Gianluca Cassese

Asset Pricing with No Exogenous Probability Measure

In this paper, we propose a model of financial markets in which agents have limited ability to trade and no probability is given from the outset. In the absence of arbitrage opportunities, assets are priced according to a probability measure that lacks countable additivity. Despite finite additivity, we obtain an explicit representation of the expected value with respect to the pricing measure, based on some new results on finitely additive measures. From this representation we derive an exact decomposition of the risk premium as the sum of the correlation of returns with the market price of risk and an additional term, the purely finitely additive premium, related to the jumps of the return process. We also discuss the implications of the absence of free lunches.

Convergence in measure under finite additivity

We investigate the possibility of replacing the topology of convergence in probability with convergence in L 1 , upon a change of the underlying measure under nite additivity. We es- tablish conditions for the continuity of linear operators and convergence of measurable sequences, including a nitely additive analogue of Koml os Lemma. We also prove several topological impli- cations. Eventually, a characterization of continuous linear functionals on the space of measurable functions is obtained.

Asset pricing in an imperfect world

In a model with no given probability measure, we consider asset pricing in the presence of frictions and other imperfections and characterize the property of coherent pricing, a notion related to (but much weaker than) the no-arbitrage property. We show that prices are coherent if and only if the set of pricing measures is non-empty, i.e., if pricing by expectation is possible. We then obtain a decomposition of coherent prices highlighting the role of bubbles. Eventually, we show that under very weak conditions, the coherent pricing of options implies a very clear representation which permits, as in Breeden and Litzenberger (J Bus 51:621–651, 1978), to extract the implied probability.

Nonparametric Estimates of Option Prices Using Superhedging

We propose a new nonparametric technique to estimate the CALL function based on the superhedging principle. Our approach does not require absence of arbitrage and easily accommodates bid/ask spreads and other market imperfections. We prove some optimal statistical properties of our estimates. As an application we first test the methodology on a simulated sample of option prices and then on the S&P 500 index options.

The Representation of Conglomerative Functionals

We prove results concerning the representation of certain linear functionals based on the notion of conglomerability, originally introduced by Dubins and de Finetti. We show that this property has some applications in probability and in statistics.

A Version of Komlós Theorem for Additive Set Functions

Abstract We provide a version of the celebrated theorem of Komlos in which, rather than random quantities, a sequence of finitely additive measures is considered. We obtain a form of the subsequence principle and some applications.

The Theorem of Halmos and Savage under Finite Additivity

Abstract We prove an extension to the finitely additive setting of the theorem of Halmos and Savage. From this we deduce extensions of classical results of Drewnowski and of Yan as well as a new characterization of weak compactness in the space of finitely additive set functions.

Conglomerability and Representations

We prove results concerning the representation of a given distribution by means of a given random quantity. The existence of a solution to this problem is related to the notion of conglomerability, originally introduced by Dubins. We show that this property has many interesting applications in probability as well as in analysis. Based on it we prove versions of the extremal representation theorem of Choquet and of Skhorohod theorem.

Quasi-martingales with a linearly ordered index set

We consider quasi-martingales indexed by a linearly order set. We show that such processes are isomorphic to a given class of (finitely additive) measures. From this result we easily derive the classical theorem of Stricker as well as the decompositions of Riesz, Rao and the supermartingale decomposition of Doob and Meyer.