Academic literature on the topic 'Multivariate Lévy models'

Abstract In this paper we consider the factor analysis for Lévy-driven multivariate price models with stochastic volatility. Our main aim is to provide conditions on the volatility process under which we can possibly reduce the dimension of the driving Lévy motion. We find that these conditions depend on a particular form of the multivariate Lévy process. In some settings we concentrate on nondegenerate symmetric α-stable Lévy motions.

AbstractThe class of spectrally positive Lévy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (e.g. between different lines of business) can be modelled with Lévy copulas. This approach is a parsimonious, efficient and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Lévy copulas seems to have primarily focussed on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Lévy copula). In reality, losses are not always associated in an identical way, and models allowing for non-exchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Lévy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Lévy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {hn, 2hn,…, nhn}, where hn ↓ 0 and nhn → ∞ as n → ∞, or at a constant time grid where hn = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

Multivariate Lévy-driven mixed moving average (MMA) processes of the type Xt = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

Multivariate Lévy-driven mixed moving average (MMA) processes of the type X t = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {h n , 2h n ,…, nh n }, where h n ↓ 0 and nh n → ∞ as n → ∞, or at a constant time grid where h n = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

We propose a consistent and computationally efficient 2-step methodology for the estimation of multidimensional non-Gaussian asset models built using Lévy processes. The proposed framework allows for dependence between assets and different tail behaviors and jump structures for each asset. Our procedure can be applied to portfolios with a large number of assets because it is immune to estimation dimensionality problems. Simulations show good finite sample properties and significant efficiency gains. This method is especially relevant for risk management purposes such as, for example, the computation of portfolio Value at Risk and intra-horizon Value at Risk, as we show in detail in an empirical illustration.

The class of marked Poisson processes and its connection with subordinated Lévy processes allow us to propose a new interpretation of multidimensional information flows and their relation to market movements. The new approach provides a unified framework for multivariate asset return models in a Lévy economy. In fact, we are able to recover several processes commonly used to model asset returns as subcases. We consider a first application example using the normal inverse Gaussian specification.

This thesis is composed of three chapters that form two parts. The first part is composed of two chapters and studies problems related to the exotic option market. In the first chapter we are interested in a numerical problem. More precisely we derive closed-form approximations for the price of some exotic options in the Black and Scholes framework. The second chapter discusses the construction of multivariate Lévy processes with and without stochastic volatility. The second part is composed of one chapter. It deals with a completely different issue. There we will study the problem of individual and temporal aggregation in panel data models.
Doctorat en sciences économiques, Orientation économie
info:eu-repo/semantics/nonPublished

Multidimensional asset models based on Lévy processes have been introduced to meet the necessity of capturing market shocks using more refined distribution assumptions compared to the standard Gaussian framework. In particular, along with accurately modeling marginal distributions of asset returns, capturing the dependence structure among them is of paramount importance, for example, to correctly price derivatives written on more than one underlying asset. Most of the literature on multivariate Lévy models focuses in fact on pricing multi-asset products, which is also the case of the model introduced in Ballotta and Bonfiglioli (2014). Believing that risk and portfolio management applications may benefit from a better description of the joint distribution of the returns as well, we choose to adopt Ballotta and Bonfiglioli (2014) model for asset allocation purposes and we empirically test its performances. We choose this model since, besides its flexibility and the ability to properly capture the dependence among assets, it is simple, relatively parsimonious and it has an immediate and intuitive interpretation, retaining a high degree of mathematical tractability. In particular we test two specifications of the general model, assuming respectively a pure jump process, more precisely the normal inverse Gaussian process, or a jump-diffusion process, precisely Merton’s jump-diffusion process, for all the components involved in the model construction. To estimate the model we propose a simple and easy-to-implement three-step procedure, which we assess via simulations, comparing the results with those obtained through a more computationally intensive one-step maximum likelihood estimation. We empirically test portfolio construction based on multivariate Lévy models assuming a standard utility maximization framework; for the exponential utility function we get a closed form expression for the expected utility, while for other utility functions (we choose to test the power one) we resort to numerical approximations. Among the benchmark strategies, we consider in our study what we call a ‘non-parametric optimization approach’, based on Gaussian kernel estimation of the portfolio return distribution, which to our knowledge has never been used. A different approach to allocation decisions aims at minimizing portfolio riskiness requiring a minimum expected return. Following Rockafellar and Uryasev (2000), we describe how to solve this optimization problem in our multivariate Lévy framework, when risk is measured by CVaR. Moreover we present formulas and methods to compute, as efficiently as possible, some downside risk measures for portfolios made of assets following the multivariate Lévy model by Ballotta and Bonfiglioli (2014). More precisely, we consider traditional risk measures (VaR and CVaR), the corresponding marginal measures, which evaluate their sensibility to portfolio weights alterations, and intra-horizon risk measures, which take into account the magnitude of losses that can incur before the end of the investment horizon. Formulas for CVaR in monetary terms and marginal measures, together with our approach to evaluate intra-horizon risk, are among the original contributions of this work.