As regards the final assertion, fix ď λ ă, and suppose that there is a λ-consistent price system p S, Qq. As S P rp λqs, Ss we have S ď and S ě p λq ą, almost surely. On the other hand, assuming that S is a Q-super-martingale implies that E Q r S s ď E Q r S s, and we arrive at a contradiction. 93

3 Remark 5.3. For later use we note that S t N tanp π pt^τqq is the so-called numéraire portfolio (see, e.g. [6]), i.e., the unique process of the form `H S verifying ` ph Sq ě, and maximizing the logarithmic utility upq supterlogp ` ph Sq qsu. The value function u above has a finite value, namely upq logpq, and, more generally, upxq logpq`logpxq, although the process S does not admit an equivalent martingale measure. In other words, log-utility optimization does make sense although the process S obviously allows for an arbitrage as S while S. We next resume two notions from [66]. The tilde indicates again that we are in the frictionless setting. Definition 5.4. Let S p S t q ďtďt be a semi-martingale. For x ą, y ą, define the sets Cpxq t X T : ď X T ď x ` p H Sq T u where H runs through the predictable, S-integrable processes such that p H Sq t ě x, for all ď t ď T, and let Dpyq ty Z T u where Z T now runs through the terminal values of super-martingale deflators p Z t q ďtďt P Z. Let us comment on the issue of non-negativity versus strict positivity in the definition of Dpyq. This corresponds to the difference between local martingale measures Q for the process S which are either assumed to be equivalent or absolutely continuous with respect to P. It is well-known in this more classical context that the closure of the set M e p Sq of equivalent local martingale measures Q involves the passage to absolutely continuous martingale measures. Similarly, to obtain the closedness of Dpq in the above theorem we have to allow for non-negative processes p Z t q ďtďt P Z rather than strictly positive processes p Z t q ďtďt P Z e. We now formulate the analogue of the results of [6] in the context of transaction costs. To stay in line with the present setting we continue to suppose that S ps t q ďtďt is a continuous process based on and adapted to pω, F, Pq equipped with a Brownian filtration pf t q ďtďt. We present a local version of the fundamental theorem of asset pricing (Theorem 5.6 below) which pertains to the notion of equivalent supermartingale deflators. Here is the corresponding primal notion in terms of arbitrage in the frictionless setting. 94

4 Definition 5.5. [6, Def. 4.] Let S p S t q ďtďt be a semi-martingale. We say that S allows for an unbounded profit with bounded risk if there is α ą such that, for every C ą, there is a predictable, S-integrable process H such that while p H Sq t ě, ď t ď T, P p H Sq ı T ě C ě α. If S does not allow for such profits, we say that S satisfies the condition pnup BRq of no unbounded profit with bounded risk. We now turn to the central result form the paper [6] of I. Karatzas and C. Kardaras. While these authors deal with the more complicated case of general semi-martingales (even allowing for convex constraints) we only deal with the case of continuous semi-martingales S. This simplifies things considerably as the problem then boils down to a careful inspection of Girsanov s formula. Fix the continuous semi-martingale S. By the Bichteler-Dellacherie theorem (see, e.g., [75] or [3]), S uniquely decomposes into S M ` A where M is a local martingale starting at M S, and A is predictable and of bounded variation starting at A. These processes M and A are continuous too and the quadratic variation process xmy t is well-defined and a.s. finite. The so-called structure condition introduced by M. Schweizer [83] states that A is a.s. absolutely continuous with respect to xmy, i.e., d S t S t dm t ` ϱ t dxmy t (5) for some predictable process pϱ t q ďtďt. If S fails to be representable in the form (5), it is well-known and easy to prove that S allows for arbitrage (in a very strong sense made precise, e.g., in [6, Def. 3.8]). The underlying idea goes as follows: if da t fails to be absolutely continuous with respect to dxmy t then one can well-define a predictable trading strategy H ph t q ďtďt which equals ` where da t ą and dxmy t and equals where da t ă and dxmy t. The strategy H clearly yields an arbitrage. We therefore may and shall assume the structure condition (5) in the sequel. The reader who is not so keen about the formalities of general continuous semi-martingales may very well think of the example of an SDE d S t S t σ t dw t ` ϱ t dt, (6) 95

5 where W is a Brownian motion σ and ϱ are predictable process such that σ t implies that ϱ t without missing anything essential in the subsequent arguments. Assuming the integrability condition ż T we may well-define the Girsanov density process " Z t exp ϱ u dm u ϱ udxmy u ϱ t dxmy t ă 8, a.s. (7) ď t ď T. (8) By Itô this is a strictly positive local martingale, such that Z S is a local martingale too (compare, e.g., [67]). In particular (8) yields an equivalent super-martingale deflator. The reciprocal Ñ Z is called the numéraire portfolio, i.e. " Ñ t exp ϱ u dm u ` ϱ udxmy u. (9) By Itô s formula Ñ is a stochastic integral on S, namely dñt d ϱ S t Ñ t, and enjoys the property of being the optimal portfolio for the log-utility maximizer. t S t For much more on this issue we refer, e.g., to []. Our aim is to characterize condition (7) in terms of the condition pnup BRq of Definition 5.5. Essentially (7) can fail in two different ways. We shall illustrate this with two proto-typical examples (compare [7]) of processes S, starting at S. First consider d S t S t dw t ` p tq dt, ď t ď, () so that ş ε ϱ t dt ă 8, for all ε ą, while ş ϱ t dt 8 almost surely. In this case it is straightforward to check directly that the sequence pñ q 8 n, n where Ñ is defined in (9), yields an unbounded profit with bounded risk, as Ñ ą and lim tñ Ñt 8, a.s. The second example is d S t S t dw t ` t dt, ď t ď, () so that ş ε ϱ t dt 8, for all ε ą. This case is trickier as now the singularity is at the beginning of the interval r, s, and not at the end. This leads to 96

6 the concept of immediate arbitrage as anlayzed in [7]. Using the law of the iterated logarithm, it is shown there (Example 3.4) that in this case, one may find an S-integrand H such that H S ě and Prp H Sq t ą s, for each t ą. For the explicit construction of H we refer to [7]. As one may multiply H with an arbitrary constant C ą this again yields an unbounded profit with bounded risk. Summing up, in both of the examples () and () we obtain an unbounded profit with bounded risk. These two examples essentially cover the general case. We have thus motivated the following local version of the Fundamental Theorem of Asset Pricing (see [6, Th. 4.] for a more general result). Theorem 5.6. Let S p S t q ďtďt be a continuous semi-martingale of the form d S t dm S t ` ϱ t dxmy t, t where pm t q ďtďt is a local martingale. The following assertions are equivalent. piq The condition pnup BRq of no unbounded profit with bounded risk holds true (Def. 5.5). pi q Locally, S satisfies the condition pnf LV Rq of no free lunch with vanishing risk. pi q The set Cpq is bounded in L pω, F, Pq. piiq The process ϱ verifying (5) and (7) exists and satisfies ş T ϱ t dxmy t ă 8, a.s. pii q The Girsanov density process Z " Z t exp ϱ u dm u ϱ udxmy u, ď t ď T, is well-defined and therefore a strictly positive local martingale. pii q The numéraire portfolio Ñ Z " Ñ t exp ϱ u dm u ` is well-defined (and therefore a.s. finite). ϱ udxmy u, ď t ď T, 97

7 piiiq The set of equivalent super-martingale deflators Z e is non-empty (ESD). piii q The set of equivalent local martingale deflators in Z e, is non-empty peldq. piii q Locally, the set of equivalent martingale measures is non-empty. Proof: piiq ô pii q ô pii q ñ piii q ô piii q ñ piiiq is obvious, and pi q ô piq holds true by Definition 5.5. piiiq ñ pi q : By definition, Cpq fails to be bounded in L if there is α ą such that, for each M ą, there is X T ` p H Sq T P Cpq such that Pr X T ě Ms ě α. () Fix Z P Z e. The strict positivity of Z T, implies that β : infter Z T A s : PrAs ě αu is strictly positive. Letting M ą β the super-martingale assumption in () we arrive at a contradiction to Er Z X s ě Er Z T XT s ě βm ą. piq ñ piiq This is the non-trivial implication. It is straightforward to deduce from piq that there is a predictable process ϱ satisfying (5) (compare [83] and the discussion preceding Theorem 5.6.) We have to show that (7) is satisfied. The reader might keep the examples () and () in mind. Define the stopping time $ & τ inf % t P r, T s :,. ϱ udxmy u Condition piiq states that Prτ ă 8s. Assuming the contrary, the set tτ ă 8u then splits into the two F τ -measurable sets " A c tτ ă 8u X lim " A d tτ ă 8u X lim tõτ tõτ ϱ udxmy u 8, ϱ udxmy u ă 8, where c refers to continuous and d to discontinuous. 98

8 If PrA c s ą it suffices to define the stopping times " τ n inf t : ϱ udxmy u ě n. For each n P N, the numéraire portfolio Ñτ n at time τ n is well-defined and given by "ż τn ż τn Ñ τn exp ϱ u dm u ` ϱ udxmy u. It is straightforward to check that Ñτ n tends to `8 a.s. on A c, which gives a contradiction to piq. We still have to deal with the case PrA c s in which case we have PrA d s ą. This is the situation of the Immediate Arbitrage Theorem. We refer to [7, Th. 3.7] for a proof that in this case we may find an S-integrable, predictable process H such that p H Sq t ą, for all τ ă t ď T almost surely on A d. This contradicts assumption piq. pii q ñ pi q : Suppose that the Girsanov density process Z is well-defined and strictly positive. We may define, for ε ą, the stopping time ) τ ε inf!t : Zt ě ε so that Prτ ε ă 8s ď ε. The stopped process S ε τ then admits an equivalent martingale measure, namely dq Z dp τε. pi q ñ piq obvious as pnup BRq is a local property. We now give a similar local version of the Fundamental Theorem of Asset Pricing in the context of transaction costs. Definition 5.7. Let S ps t q ďtďt be a strictly positive, continuous process. We say that S allows for an obvious arbitrage if there are α ą and r, T sy t8u-valued stopping times σ ď τ with Prσ ă 8s Prτ ă 8s ą such that either paq S τ ě p ` αqs σ, a.s. on tσ ă 8u, or pbq S τ ď S `α σ, a.s. on tσ ă 8u. We say that S allows for an obvious immediate arbitrage if, in addition, we have paq S t ě S σ, for t P σ, τ, a.s. on tσ ă 8u, or pbq S t ď S σ, for t P σ, τ, a.s. on tσ ă 8u. 99

9 We say that S satisfies the condition pnoaq (resp. pnoiaq) of no obvious arbitrage (resp. no obvious immediate arbitrage) if no such opportunity exists. It is indeed rather obvious how to make an arbitrage if pnoaq fails, provided the transaction costs ă λ ă are smaller than α. Assuming e.g. condition paq, one goes long in the asset S at time σ and closes the position at time τ. In case of an obvious immediate arbitrage one is in addition assured that during such an operation the stock price will never fall under the initial value S σ. In particular this gives an unbounded profit with bounded risk under transaction costs λ. In the case of condition pbq one does a similar operation by going short in the asset S. Next we formulate an analogue of Theorem 5.6 in the setting of transaction costs. Theorem 5.8. Let S ps t q ďtďt be a strictly positive, continuous process. The following assertions are equivalent. piq Locally, there is no obvious immediate arbitrage pn OIAq. pi q Locally, there is no obvious arbitrage pnoaq. pi q Locally, for each ă λ ă, the process S does not allow for an arbitrage under transaction costs λ, i.e. C X L ` tu, (3) where C is the cone given by Definition 4.6 for the stopped process S τ. pi 3 q Locally, for each ă λ ă, the process S does not allow for a free lunch with vanishing risk under transaction costs λ, i.e. C X L 8 X L 8` tu, (4) where the bar denotes the closure with respect to the norm topology of L 8. pi 4 q Locally, for each ă λ ă, the process S does not allow for a free lunch under transaction costs λ, i.e. C X L 8 X L 8` tu, (5) where now the bar denotes the closure with respect to the weak star topology of L 8.

10 piiq Locally, for each ă λ ă, the condition pcp S λ q of existence of a λ-consistent price system holds true. pii q For each ă λ ă the condition pcld λ q of existence of a λ- consistent local martingale deflator holds true. Proof: pi 4 q ñ pi 3 q ñ pi q ñ pi q ñ piq is straight-forward, as well as piiq ô pii q. piq ñ piiq: As assumption piiq is a local property we may assume that S satisfies (N OIAq. To prove piiq we do a similar construction as in ([4], Proposition.): we suppose in the sequel that the reader is familiar with the proof of [4], Proposition. and define the preliminary stopping time ϱ by ϱ inf! t ą : St S ) ě ` λ or St S ď. `λ In fact, in [4] we wrote ε instead of λ which does not matter as both quantities are arbitrary small. 3 Define the sets Ā`, Ā, and Ā as Ā` t ϱ ă 8, S ϱ p ` λqs u, (6) Ā ϱ ă 8, S ϱ `λ S (, (7) Ā t ϱ 8u. (8) It was observed in [4] that assumption pnoaq by definition rules out the cases PrĀ` s and PrĀ s. But under the present weaker assumption pnoiaq we cannot a priori exclude the possibilities PrĀ` s and PrĀ s. To refine the argument from [4] in order to apply to the present setting, we distinguish two cases. Either we have PrĀ` s ă and PrĀ s ă ; in this case we let ϱ ϱ and proceed exactly as in the proof of ([4], Proposition.) to complete the first inductive step. The second case is that one of the probabilities PrĀ` s or PrĀ s equals one. We assume w.l.g. PrĀ` s, the other case being similar. Define the real number β ď as the essential infimum of the random S variable min t ďtď ϱ S. We must have β ă, otherwise the pair p, ϱ q would define an immediate obvious arbitrage. We also have the obvious inequality β ě. `λ We define, for ą γ ě β the stopping time ϱ γ inf! t ą : S t S ě ` λ or St S ) ď γ.

11 Defining Āγ,` γ ts ϱ a.s. partition of Ā` ą γ ą β. We claim that lim γœβ PrĀγ, p ` λqs u and Āγ, γ S ϱ γs ( we find an into the sets Āγ,` and Āγ,. Clearly PrĀγ, s ą, for s. Indeed, supposing that this limit were positive, we again could find an obvious immediate arbitrage as in this case we have that PrĀβ, s ą. Hence the pair of stopping times σ ϱ β. β βs u ` 8 β p`λqs u ts ϱ ts ϱ and τ ϱ. β βs u ` 8 β p`λqs u ts ϱ ts ϱ would define an obvious immediate arbitrage. We thus may find ą γ ą β such that PrĀγ, s ă. After having found this value of γ we can define the stopping time ϱ in its final form as ϱ : ϱ γ. Next we define, similarly as in (6) and (7) the sets A` tϱ ă 8, S ϱ p ` λqs u A tϱ ă 8, S ϱ γs u to obtain a partition of Ω into two sets of positive measure. As in [4] we define a probability measure Q on F ϱ by letting dq dp to be constant on these two sets, where the constants are chosen such that Q ra` s β and Q `λ β ra s λ. We then may define the Q `λ β -martingale p S t q ďtďϱ by S t E Q rs ϱ F t s, ď t ď ϱ, to obtain a process remaining in the interval rγs, p ` λqs s. The above weights for Q were chosen in such a way to obtain S E Q rs ϱ s S. This completes the first inductive step similarly as in [4]. Summing up, we obtained ϱ, Q and p S t q ďtďϱ precisely as in the proof of ([4], Proposition.) with the following additional possibility: it may happen that ϱ does not stop when S t first hits p ` λqs or S, but rather when S `λ t first hits p ` λqs or βs, for some ă β ă. In this case we have `λ PrA s and we made sure that PrA s ă, i.e., we have a control on the probability of ts ϱ βs u.

12 We now proceed as in [4] with the inductive construction of ϱ n, Q n and p S t q ďtďϱn. The new ingredient is that again we have to take care (conditionally on F ϱn ) of the additional possibility PrAǹ s or PrAń s. Supposing again w.lg. that we have the first case, we deal with this possibility precisely as for n above, but now we make sure that PrAń s ă n. This completes the inductive step and we obtain, for each n P N, an equivalent probability measure Q n on F ϱn and a Q n -martingale p S t q ďtďϱn taking values in the bid ask spread pr S `λ t, p ` λqs t sq ďtďϱn. We note in passing that there is no loss of generality in having chosen this normalization of the bid ask spread instead of the usual normalization rp λ qs, S s by passing from S to S p λ qs and from λ to λ λ. There is one more thing to check to complete the proof of piiq : we have to show that the stopping times pϱ n q 8 n increase almost surely to infinity. This is verified in the following way: suppose that pϱ n q 8 n remains bounded on a set of positive probability. On this set we must have that Sϱ n` S ϱn equals p`λq or, except for possibly finitely many `λ n s. Indeed, the above requirement PrAń s ă n makes sure that a.s. the novel possibility of moving by a value different from p ` λq or can only happen finitely many times. Therefore `λ we may, as in [4], conclude from the continuity and strict positivity of the trajectories of S that ϱ n increases a.s. to infinity which completes the proof of piiq. piiq ñ pi 4 q As piiq as well as pi 4 q are local properties holding true for each ă λ ă, it will suffice to show that pcp S λ q implies (5), for fixed ă λ ă. Let p S, Qq be a λ-consistent price system and define the half-space H of L 8 pω, F, Pq H ϕ T P L 8 : E Q rϕ T s ď (, which is σ -closed and satisfies H X L 8` tu. It follows from Proposition 4.5 that, for all self-financing, admissible trading strategies pϕ t, ϕ t q ďtďt we have that pϕ t Z t ` ϕ t Z t q ďtďt is a super-martingale under Q, which implies that C X L 8. Hence (5) holds true. Recall Theorem 4. from the previous section. It states the polarity between the sets Cpxq and Dpyq in L `. This result which will turn out to be the basis of the duality theory of portfolio optimization in the next. The crucial hypothesis in Theorem 4. is the assumption of pcp S λ q, for each ă λ ă. It turns out that it is sufficient to impose this hypothesis only locally i.e. under one of the conditions listed in Theorem 5.8. The proof is rather standard but somewhat lengthy and was carried out in detail in 3

13 [8] and []. Here we content ourselves to simply stating this result without going through the proof. Theorem 5.9. Suppose that the continuous, strictly positive process S ps t q ďtďt satisfies condition pcp S λ q locally, for each ă λ ă. Fix ă λ ă. piq The sets Apxq, Cpxq, Bpyq, Dpyq defined in Definition 4. are convex, closed (w.r to convergence in measure) subsets of L pr q and L `prq respectively. The sets Cpxq and Dpyq are also solid. piiq Fix x ą, y ą and ϕ T P L`pRq. We have ϕ T P Cpxq iff xϕ T, Z T y ď xy, (9) for all Z T P Dpyq. In fact, we also have sup E Q rϕ p S,QqP T s xy. () CP S λ pii q We have Z T P Dpyq iff xϕ T, Z T y ď xy () for all ϕ T P Cpxq. piiiq The sets Apq and Cpq are bounded in L pr q and L prq respectively and contain the constant functions p, q (resp. ). 4