Multivariate Subordination Research Articles

Necessity of weak subordination for some strongly subordinated Lévy processes

AbstractConsider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

Correlating L\xe9vy processes with self-decomposability: applications to energy markets

Based on the concept of self-decomposability, we extend some recent multidimensional Lévy models built using multivariate subordination. Our aim is to construct multivariate Lévy processes that can model the propagation of the systematic risk in dependent markets with some stochastic delay instead of affecting all the markets at the same time. To this end, we extend some known approaches keeping their mathematical tractability, study the properties of the new processes, derive closed-form expressions for their characteristic functions and detail how Monte Carlo schemes can be implemented. We illustrate the applicability of our approach in the context of gas, power and emission markets focusing on the calibration and on the pricing of spread options written on different underlying commodities.

Asymptotic Behavior of Eigenvalues of Variance-Covariance Matrix of a High-Dimensional Heavy-Tailed Lévy Process

In this paper, we study the limiting behavior of eigenvalues of the variance-covariance matrix of a random sample from a multivariate subordinator heavy-tailed Levy process, and use large deviations of a heavy-tailed stochastic process to derive the limit distributions of its components. We confine our study to multivariate Levy processes with regularly varying random components and possibly different indices of regularity. Assuming that the product of increments of the marginal components are also regularly varying random variables, we show that the product of two dependent regularly varying Log-Gamma random variables with integer-valued shape parameters is also a regularly varying random variable with index depending on the correlation between the original variables. This result enables us to derive the limiting tail behavior of sample variance-covariance matrix from a multivariate Levy process having Log-Gamma components with integer-valued shape parameters and different indices of regularity.

Financial modelling applying multivariate Lévy processes: New insights into estimation and simulation

In general, daily or intra-day stock returns are fat-tailed and heavily skewed. Lévy processes fulfil these modelling requirements and produce marginal distributions with finite variances. An extensive body of literature looks into the fittings and applications of single processes. In contrast, our analysis of multivariate Lévy models finds applications in pricing multivariate options or in portfolio and risk management. We use the technique of multivariate subordination and conduct a large simulation study on the fitting of the αρGH, αρNIG, and αρVG models in order to identify the best fitting method for multivariate Lévy processes, as well as the best multivariate model overall. Our findings confirm previous results in the literature, namely that the MLE is the best estimation approach in a two-step fitting procedure and the αρGH model is the best multivariate model. It reveals that also the χ2 method is appropriate.

Multivariate asset‐pricing model based on subordinated stable processes

AbstractIn this paper, we consider a multidimensional time‐changed stochastic process in the context of asset‐pricing modeling. The proposed model is constructed from stable processes, and its construction is based on two popular concepts: multivariate subordination and Lévy copulas. From a theoretical point of view, our main result is Theorem 1, which yields a simulation method from the considered class of processes. Our empirical study shows that the model represents the correlation between asset returns quite well. Moreover, we provide some evidence that this model is more appropriate for describing stock prices than classical time‐changed Brownian motion, at least if the cumulative amount of transactions is used for a stochastic time change.

A note on the multivariate generalized asymmetric Laplace motion

In this note, we use multivariate subordination to introduce a multivariate extension of the generalized asymmetric Laplace motion. The class introduced provides a unified framework for several multivariate extensions of the popular variance gamma process. We also show that the associated time one distribution extends the multivariate generalized asymmetric Laplace distributions proposed in the statistical literature.

Weak subordination of multivariate Lévy processes and variance generalised gamma convolutions

Subordinating a multivariate Lévy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new Lévy process, provided the subordinator and the subordinate are independent processes. The variance-gamma model in finance was generated accordingly from a Brownian motion and a gamma process. Alternatively, multivariate subordination can be used to create Lévy processes, but this requires the subordinate to have independent components. In this paper, we show that there exists another operation acting on pairs $(T,X)$ of Lévy processes which creates a Lévy process $X\odot T$. Here, $T$ is a subordinator, but $X$ is an arbitrary Lévy process with possibly dependent components. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. We illustrate our methods giving a weak formulation of the variance-$\boldsymbol{\alpha}$-gamma process that exhibits a wider range of dependence than using traditional subordination. Also, the variance generalised gamma convolution class of Lévy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.

A recursive pricing method for autocallables under multivariate subordination

In this paper we develop a new class of models for pricing autocallables based on multivariate subordinate Orstein Uhlenbeck (OU) processes. Starting from d independent OU processes and an independent d-dimensional Levy subordinator, we construct a new process by time changing each of the OU processes with a coordinate of the Levy subordinator. The prices of underlying assets are then modeled as an exponential function of the subordinate processes. The new models introduce state-dependent jumps in the asset prices and the dependence among jumps is governed by the Levy measure of the d-dimensional subordinator. By employing the eigenfunction expansion technique, we are able to derive the analytical formulas for the worst-of autocallable prices. We also numerically implement a specific model and test its sensitivity to some of the key parameters of the model.

A note on Marked Point Processes and multivariate subordination

The aim of this paper is to state a correspondence between marked Poisson processes and multivariate subordinated Lévy processes. We prove that, under suitable conditions, marked Poisson processes are in law subordinated Brownian motions and we provide their Lévy triplet and characteristic function. We introduce the class of multivariate Gaussian marked Poisson processes and prove that–in law–they are multivariate subordinated Brownian motions.

Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing

We unify and extend a number of approaches related to constructing multivariate Madan–Seneta Variance-Gamma models for option pricing. Complementing Grigelionis’ (2007) class, an overarching model is derived by subordinating multivariate Brownian motion to a subordinator from Thorin’s (1977) [58, 59] class of generalised Gamma convolutions. Multivariate classes developed by Pérez-Abreu and Stelzer (2014), Semeraro (2008) and Guillaume (2013) are submodels. The classes are shown to be invariant under Esscher transforms, and quite explicit expressions for canonical measures are obtained, which permit applications such as option pricing using PIDEs or tree based methodologies. We illustrate with best-of and worst-of European and American options on two assets.

Integrability of multivariate subordinated Lévy processes in Hilbert space

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].

Constructing Markov Processes with Dependent Jumps by Multivariate Subordination: Applications to Multi-Name Credit-Equity Modeling

This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes Xi with killing and an independent d-dimensional time change T, we construct a new process by time changing each of the Markov processes Xi with a coordinate Ti. When T is a d-dimensional Levy subordinator, the time changed process Yi:=Xi(Ti(t)) is a time-homogeneous Markov process with state-dependent jumps and killing in the product of the state spaces of Xi. The dependence among jumps of its components is governed by the d-dimensional Levy measure of the subordinator. When T is a d-dimensional additive subordinator, Y is a time-inhomogeneous Markov process. When Ti= ∫0tVsi ds with Vi forming a multi-variate Markov process, (Yi,Vi) is a Markov process, where each Vi plays a role of stochastic volatility of Yi. This construction provides a rich modeling architecture for building multivariate models in finance with time-dependent and state-dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multi-name unified credit-equity models and correlated commodity models.

MULTIVARIATE SUBORDINATION OF MARKOV PROCESSES WITH FINANCIAL APPLICATIONS

This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes with killing and an independent d‐dimensional time change , we construct a new process by time, changing each of the Markov processes with a coordinate . When is a d‐dimensional Lévy subordinator, the time changed process is a time‐homogeneous Markov process with state‐dependent jumps and killing in the product of the state spaces of . The dependence among jumps of its components is governed by the d‐dimensional Lévy measure of the subordinator. When is a d‐dimensional additive subordinator, Y is a time‐inhomogeneous Markov process. When with forming a multivariate Markov process, is a Markov process, where each plays a role of stochastic volatility of . This construction provides a rich modeling architecture for building multivariate models in finance with time‐ and state‐dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multiname unified credit‐equity models and correlated commodity models.

The αVG model for multivariate asset pricing: calibration and extension

Luciano and Semeraro proposed a class of multivariate asset pricing models where the asset log-returns are modeled by a multivariate Brownian motion time-changed by a multivariate subordinator which consists of the weighted sum of a common and an idiosyncratic subordinator. In the original setting, Luciano and Semeraro imposed some constraints on the subordinator parameters such that the multivariate subordinator is of the same subordinator sub-class as its components, leading to asset log-returns of a particular Levy type. This restriction leads to marginal characteristic functions which are independent on the common subordinator setting. In this paper, we propose to extend the original model by relaxing the constraints on the subordinator parameters, leading to marginal characteristic functions which become a function of the whole parameter set. Under this generalized version, the volatility of the log-returns depends on both the common and idiosyncratic subordinator settings, and not only on the idiosyncratic one, which makes the generalized model more in line with the empirical evidence of the presence of both an idiosyncratic and a common component in the business clock. For the numerical study, we compare the calibration fit of both univariate option surfaces and market implied correlations for a period extending from the 2nd of June 2008 until the 30th of October 2009 under the two model settings and assess the calibration risk arising from different calibration procedures by pricing traditional multivariate exotic options. In particular we show that the decoupling calibration procedure fails to accurately replicate the market dependence structure under the original model for highly correlated asset returns and we propose an alternative methodology which rests on a joint calibration of the univariate and the dependence structure and which leads to an accurate fit of the market reality under both the generalized and original models.

A Multivariate Pure-Jump Model with Multi-Factorial Dependence Structure

In this work we propose a new approach to build multivariate pure jump processes. We introduce linear and nonlinear dependence, without restrictions on marginal properties, by imposing a multi-factorial structure separately on both positive and negative jumps. Such a new approach provides higher flexibility in calibrating nonlinear dependence than in other comparable Levy models in the literature. Using the notion of multivariate subordinator, this modeling approach can be applied to the class of univariate Levy processes which can be written as the difference of two subordinators. A common example in the financial literature is the variance gamma process, which we extend to the multivariate (multi-factorial) case. The model is tractable and a straightforward multivariate simulation procedure is available. An empirical analysis documents an accurate multivariate fit of stock index returns in terms of both linear and nonlinear dependence. An example of multi-asset option pricing emphasizes the importance of the proposed multivariate approach.

A MULTIVARIATE PURE-JUMP MODEL WITH MULTI-FACTORIAL DEPENDENCE STRUCTURE

In this work we propose a new approach to build multivariate pure jump processes. We introduce linear and nonlinear dependence, without restrictions on marginal properties, by imposing a multi-factorial structure separately on both positive and negative jumps. Such a new approach provides higher flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. Using the notion of multivariate subordinator, this modeling approach can be applied to the class of univariate Lévy processes which can be written as the difference of two subordinators. A common example in the financial literature is the variance gamma process, which we extend to the multivariate (multi-factorial) case. The model is tractable and a straightforward multivariate simulation procedure is available. An empirical analysis documents an accurate multivariate fit of stock index returns in terms of both linear and nonlinear dependence. An example of multi-asset option pricing emphasizes the importance of the proposed multivariate approach.

Multivariate time changes for Lévy asset models: Characterization and calibration

We build a theoretical framework for multivariate subordination of Brownian motions, with a common and an idiosyncratic component. This follows economic intuition and introduces generalizations of some well known multivariate Lévy processes for financial applications: the compound Poisson, NIG, Variance Gamma and CGMY. In most cases we obtain the characteristic function in closed form. The extension is first kept parsimonious, by adding one parameter only. The empirical fit of (linear) dependence is then increased, by allowing for dependent Brownian motions.

A MULTIVARIATE VARIANCE GAMMA MODEL FOR FINANCIAL APPLICATIONS

In this paper we subordinate a multivariate Brownian motion with independent components by a multivariate gamma subordinator. The resulting process is a generalization of the bivariate variance gamma process proposed by Madan and Seneta [7], mentioned in Cont and Tankov [4] and calibrated in Luciano and Schoutens [5] as a price process. Our main contribution here is to introduce a multivariate subordinator with gamma margins. We investigate the process, determine its Lévy triplet and analyze its dependence structure. At the end we propose an exponential Lévy price model.

Extending Time-Changed LLvy Asset Models Through Multivariate Subordinators

The traditional multivariate Levy 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 Levy 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. First we characterize the subordinator, then the time changed processes via their Levy measure and characteristic exponent. Finally we 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 of major stock indices.

Multivariate subordination, self-decomposability and stability

Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.