Multivariate Stable Futures Prices

Abstract

This paper introduces new techniques for modeling financial data under the assumption that the data belong to the domain of attraction of a multivariate stable Pareto law. We provide tail estimators for the index of stability parameter "a" and the corresponding spectral measure. These estimators are then applied to test the associtation of the individual components and to compute estimates of portfolio risk and the covariation of commodities. A practical example is given using DM-dollar and JY-dollar exchange rates data. Copyright 1995 Blackwell Publishers.

On Tail Index Estimation for Dependent, Heterogeneous Data

“…LEMMA B.5 Let fX n;t g be a mean-zero stochastic array with n := jj P n t=1 X n;t jj 2 > 0 uniformly in n. De…ne Z n;i := P ikn t=(i 1)kn+ln+1 X n;t andF n;i := (fE n; : ik n g) and let the sequences fl n ; k n ; r n g be as in (13). Then P n t=1 X n;t = n =) N (0; 1) under the following conditions:…”

“…The so-called Hill-estimator has been used pervasively in the applied …nance, economics, statistics and telecommunications literatures. Consider Akgiray and Booth (1988), Cheng and Rachev (1995), Quintos et al (2001), Resnick and Rootzén (2000), Chan et al (2007), and Hill (2008a), to name a few. For alternative estimation techniques consult Pickands (1975), Smith (1987), Rootzén et al (1990), Smith and Weissman (1994), Drees et al (2004), Csörgö and Viharos (1995), Beirlant et al (2005) and Iglesias and Linton (2008).…”

Modeling Financial Data with Stable Distributions

“…Because of numerical problems with the 1-parameterization, we will always use the 0-parameterization in estimation. If desired, the parameter δ 1 can be estimated by using (4). There are several methods available for this basic estimation problem: a quan-tile method of McCulloch (1986), a fractional moment method of Ma and Nikias (1995), sample characteristic function (SCF) method of Kogon-Williams (1998) based on ideas of Koutrouvelis, and maximum likelihood (ML) estimation of DuMouchel (1971) and Nolan (2001).…”

Indirect estimation of elliptical stable distributions

“…The choice of the Student's t motivated by Demarta and McNeil (2005), Frahm, Junker and Szimayer (2005) and Frahm (2006) whom suggest that this distribution as a reference model for elliptically contoured distributions. 8 In actual facts, it is not strictly necessary to work with standard maximum likelihood, and more efficient algorithms such as the EM (Meng and van Dijk, 1995) can be employed. However, as we will point out later, we will follow a gradient based approach according to which the estimation of the auxiliary model is required only once and it is instead crucial the availability of an analytic version of the gradient; for this reason we believe that ML is more appropriate in this context.…”

“…Though theoretically possible, it is not clear how to make these methods operational for the estimation of high dimension processes. The second approach is based on one dimensional projections of the multivariate process (Nolan, Panorska and McCulloch, 2001;Rachev and Xin, 1993;and Cheng and Rachev, 1995). The only paper, to our knowledge, that estimates all the parameters of the multivariate stable distribution is Nolan (2005) who extends above mentioned results based on projections to the location and tail index.…”