Consistent Price Systems and Arbitrage Opportunities of the Second Kind in Models with Transaction Costs.

In contrast with the classical models of frictionless financial markets, market models with proportional transaction costs, even satisfying usual no-arbitrage properties, may admit arbitrage opportunities of the second kind. This means that there are self-financing portfolios with initial endowments laying outside the solvency region but ending inside. Such a phenomenon was discovered by M. R´asonyi in the discrete-time framework. In this note we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for the property which we call No Free Lunch of the 2nd Kind, NFL2. We provide a number of equivalent conditions elucidating, in particular, the financial meaning of the property B which appeared as an indispensable “technical” hypothesis in previous papers on hedging (super-replication) of contingent claims under transaction costs. We show that it is equivalent to another condition on the “richness” of the set of consistent price systems, close to the condition PCE introduced by R´asonyi. In the last section we deduce the R´asonyi theorem from our general result using specific features of discrete-time models.Consistent price systems; No Free Lunch; Arbitrage; Transaction costs; Martingales; Set-valued processes;

Mean square error for the Leland-Lott hedging strategy: convex pay-offs.

Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim V T using the classical Black–Scholes formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients k n =k 0 n −α , where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility $\widehat{\sigma}_{n}$ in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value $V^{n}_{T}$ to the pay-off V T . In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary pay-off V T =G(S T ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n −1/2 in L 2 and find the first order term of the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are $t_{i}^{n}=g(i/n)$, where the strictly increasing scale function g:[0,1]→[0,1] and its inverse f are continuous with their first and second derivatives on the whole interval, or g(t)=1−(1−t) β , β≥1. We show that the sequence $n^{1/2}(V_{T}^{n}-V_{T})$ converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.Diffusion approximation; Martingale limit theorem; European option; approximate hedging; transaction costs; Leland-Lott strategy; Black-Scholes formula;

Essential Supremum in a d-dimensional Real Space with Respect to a Random Cone and Applications

The goal of this paper is to introduce the notion of essential supremum of a family of multi-dimensional random variables with respect to a random convex cone. We give two applications in mathematical finance; We determine the ''minimal'' portfolio process super-hedging an American claim in the Kabanov dicrete-time model with transaction costs. For the same model, we construct a dynamic risk measure in a continuous-time setting. At last, we solve a Skorokhod problem with oblique reflection.ou

Why the Market's Participants in the Modigliani-Miller Model are Markowitz Rational?

The seminal works of Modigliani-Miller and Markowitz-Sharpe remain the cornerstone of financial theory. We reconcile these seemingly distinct approaches, in a unified theorem, by showing that the agents acting on the market defined by Modigliani-Miller are Markowitz rational when deriving the arbitrage reasoning in terms of Sharpe ratios. As a main policy implication, we show that government guarantees modify market's equilibrium as they provoke arbitrage opportunities

A Model of Self-Regulation in Banking Industry

This article derives a model of self-regulation where banks issue insurance products to hedge their leverage ratio. This approach is an alternative policy to Basel regulation for controlling systemic risk without increasing equity level. We show some conditions under which the model can be applied to each of the 22 banks of 5 major countries from 2005 to 2012

On Supremal and Maximal Sets with Respect to Random Partial Orders

The paper deals with de nition of supremal sets in a rather general frameworkwhere deterministic and random preference relations (preorders) and partialorders are de ned by continuous multi-utility representations. It gives ashort survey ofnonnonouirechercheInternationa

The Impact of Speculation on Firms' Capital Structure

Modigliani and Miller (1958) suppose that "nothing is involved in the way of specific assumptions about investor attitudes or behavior other than that investors behave consistently and prefer more income to less income, ceteris paribus." Moreover, it is assumed that arbitrage opportunities may only occur in presence of two firms having distinct values but same pro file. The natural question is why such investors dismiss speculative opportunities between firms having same values with distinct capital structures?no

VECTOR-VALUED COHERENT RISK MEASURE PROCESSES

Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results, we study different notions of time consistency and we give examples of vector-valued risk measure processes