Giuliana Regoli

Correction of incoherent conditional probability assessments

In this paper we deep in the formal properties of an already stated discrepancy measure between a conditional assessment and the class of unconditional probability distributions compatible with the assessment domain.

A class of bivariate exponential distributions

We introduce a class of absolutely continuous bivariate exponential distributions, generated from quadratic forms of standard multivariate normal variates. This class is quite flexible and tractable, since it is regulated by two parameters only, derived from the matrices of the quadratic forms: the correlation and the correlation of the squares of marginal components. A simple representation of the whole class is given in terms of 4-dimensional matrices. Integral forms allow evaluating the distribution function and the density function in most of the cases. The class is introduced as a subclass of bivariate distributions with chi-square marginals; bounds for the dimension of the generating normal variable are underlined in the general case. Finally, we sketch the extension to the multivariate case.

A general setting for symmetric distributions and their relationship to general distributions

A standard method of obtaining non-symmetrical distributions is that of modulating symmetrical distributions by multiplying the densities by a perturbation factor. This has been considered mainly for central symmetry of a Euclidean space in the origin. This paper enlarges the concept of modulation to the general setting of symmetry under the action of a compact topological group on the sample space. The main structural result relates the density of an arbitrary distribution to the density of the corresponding symmetrised distribution. Some general methods for constructing modulating functions are considered. The effect that transformations of the sample space have on symmetry of distributions is investigated. The results are illustrated by general examples, many of them in the setting of directional statistics.

Option Pricing in Incomplete Markets Based on Partial Information

In this paper we describe a new approach for the valuation problem in incomplete markets with m ≥ 1 stocks which can be used when the available information about the uncertainty model is only a partial conditional probability assessment p. We select a risk neutral probability minimizing a discrepancy measure between p and the convex set of all possible risk neutral probabilities.

Risk Neutral Valuations Based on Partial Probabilistic Information

In a viable single-period model with one stock and k ≥ 2 scenarios the completeness of the market is equivalent to the uniqueness of the risk neutral probability; this equivalence allows to price every derivative security with a unique fair price. When the market is incomplete, the set of all possible risk neutral probabilities is not a singleton and for every non attainable derivative security we have a bid-ask interval of possible prices. In literature, different methods have been proposed in order to select a unique risk neutral probability starting with the real world probability p. Contrary to the complete case, in all these models \(\textbf{p}\) is really used for the option pricing and its elicitation is a crucial point for every criterion used to select a risk neutral probability. We propose a method for the valuation problem in incomplete markets which can be used when p is a partial conditional probability assessment as well as when we have different expert opinions expressed through conditional probability assessments. In fact, it is not always possible to elicit a probability distribution p over all the possible states of the world: the information that we have could be partial, conditional or even not coherent. Therefore we will select a risk neutral probability by minimizing a discrepancy measure introduced in [2] and analized in [3] between p and the set of all possible risk neutral probability, where p can be a partial conditional probability assessments or it can be given by the fusion of different expert opinions.