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

Lancelot F. James|ArXiv.org|Mar 3, 2005

Stochastic processes and financial applications25 references5 citations

TL;DR

This paper develops a tractable likelihood-based framework for continuous-time stochastic volatility models, including Ornstein-Uhlenbeck processes, by representing integrated volatility as a linear functional of a Poisson random measure. Using an Esscher transform and a special case of the Weber-Sonine formula, it derives explicit marginal likelihoods via multidimensional Fourier-cosine transforms, enabling full Bayesian inference without simulation and allowing exact numerical integration for models with leverage effects.

ABSTRACT

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.

Motivation & Objective

• To address the lack of tractable likelihood analysis for continuous-time stochastic volatility models based on integrated volatility.
• To extend Barndorff-Nielsen and Shephard's OU-based models to a broader class of likelihood-based models using Poisson random measures.
• To enable full Bayesian posterior analysis of the integrated volatility process without requiring simulation of random measures.
• To derive explicit expressions for likelihoods incorporating leverage effects in price-volatility dynamics.
• To show that classical numerical integration suffices for inference, even in general cases, avoiding reliance on Lévy density knowledge.

Proposed method

• Represent the integrated volatility process τ(t) as a linear functional of a common unobserved Poisson random measure.
• Apply an Esscher-type transform to the Poisson random measure to facilitate likelihood computation.
• Use a special case of the Weber-Sonine formula involving Bessel functions and cosine transforms to express the marginal likelihood.
• Derive the posterior predictive density of log prices using the Fourier-cosine representation of the likelihood.
• Employ characteristic function identities for normal variables to handle correlated jump components in leverage models.
• Express the likelihood in terms of an integral over ℝⁿ involving complex-valued functions Υₙ and Ωₙ, enabling numerical evaluation.

Experimental results

Research questions

• RQ1 Can a general class of likelihood-based continuous-time stochastic volatility models be analyzed without relying on MCMC simulation?
• RQ2 Can explicit expressions for the marginal likelihood be derived when integrated volatility is a linear functional of a Poisson random measure?
• RQ3 How can leverage effects between price and volatility jumps be incorporated into a tractable likelihood framework?
• RQ4 Is it possible to perform Bayesian inference on the integrated volatility process using only classical numerical integration?
• RQ5 Can inference be conducted without explicit knowledge of the Lévy density, relying instead on the Laplace transform of the underlying random variable?

Key findings

• The marginal likelihood of the observed log-returns can be expressed as a multidimensional Fourier-cosine transform, enabling tractable computation.
• The posterior density of the log price given the data is explicitly derived, which may support applications in option pricing.
• For models with leverage effects, the likelihood is expressed as an integral involving complex-valued functions derived from characteristic functions of normal variables.
• Inference for certain subclasses requires only a finite number of independent latent random variables, simplifying computation.
• The framework allows exact likelihood evaluation using classical numerical integration, avoiding simulation of random measures.
• The method remains valid even when the Lévy density is unknown, as long as the Laplace transform (or Lévy exponent) is known.