Christophe Stricker

No-arbitrage criteria for financial markets with efficient friction

Abstract. We consider a multi-asset discrete-time model of a financial market with proportional transaction costs and efficient friction and prove necessary and sufficient conditions for the absence of arbitrage. Our main result is an extension of the Dalang–Morton–Willinger theorem. As an application, we establish a hedging theorem giving a description of the set of initial endowments which allows to super-replicate a given contingent claim.

On the closedness of sums of convex cones in \(L^0\) and the robust no-arbitrage property

Abstract. This note is a natural complement to our previous work where we studied no-arbitrage criteria for markets with efficient friction. We discuss, in our general geometric framework, the recent result of Walter Schachermayer on a necessary and sufficient condition for the existence of strictly consistent price systems and give its quick proof.

Minimal Hellinger martingale measures of order q

Abstract This paper proposes an extension of the minimal Hellinger martingale measure (MHM hereafter) concept to any order q≠1 and to the general semimartingale framework. This extension allows us to provide a unified formulation for many optimal martingale measures, including the minimal martingale measure of Föllmer and Schweizer (here q=2). Under some mild conditions of integrability and the absence of arbitrage, we show the existence of the MHM measure of order q and describe it explicitly in terms of pointwise equations in ℝd. Applications to the maximization of expected power utility at stopping times are given. We prove that, for an agent to be indifferent with respect to the liquidation time of her assets (which is the market’s exit time, supposed to be a stopping time, not any general random time), she is forced to consider a habit formation utility function instead of the original utility, or equivalently she is forced to consider a time-separable preference with a stochastic discount factor.

Hedging of Contingent Claims under Transaction Costs

We consider a general framework covering models of financial markets with transaction costs. Assuming that the solvency cones are proper and evolve in time continuously we prove a hedging theorem describing the set of initial endowments allowing to hedge a vector-valued contingent claim by a self-financing portfolio.

Remarks on the true No-arbitrage Property

We discuss conditions of absence of arbitrage in the classical sense (the “true” NA property) for the model given by a family of continuous value processes. In particular, we obtain a criterion for the NA property in a market model with countably many securities with continuous price processes. This result generalizes the well-known criteria due to Levental-Skorohod and Delbaen-Schachermayer.

On Martingale Selectors of Cone-Valued Processes

We discuss a result of Guasoni, Rasonyi, and Schachermayer on the existence of martingale selectors for a class of continuous cone-valued processes. The setting includes that arising in models of financial markets with transaction costs.

The Dalang–Morton–Willinger Theorem Under Delayed and Restricted Information

We extend the classical no-arbitrage criteria to the case of a model where the investor’s decisions are based on a partial information (e.g., because of delay or round-off errors), that is the portfolio strategies are predictable with respect to a subfiltration. Our main result is a ramification of the famous Dalang–Morton– Willinger theorem: the model is arbitrage-free if and only if there exists an equivalent probability measure P̃ such that the optional projection of the price process with respect to P̃ is a P̃ -martingale.

On the law of one price

Abstract.We consider the standard discrete-time model of a frictionless financial market and show that the law of one price holds if and only if there exists a martingale density process with strictly positive initial value. In contrast to the classical no-arbitrage criteria, this density process may change its sign. We also give an application to the CAPM.

No-arbitrage criteria for financial markets with transaction costs and incomplete information

This note deals with criteria of absence of arbitrage opportunities for an investor acting in a market with frictions and having a limited access to the information flow. We develop a mathematical scheme covering major models of financial markets with transaction costs and prove several results including a criterion for the robust no-arbitrage property and a hedging theorem.

Componentwise and Vector Stochastic Integration with Respect to Certain Multi-Dimensional Continuous Local Martingales

In a previous paper we provided a condition under which the componentwise and the vector stochastic integration with respect to a given continuous ℝ d -local martingale are the same notions. Here, we improve this result, taking more general volatility matrices as the previous ones which were non-singular.