A Multivariate Time-Changed Lévy Model for Financial Applications

The purpose of this paper is to define a bivariate L´evy process by subordination of a Brownian motion. In particular we investigate a generalization of the bivariate Variance Gamma process proposed in Luciano and Schoutens [8] as a price process. Our main contribution here is to introduce a bivariate subordinator with correlated Gamma margins. We characterize the process and study its dependence structure. At the end wealso propose an exponential Lévy price model based on our process.Levy processes, multivariate subordinators, dependence, multivariate asset modelling.

Refinement Derivatives and Values of Games

A definition of set-wise differentiability for set functions is given through refining the partitions of sets. Such a construction is closely related to the one proposed by Rosenmuller (1977) as well as that studied by Epstein (1999) and Epstein and Marinacci (2001). We present several classes of TU games which are differentiable and study differentiation rules. The last part of the paper applies refinement derivatives to the calculation of value of games. Following Hart and Mas-Colell (1989), we define a value operator through the derivative of the potential of the game. We show that this operator is a truly value when restricted to some appropriate spaces of games. We present two alternative spaces where this occurs: the spaces pM( ) and POT2. The latter space is closely related to Myerson's balanced contribution axiom.TU games; large games; non-additive set functions; value; derivatives

A Generalized Normal Mean Variance Mixture for Return Processes in Finance

Time-changed Brownian motions are extensively applied as mathematical models for asset returns in Finance. Time change is interpreted as a switch to trade-related business time, different from calendar time. Time-changed Brownian motions can be generated by infinite divisible normal mixtures. The standard multivariate normal mean variance mixtures assume a common mixing variable. This corresponds to a multidimensional return process with a unique change of time for all assets under exam. The economic counterpart is uniqueness of trade or business time, which is not in line with empirical evidence. In this paper we propose a new multivariate definition of normal mean-variance mixtures with a flexible dependence structure, based on the economic intuition of both a common and an idiosyncratic component of business time. We analyze both the distribution and the related process. We use the above construction to introduce a multivariate generalized hyperbolic process with generalized hyperbolic margins. We conclude with a stock market example to show the ease of calibration of the model.multivariate normal mean variance mixtures, multivariate generalized hyperbolic distributions, Levy processes, multivariate subordinators

Multivariate Variance Gamma and Gaussian dependence: a study with copulas

This paper explores the dynamic dependence properties of a Levy process, the Variance Gamma, which has non Gaussian marginal features and non Gaussian dependence. In a static context, such a non Gaussian dependence should be represented via copulas. Copulas, however, are not able to capture the dynamics of dependence. By computing the distance between the Gaussian copula and the actual one, we show that even a non Gaussian process, such as the Variance Gamma, can "converge" to linear dependence over time. Empirical versions of different dependence measures confirm the result.

Extending Time-Changed Lévy Asset Models Through Multivariate Subordinators

The traditional multivariate Lévy process constructed by subordinating a Brownian motion through a univariate subordinator presents a number of drawbacks, including the lack of independence and a limited range of dependence. In order to face these, we investigate multivariate subordination, with a common and an idiosyncratic component. We introduce generalizations of some well known univariate Lévy processes for financial applications: the multivariate compound Poisson, NIG, Variance Gamma and CGMY. In all these cases the extension is parsimonious, in that one additional parameter only is needed. We characterize first the subordinator, then the time changed processes via their Lévy measure and characteristic exponent. We further study the subordinator association, as well as the subordinated processes linear and non linear dependence. We show that the processes generated with the proposed time change can include independence and that they span the whole range of linear dependence. We provide some examples of simulated trajectories,scatter plots and both linear and non linear dependence measures. The input data for these simulations are calibrated values for major stock indices.Lévy processes, multivariate subordinators, dependence (association, correlation), multivariate asset modelling

Single and joint default in a structural model with purely discontinuous assets

Structural models of credit risk are known to present both vanishing spreads at very short maturities and a poor spread fit over longer maturities. The former shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering discontinuous assets. In this paper we resort to a pure jump process of the Variance-Gamma type. First we calibrate the corresponding Merton type structural model to single-name data for the DJ CDX NA IG and CDX NA HY components. By so doing, we show that it circumvents also the diffusive structural models difficulties over longer horizons. In particular, it corrects for underprediction of low risk spreads and overprediction of high risk ones. Then we extend the model to joint default, resorting to a recent formulation of the VG multivariate model and without superimposing a copula choice. We fit default correlation for a sample of CDX NA names, using equity correlation. The main advantage of our joint model with respect to the existing non diffusive ones is that it allows calibration without the equicorrelation assumption, but still in a parsimonious way. As an example of the default assessments which the calibrated model can provide, we price a FtD swap.credit risk, structural models, Lévy asset prices, default probability, joint default.

Computational and Analytical Bounds for Multivariate Bernoulli Distributions

Building on a new but simple method to characterize multivariate Bernoulli variables with given means, we investigate their dependence structure. We evaluate on some computational examples whether the assumption of exchangeability is binding. This is useful in applications where exchangeability is a standard assumption, such as credit risk