arXiv subjects

Lancelot F. James

Publications and source records attributed to Lancelot F. James.

43 records · Page 3Linked to original sources

arXiv2005

Analysis of a Class of Likelihood Based Continuous Time Stochastic Volatility Models including Ornstein-Uhlenbeck Models in Financial Economics

In a series of recent papers Barndorff-Nielsen and Shephard introduce an attractive class of continuous time stochastic volatility models for financial assets where the volatility processes are functions of positive Ornstein-Uhlenbeck(OU) processes. This models are known to be substantially more flexible than Gaussian based models. One current problem of this approach is the unavailability of a tractable exact analysis of likelihood based stochastic volatility models for the returns of log prices of stocks. With this point in mind, the likelihood models of Barndorff-Nielsen and Shephard are viewed as members of a much larger class of models. That is likelihoods based on n conditionally independent Normal random variables whose mean and variance are representable as linear functionals of a common unobserved Poisson random measure. The analysis of these models is facilitated by applying the methods in James (2005, 2002), in particular an Esscher type transform of Poisson random measures; in conjunction with a special case of the Weber-Sonine formula. It is shown that the marginal likelihood may be expressed in terms of a multidimensional Fourier-cosine transform. This yields tractable forms of the likelihood and also allows a full Bayesian posterior analysis of the integrated volatility process. A general formula for the posterior density of the log price given the observed data is derived, which could potentially have applications to option pricing. We extend the models to include leverage effects in section 5. It is shown that inference does not necessarily require simulation of random measures. Rather, classical numerical integration can be used in the most general cases.

math.ST↗

arXiv2005

Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and Beta-Gamma processes

Suppose that P_θ(g) is a linear functional of a Dirichlet process with shape θH, where θ>0 is the total mass and H is a fixed probability measure. This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and a posterior calculus for Gamma processes to ascertain properties of linear functionals of Dirichlet processes. In particular, in conjunction with a Gamma identity, we show easily that a generalized Cauchy-Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, Beta-Gamma processes. This represents a generalization of an identity due to Cifarelli and Regazzini, which is also known as the Markov-Krein identity for mean functionals of Dirichlet processes. These results also provide new explanations and interpretations of results in the literature. The identities are analogues to quite useful identities for Beta and Gamma random variables. We give a result which can be used to ascertain specifications on H such that the Dirichlet functional is Beta distributed. This avoids the need for an inversion formula for these cases and points to the special nature of the Dirichlet process, and indeed the functional Beta-Gamma calculus developed in this paper.

math.ST↗

arXiv2005

Baysian inference via classes of normalized random measures

One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. Here we exploit theoretical properties of Poisson random measures in order to provide a comprehensive Bayesian analysis of random probabilities which are obtained by an appropriate normalization. Specifically we achieve explicit and tractable forms of the posterior and the marginal distributions, including an explicit and easily used description of generalizations of the important Blackwell-MacQueen Pólya urn distribution. Such simplifications are achieved by the use of a latent variable which admits quite interesting interpretations which allow to gain a better understanding of the behaviour of these random probability measures. It is noteworthy that these models are generalizations of models considered by Kingman (1975) in a non-Bayesian context. Such models are known to play a significant role in a variety of applications including genetics, physics, and work involving random mappings and assemblies. Hence our analysis is of utility in those contexts as well. We also show how our results may be applied to Bayesian mixture models and describe computational schemes which are generalizations of known efficient methods for the case of the Dirichlet process. We illustrate new examples of processes which can play the role of priors for Bayesian nonparametric inference and finally point out some interesting connections with the theory of generalized gamma convolutions initiated by Thorin and further developed by Bondesson.

math.ST↗

arXiv2005

A note on exact likelihoods of the Carr-Wu models for leverage effects and volatility in financial economics

Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most stochastic processes used in traditional option pricing models can be cast as special cases of time-changed Lévy processes. In particular these are models which can be tailored to exhibit correlated jumps in both the log price of assets and the instantaneous volatility. Naturally similar to a recent work of Barndorff-Nielsen and Shephard (2001a, b), such models may be used in a likelihood based framework. These likelihoods are based on the unobserved integrated volatility, rather than the instantaneous volatility. James (2005) establishes general results for the likelihood and estimation of a large class of such models which include possible leverage effects. In this note we show that exact expressions for likelihood models based on generalizations of Carr and Wu (2005) and Huang and Wu (2005), follow essentially from the arguments in Theorem 5.1 in James (2005) with some slight modification. This serves to formally verify a claim made by James (2005).

math.ST↗

arXiv2005

A Class of Generalized Hyperbolic Continuous Time Integrated Stochastic Volatility Likelihood Models

This paper discusses and analyzes a class of likelihood models which are based on two distributional innovations in financial models for stock returns. That is, the notion that the marginal distribution of aggregate returns of log-stock prices are well approximated by generalized hyperbolic distributions, and that volatility clustering can be handled by specifying the integrated volatility as a random process such as that proposed in a recent series of papers by Barndorff-Nielsen and Shephard (BNS). The BNS models produce likelihoods for aggregate returns which can be viewed as a subclass of latent regression models where one has n conditionally independent Normal random variables whose mean and variance are representable as linear functionals of a common unobserved Poisson random measure. James (2005b) recently obtains an exact analysis for such models yielding expressions of the likelihood in terms of quite tractable Fourier-Cosine integrals. Here, our idea is to analyze a class of likelihoods, which can be used for similar purposes, but where the latent regression models are based on n conditionally independent models with distributions belonging to a subclass of the generalized hyperbolic distributions and whose corresponding parameters are representable as linear functionals of a common unobserved Poisson random measure. Our models are perhaps most closely related to the Normal inverse Gaussian/GARCH/A-PARCH models of Brandorff-Nielsen (1997) and Jensen and Lunde (2001), where in our case the GARCH component is replaced by quantities such as INT-OU processes. It is seen that, importantly, such likelihood models exhibit quite different features structurally. One nice feature of the model is that it allows for more flexibility in terms of modelling of external regression parameters.

math.ST↗

arXiv2003

Functionals of Dirichlet processes, the Markov Krein Identity and Beta-Gamma processes

This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and less known results for the gamma process, to address the formidable problem of assessing the distribution of linear functionals of Dirichlet processes. In particular, in conjunction with a gamma identity, we show easily that a generalized Cauchy-Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, beta-gamma processes. This represents a generalization of the Markov-Krein identity for mean functionals of Dirichlet processes. A prior to posterior analysis of beta-gamma processes is given that not only leads to an easy derivation of the Markov-Krein identity, but additionally yields new distributional identities for gamma and beta-gamma processes. These results give new explanations and intepretations of exisiting results in the literature. This is punctuated by establishing a simple distributional relationship between beta-gamma and Dirichlet processes.

math.PR↗

arXiv2002

Poisson Process Partition Calculus with applications to Exchangeable models and Bayesian Nonparametrics

This article discusses the usage of a partiton based Fubini calculus for Poisson processes. The approach is an amplification of Bayesian techniques developed in Lo and Weng for gamma/Dirichlet processes. Applications to models are considered which all fall within an inhomogeneous spatial extension of the size biased framework used in Perman, Pitman and Yor. Among some of the results; an explicit partition based calculus is then developed for such models, which also includes a series of important exponential change of measure formula. These results are applied to obtain results for Levy-Cox models, identities related to the two-parameter Poisson-Dirichlet process and other processes, generalisations of the Markov-Krein correspondence, calculus for extended Neutral to the Right processes, among other things.

math.PR↗