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Fundamental Theorem of Asset Pricing under Transaction costs and Model uncertainty
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48104, email@example.com
Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, firstname.lastname@example.org
We prove the Fundamental Theorem of Asset Pricing for a discrete time financial market where trading is subject to proportional transaction cost and the asset price dynamic is modeled by a family of probability measures, possibly non-dominated. Using a backward-forward scheme, we show that when the market consists of a money market account and a single stock, no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of consistent price systems. We also show that when the market consists of multiple dynamically traded assets and satisfies efficient friction, strict no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of strictly consistent price systems.
Key words: transaction costs, non-dominated collection of probability measures, Fundamental Theorem of Asset Pricing, martingale selection problem
MSC2000 subject classification: Primary: 60G42, 91B28; secondary: 03E15
OR/MS subject classification: Primary: finance: asset pricing; secondary: probability
History: Received on May 13, 2014; Second version received on May 8, 2015; Accepted on August 29, 2015.
The Fundamental Theorem of Asset Pricing (FTAP), as suggested by its name, is one of the most important theorems in mathematical finance and has been established in many different settings: discrete and continuous, with and without transaction costs. It relates no-arbitrage concepts to the existence of certain fair pricing mechanisms, which provides the rationale for why in duality theory, it is often reasonable to assume that the dual domain is non-empty.
We consider a discrete time, finite horizon financial market consisting of stocks and a zero-interest money market account. When the market is frictionless and modeled by a single probability measure, the classical result by Dalang-Morton-Willinger  asserts there is no-arbitrage if and only if the set of equivalent martingale measures is not empty. With proportional transaction costs, the set of martingale measures is replaced by the set of consistent price systems (CPS’s) or strictly consistent price systems (SCPS’s). Equivalence between no-arbitrage and existence of a CPS is established by Kabanov and Stricker  for finite probability space , and by Grigoriev  when the dimension is two. Such equivalence in general does no hold for infinite spaces and higher dimensions (see Section 3 of Schachermayer  and page 128-129 of Kabanov and Safarian  for counter examples). For such an equivalence one needs the notion of robust no-arbitrage introduced by Schachermayer in , where he showed that this notion is equivalent to the existence of an SCPS. Alternatively, robust no-arbitrage can be replaced by strict no-arbitrage plus efficient friction (see Kabanov et al. ). There exist a few different proofs of the FTAP under transaction costs. Besides the proofs of Schachermayer  and Kabanov et al.  which rely on the closedness of the set of hedgeable claims and a separation argument, there is a utility-based proof by Smaga  and proofs based on random sets by Rokhlin .
In recent years, model uncertainty has gained a lot of interest in financial mathematics since it corresponds to a more realistic modeling of the financial market. By model uncertainty, we mean a family of probability measures, usually non-dominated. Each member of represents a possible model for the stock. One should think of as being obtained from market data. We have a collection of measures rather than a single one because we do not have point estimates but confidence intervals. The non-dominated case is generally much harder because the classical separation argument does not work. In a market model without transaction cost, the recent work by Bouchard and Nutz  used a local analysis to establish the equivalence between the absence of arbitrage in a quasi-sure sense and the existence of an “equivalent” family of martingale measures. Acciaio et al.  obtained a different version of the FTAP by working with a different no-arbitrage condition which excludes model-independent arbitrage, i.e. arbitrage in a sure sense only.
In this paper, we prove the FTAP when both proportional transaction cost and model uncertainty are present. We start with a financial market consisting of a money market account and a single stock and introduce the no-arbitrage concept NA. We prove its equivalence to the existence of a family of CPS’s, and later extend our results to a market with multiple assets where NA is replaced by NA plus efficient friction, and CPS is replaced by SCPS. Similar to the classical theory (in the set-up of which there is a dominating measure), the two-dimensional case is different from the multi-dimensional case in that no-arbitrage alone is sufficient to imply the existence of a CPS. That is the market model need not satisfy the efficient friction hypothesis. Hence the two-dimensional case is worth a separate study, and will be the main focus of this paper. On a related note, when there is a single stock, Dolinsky and Soner  proved the super-hedging theorem (by first discretizing the state space and then taking a limit) and stated the FTAP as a corollary. We follow a different methodology and state the result for an arbitrary collection of probability measures instead of the collection of all probability measures as they do. Thanks to our method we are able to work with a more general structure on the proportional transaction cost instead of taking it a constant, which is useful as this proportion in real markets is likely to change with changing market conditions over time. Our contribution can be seen as an extension of Bouchard and Nutz  to the transaction cost case, and a generalization of the results of Rokhlin  on the martingale selection problem to the non-dominated case.
Similar to Bouchard and Nutz , we proceed in a local fashion: first obtain a CPS for each single-period model and then do pasting using a suitable measurable selection theorem. Although our proof follows the ideas in , the multi-period case turns out to be quite different when transaction costs are added. A distinct feature for frictionless markets is that the absence of arbitrage for the multi-period market is equivalent to the absence of arbitrage in all single-period markets. So it is enough to look at each single period separately and paste the martingale measures together. This equivalence, however, breaks down in the presence of transaction cost. A simple example is the following two-period market: where an underline denotes the bid price and an overline denotes the ask price. Each period is arbitrage-free, but buying at time 0 and selling at time 2 is an arbitrage for the two-period market. So we cannot in general paste two one-period martingale measures to get a two-period martingale measure; in particular, the endpoints of the underlying martingales constructed for each single period may not match. We need to solve a non-dominated martingale selection problem. The martingale selection problem when is a singleton was studied by Rokhlin in a series of papers [15, 17, 16] using the notion of support of regular conditional upper distribution of set-valued maps. In our case, it is difficult to talk about conditional distribution due to the lack of dominating measure. Nevertheless, we got some inspiration from Rokhlin [17, 16] and Smaga  and developed a backward-forward scheme:
Backward recursion: modify the original bid-ask prices backward in time by potentially more favorable ones to account for the missing future investment opportunities;
Forward extension: extend the CPS forward in time in the modified market.
When there is no dominating measure, the backward-forward scheme brings some measurability issues. On one hand, neither Borel nor universal measurability is preserved under the backward recursion. This can be circumvented by working with lower semi-analytic bid prices and upper semi-analytic ask prices. On the other hand, the proofs of Lemmas 4.6 and 4.8 in Bouchard and Nutz  rely critically on the stock prices being Borel measurable. However, our modified market only has semi-analytic bid-ask prices, which makes measurable selection in many places challenging (see Remarks 6 and 7). To overcome this difficulty, we break up the selection of a CPS into two steps: first select a pair of measures using the semi-analyticity of the bid and ask prices separately, and then find a convex weight function which plays an important role in boundary extension, which is an additional step that needs to be taken care of in dimension two because of the lack of the efficient friction hypothesis. A measurable one-period CPS can be constructed out of the measurable triplet and the bid-ask prices of the modified market.
Similar measurability issues exist in the multi-dimensional case. Fortunately, it turns out that doing backward recursion on the dual cones of the solvency cones preserves graph analyticity which is enough to make measurable selection in the forward extension possible. Here at each step in the forward extension, we do not select a one-period SCPS with a prescribed initial value directly, but a vector of equivalent measures related to via . This allows us to use the fact that there exists a jointly Borel measurable function , which is key to measurable selection.
We point out that during the review process, a paper by Bouchard and Nutz  which is closely related to ours came out. There the authors propose a different notion of no-arbitrage (a quasi-sure version of no-arbitrage of the second kind), denoted by NA, with which they proved a multi-dimensional version of the FTAP under the extra assumption of efficient and bounded friction. We find their no-arbitrage notion a bit too strong. It means that the market is already in a good form for martingale extension, so that the backward recursion is avoided. A simple one-period, single stock market which satisfies NA, thus reasonable in our opinion, fails NA. Despite that NA fails for the original market, it holds for the corresponding modified market. The multi-dimensional part of our paper is a generalization of  to allow for more general market structure. Apart from the no-arbitrage concept used, our paper also differs in two other ways. First, we do not assume bounded friction and ; efficient friction can also be dropped in the two-dimensional case. Second, our measurability assumption is weaker; we are able to handle markets with universally measurable solvency cones as long as their dual cones have analytic graphs.
The rest of the paper is organized as follows. Sections id1, 2 and id1 are devoted to the two-dimensional case. In Section id1, we introduce the probabilistic framework, set up the financial market model, and state the main result. In Section 2, we prove some one-period results which serve as the building blocks. In Section id1, we present the backward-forward schemes for a multi-period market and prove the main theorem. Section id1 extends our methodology and results to the multi-dimensional case. A few technical lemmas are collected in Appendix.
We follow the notation in Bouchard and Nutz . Let be the time horizon. Let be a Polish space and be the -fold Cartesian product with the convention that is a singleton. Denote by the Borel sigma-algebra on , and by the universal completion of . We write for . When we say a stochastic process is predictable or adapted, we mean with respect to the filtration . Let denote the set of all probability measures on . For each and , we are given a nonempty convex set , representing the set of possible models for the -th period. We assume (considered as set-valued maps from to ) has an analytic graph. This assumption ensures that admits a universally measurable selector. Define the uncertainty set of the multi-period market by
Consider a financial market consisting of a money market account with zero interest rate, and a stock with strictly positive bid price and ask price . are assumed to be lower semi-analytic (l.s.a.) and upper semi-analytic (u.s.a.), respectively, for all . A self-financing111We allow agents to throw away non-negative quantities of the assets. portfolio process is an -valued predictable process satisfying and for all , where . Denote by the set of self-financing portfolio processes. Let . is interpreted as the set of hedgeable claims (in terms of physical units) from zero initial endowment.
We say NA() holds if for all , -quasi-surely (q.s.)222A set is -polar if it is -null for all . A property is said to hold -q.s. if it holds outside a -polar set. implies -q.s..
NA() implies NA(). Indeed, let be such that -q.s. hence -a.s. for all . For each , NA() implies -a.s.. Since this holds for all , -q.s. and NA() holds. The reverse direction is not true. Consider a one-period market with and , . Let be the Dirac measures concentrated on and , respectively. Then it is easy to see that there is arbitrage under both and , but not under .333“conv” stands for convex hull.
A pair is called a consistent price system (CPS) if is a -martingale in the filtration and -a.s.. Denote the set of all consistent price systems by .
The main theorem of this paper is given below.
The following are equivalent:
.444The notation is taken from . It means for some .
Let be the collection of the first components of that are dominated by some measure in . Theorem 1(ii) is equivalent to saying are equivalent in terms of polar sets. When is a singleton, we recover the classical result of the existence of an equivalent measure.
In this section, we prove the more difficult direction of the FTAP (i.e. no-arbitrage implies the existence of CPS’s) for a one-period market. To prepare for multi-period case, we also discuss how to construct martingales with prescribed initial values.
Let be a measurable space with filtration and . Let be a nonempty convex set. The bid and ask price processes of the stock are given by constants and -measurable random variables , respectively. Note that NA for this one-period market can be stated in the equivalent form: , -q.s. implies -q.s..
For each , define
is nonempty by Lemma 11.
Suppose satisfying and . Then , such that and
Let . We first show, along the same lines in the first paragraph on page 13 of , that there exist (not necessarily equivalent) such that
Let . We have . Define , use Lemma 11 to replace by such that , and further replace by defined by . It can be checked that and for small enough. Similar, we obtain the existence of .
It remains to replace , by , with the additional requirement that . To this end, let . It is easy to see that is a set of equivalent measures contained in . Moreover, for sufficiently close to 1, and for sufficiently close to zero. \@endparenv
Suppose NA holds. Then .
Let and consider three cases:
Case 1. -q.s.. In this case, NA implies -q.s.. We choose the CPS to be , and .
Case 2. -q.s.. In this case, NA implies -q.s.. We choose the CPS to be , and .
Case 3. such that and . In this case, let be given by Lemma 1. We can find such that
We have , -a.s., and . \@endparenv
We can directly use (defined in the proof of Lemma 1) instead of to construct the CPS in Case 3. The equivalence of and only matters in the multi-period case where it is important for us to construct martingales that stay away from the boundary of the bid-ask spread whenever possible (so that they are extendable to the next period). If , defined by (2) will satisfy -a.s..555“ri” stands for relative interior.
Recall that when we go to the multi-period case, we cannot directly paste two single-period CPSs, but to first make sure the starting point of the current-period martingale matches the terminal point of its parent. In other words, we are interested in constructing martingales with certain prescribed initial values. Proposition 2 gives a set of starting points that admit a martingale extension.
For a random variable and a nonempty family of probability measures on , the support of the distribution of under , denoted by supp, is the smallest closed set such that . Equivalently, a point belongs to if and only if every open ball around has positive measure under some member of .
Suppose . Then , such that and
We can find and such that . By definition of support, satisfying and . Lemma 1 applied to the market yields the desired and . \@endparenv
It is possible to show that and , and such that . For the proof, simply consider three cases: 1) with , 2) with , and 3) , . We state the one-period result in terms of a pair of measures because it is more amenable for measurable selection when moving to the multi-period case (see Remark 7). Once we have a pair of measures, a CPS can be easily constructed by means of (1) and (2).
In this section, we prove the FTAP for a multi-period market through a backward-forward scheme. Back to the setup in introduction, the set is defined as the product of the nonempty convex sets which have analytic graphs. Throughout this section, we also assume is l.s.a. and is u.s.a.. The reason for working with semi-analytic price processes is that we need a property that can be preserved under the backward recursion (3). Both Borel and universal measurability are not preserved (see Remark 5). This problem does not exist in market without transaction cost since there is no need to redefine the stock price, nor does it matter when there is a dominating measure , since we can always modify a universally measurable map on a -null set to make it Borel measurable. Finally, for a map on , we will often see it as a map on and write .
Define processes recursively by and
for , .
For each , is l.s.a. and is u.s.a..
We only show the lower semi-analyticity of . A symmetric argument gives the upper semi-analyticity of . is l.s.a. by assumption. Suppose is l.s.a., we deduce the lower semi-analyticity of . Let , we have
It can be checked that if is an analytic set, then is an u.s.a. function. Hence is u.s.a. by the induction hypothesis. By [2, Proposition 7.48], we also have is u.s.a.. It follows that is the projection of the intersection of two analytic sets, thus also analytic. This shows is l.s.a.. , being the maximum of two l.s.a. functions, is also l.s.a.. \@endparenv
Neither Borel nor universal measurability are preserved under the backward recursion. To see this, similar to Remark 4.4 of Bouchard and Nutz , consider , and for some . Then . If is Borel, then is Borel measurable. But is not Borel measurable in general because the projection of a Borel set may not be Borel. Similarly, if is universally measurable, then is universally measurable. But is not universally measurable in general because the projection of a universally measurable set may not be universally measurable.
Apart from preserving semi-analyticity, the recursively defined -market666 refers to a multi-period market with bid price and ask price for all . has two nice properties. First, its spread is not too wide: at least all points in the interior of admits a martingale extension to the next period -q.s., although there are delicate issues when the point lies on the boundary of the spread. Second, its spread is not too narrow either, in the sense that it still satisfies NA when the original market does. In summary, this new market fits our needs perfectly. The general idea of proving the nontrivial implication of multi-period FTAP is to replace the original market by the modified market , and do martingale extension in the modified market. Interior extension is not too hard in view of Proposition 2; the challenging part is boundary extension. It turns out that boundary extension is possible if we avoid hitting boundaries as much as we can from the beginning.
Before proving the main theorem, we need two crucial lemmas.
Let NA hold for the original market . Then NA also holds for the modified market . In particular, -q.s. for all .
We prove NA for the modified market by backward induction. Let
denotes the -th intermediate market obtained in the backward recursion procedure. We show NA for implies that for . It suffices to show that for any self-financing portfolio process in , there exists a self-financing portfolio process in with equal or better terminal position.
Let be a self-financing portfolio process in with and -q.s.. Consider another portfolio process defined by ,
That is, we follow up to time , stick to its stock position whenever the transaction at time can be carried out in , and postpone the transaction to time if it is not admissible in , and follow the stock position of again afterwards. Clearly, is self-financing in , and . We want to show -q.s.. It suffices to show . During the -th and -th periods, and are trading the same total number of shares, just at different times. So we only need to check that faces a trading price as favorable as, if not more favorable than the one faced by . By our construction of , when , we must have -q.s.. Fubini’s theorem implies -q.s. on . Similarly, -q.s. on . Therefore, has price disadvantage only on a -polar set. \@endparenv
Unlike Bouchard and Nutz , we do not attempt to show the set
is universally measurable and -polar. In fact, the measurability of depends on the measurability of the set-value maps supp, supp. With , only known to be semi-analytic, the measurability of the support maps is questionable. Even if they are universally measurable, to show is -polar by a contradiction argument similar to , one needs to construct an arbitrage strategy and a measure under which a profit can be realized with positive probability. The construction of such a measure involves a measurable selection from e.g. the set-valued map . However, when is l.s.a. and is u.s.a., we have that is l.s.a. and is co-analytic. So graph fails to be analytic in general. By assuming has a Borel graph, we can make graph() co-analytic. But we are not aware of a selection theorem that applies to co-analytic set, unless one is willing to assume -determinancy777-determinancy refers to the principle that every analytic game is determined. It cannot be proved in the standard ZFC axioms (Zermelo-Fraenkel with the Axiom of Choice). (see Kechris [13, Corollary 36.21]).
Let and , be Borel. Let
Then has an analytic graph and there exist universally measurable selectors for on the universally measurable set . In addition, there exists a universally measurable function such that
We first show has an analytic graph. The proof is very similar to that of Bouchard and Nutz [4, Lemma 4.8]. So we shall be brief. Let
By Bertsekas and Shreve [2, Proposition 7.48], is u.s.a. and is l.s.a.. So has an analytic graph. Let
where we choose a version of the Radon-Nikodym derivatives (using absolutely continuous parts) that are jointly Borel measurable (see Dellacherie and Meyer [6, Theorem V.58] and the remark after it). Bertsekas and Shreve [2, Propositions 7.26, 7.29] then imply is Borel. So is Borel. Hence has an analytic graph. An application of Jankov-von Neumann Selection Theorem (Lemma 14) yields the desired universally measurable selectors , and for on . Outside this set, define .
Next, we construct a universally measurable weight function . By Bertsekas and Shreve [2, Proposition 7.46], the maps and are universally measurable. We have and on . Define on and on . Then is universally measurable, -valued and satisfies (5). \@endparenv
Lemma 4 is the measurable version of Proposition 2. By Remark 4, in a one-period market, one can directly construct a CPS with the martingale component being a convex combination of the bid-ask prices. Thus it may be natural to work with
where . The problem is that when is l.s.a. and is u.s.a., is only universally measurable and is not analytic in general, which makes measurable selection difficult. To overcome this issue, we break up the selection of a CPS into two steps: first select a pair of measures using the lower semi-analyticity of and the upper semi-analyticity of separately, and then find a convex weight . A martingale can be constructed out of these selectors as an adapted convex combination of the bid-ask prices (see (8)).
We are now ready to prove our main results.
. (i) (ii): We first replace the original -market by the modified -market which lies inside , is still semi-analytic by Lemma 2 and satisfies NA() by Lemma 3. It suffices to prove (ii) for the modified market because any CPS for the modified market is a CPS for the original market. Let us prove an auxiliary claim that (i) implies the following:
, , with and an adapted process , valued in , such that if , if , and if , . Moreover, letting
with the convention that , we have on the set -q.s.. Similarly, letting
we have on the set -q.s..
We do induction on the number of periods in the market. When there is only one period, for any , we set when