Irene Klein

Asymptotic arbitrage with small transaction costs

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for a sequence of financial markets with small proportional transaction costs λn on market n, in terms of contiguity properties of sequences of equivalent probability measures induced by λn-consistent price systems. These results are analogous to the frictionless case; compare (Kabanov and Kramkov in Finance Stoch. 2:143–172, 1998; Klein and Schachermayer in Theory Probab. Appl. 41:927–934, 1996). Our setting is simple, each market n contains two assets. The proofs use quantitative versions of the Halmos–Savage theorem (see Klein and Schachermayer in Ann. Probab. 24:867–881, 1996) and a monotone convergence result for nonnegative local martingales. Moreover, we study examples of models which admit a strong asymptotic arbitrage without transaction costs, but with transaction costs λn>0 on market n; there does not exist any form of asymptotic arbitrage. In one case, (λn) can even converge to 0, but not too fast.

No asymptotic free lunch reviewed in the light of Orlicz spaces

No asymptotic free lunch (NAFL) was introduced in [11] and led to a general version of the Fundamental Theorem of Asset Princing (FTAP) for large financial markets. The present note observes that NAFL can be defined in a natural way using Orlicz spaces. This gives a transparent proof of the FTAP-result.

A COMMENT ON MARKET FREE LUNCH AND FREE LUNCH

Frittelli (2004) introduced a market free lunch depending on the preferences of the agents in the market. He characterized no arbitrage and no free lunch with vanishing risk in terms of no market free lunch (the difference comes from the class of utility functions determining the market free lunch). In this note we complete the list of characterizations and show directly (using the theory of Orlicz spaces) that no free lunch is equivalent to the absence of market free lunch with respect to monotone concave utility functions.

Market free lunch and large financial markets

The main result of the paper is a version of the fundamental theorem of asset pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover, various notions of no asymptotic arbitrage are characterized in terms of no asymptotic market free lunch; the difference lies in the set of utilities. In particular, it is shown directly that no asymptotic market free lunch with respect to monotone concave utilities is equivalent to no asymptotic free lunch. In principle, the paper can be seen as the large financial market analogue of [Math. Finance 14 (2004) 351--357] and [Math. Finance 16 (2006) 583--588].

No Arbitrage Theory for Bond Markets

We investigate default-free bond markets and relax assumptions on the numeraire, which is typically chosen to be the bank account. Considering numeraires different from the bank account allows us to study bond markets where the bank account process is not a valid numeraire or does not exist at all. We argue that this feature is not the exception, but rather the rule in bond markets when starting with, e.g., terminal bonds as numeraires. Our setting are general cadlag processes as bond prices, where we employ directly methods from large financial markets. Moreover, we do not restrict price processes to be semimartingales, which allows for example to consider markets driven by fractional Brownian motion. In the core of the article we relate the appropriate no arbitrage assumptions (NAFL), i.e. no asymptotic free lunch, to the existence of an equivalent local martingale measure with respect to the terminal bond as numeraire, and no arbitrage opportunities of the first kind (NAA1) to the existence of a supermartingale deflator, respectively. In all settings we obtain existence of a generalized bank account as a limit of convex combinations of roll-over bonds. The theory is illustrated by several examples.

Asymptotic proportion of arbitrage points in fractional binary markets

A fractional binary market is a binary model approximation for the fractional Black–Scholes model, which Sottinen constructed with the help of a Donsker-type theorem. In a binary market the non-arbitrage condition is expressed as a family of conditions on the nodes of a binary tree. We call “arbitrage points” the nodes which do not satisfy such a condition and “arbitrage paths” the paths which cross at least one arbitrage point. In this work, we provide an in-depth analysis of the asymptotic proportion of arbitrage points and arbitrage paths. Our results are obtained by studying an appropriate rescaled disturbed random walk.

Asymptotic Arbitrage with Small Transaction Costs

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for a sequence of financial markets with small proportional transaction costs in terms of contiguity properties of sequences of equivalent probability measures induced by consistent price systems. These results are analogous to the frictionless case. Our setting is simple, each market contains two assets. The proofs use quantitative versions of the Halmos-Savage Theorem and a monotone convergence result of nonnegative local martingales. Moreover, we study examples of models which admit a strong asymptotic arbitrage without transaction costs; but with transaction costs there does not exist any form of asymptotic arbitrage.