This paper examines a class of continuous-time models with stochastic volatility that incorporate jumps in returns and volatility. We develop a likelihood-based es- timation strategy and provide estimates of model parameters, spot volatility, jump times and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997 and 1998.

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Introduction Estimating functions provide a general framework for finding estimators and studying their properties in many di#erent kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obtained by equating the estimating function to zero and solving the resulting estimating equation with respect to the parameter. The idea of using estimating equations is an old one and goes back at least to Karl Pearson's introduction of the method of moments. The term estimating function may have been coined by Kimball (1946). The estimating function approach has turned out to be very useful in obtaining, improving and studying estimators for discretely sampled parametric di#usion-type models, where the likelihood function is usually not explicitly known. Estimating functions are often constructed by combining relationships (dependent on the unknown parameter) between an observa

In this paper, we adapt recently developed simulation-based sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion pro-cesses. The estimation framework involves the introduction of m −1 latent data points between every pair of observations. Sequential MCMC methods are then used to sam-ple the posterior distribution of the latent data and the model parameters on-line. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. short-term interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. short-term interest rate. 1

This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be nonstationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data.

This paper is concerned with the Bayesian estimation of stochastic rate constants in the context of dynamic models of intra-cellular processes. The underlying discrete stochastic kinetic model is replaced by a di#usion approximation (or stochastic di#erential equation approach) where a white noise term models stochastic behaviour and the model is identified using equispaced time course data. The estimation framework involves the introduction of m-1 latent data points between every pair of observations. MCMC methods are then used to sample the posterior distribution of the latent process and the model parameters

This article develops a direct filtration-based maximum likelihood methodology for estimating the parameters and realizations of latent affine processes. Filtration is conducted in the transform space of characteristic functions, using a version of Bayes ’ rule for recursively updating the joint characteristic function of latent variables and the data conditional upon past data. An application to daily stock market returns over 1953-96 reveals substantial divergences from EMM-based estimates; in particular, more substantial and time-varying jump risk. The implications for pricing stock index options are examined. 3 “The Lion in Affrik and the Bear in Sarmatia are Fierce, but Translated into a Contrary Heaven, are of less Strength and Courage.” Jacob Ziegler; translated by Richard Eden (1555) While models proposing time-varying volatility of asset returns have been around for thirty years, it has proven extraordinarily difficult to estimate the parameters of the underlying volatility process,

This paper provides a methodology for computing optimal filtering distributions in discretely observed continuous-time jump-diffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing model diagnostics such as likelihood ratios, and parameter estimation. Our approach combines time-discretization schemes with Monte Carlo methods to compute the optimal filtering distribution. Our approach is very general, applying in multivariate jump-diffusion models with nonlinear characteristics and even non-analytic observation equations, such as those that arise when option prices are available. We provide a detailed analysis of the performance of the filter, and analyze four applications: disentangling jumps from stochastic volatility, forecasting realized volatility, likelihood based model comparison, and filtering using both option prices and underlying returns.

In this paper we introduce novel particle filters for a class of partially-observed continuous-time dynamic models where the signal is given by a multivariate diffusion process. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Unlike available methods, our particle filters do not require approximations of the transition and/or the observation density using time-discretisations. Instead, they build on recent methodology for the exact simulation of diffusion process and the unbiased estimation of the transition density as described in the recent article Beskos et al. (2005c). In particular, we require the Generalised Poisson Estimator, which is a substantial generalisation of the Poisson Estimator (Beskos et al., 2005c), and it is introduced in this paper. Thus, our filters avoid the systematic biases caused by time-discretisations and they have significant computational advantages over alternative continuous-time filters. These advantages are supported by a central limit theorem which is established in this paper. Keywords: Continuous-time filtering, Exact Algorithm, Central Limit Theorem, Cox Process 1

We consider analysis of complex stochastic models based upon partial information. MCMC and reversible jump MCMC are often the methods of choice for such problems, but in some situations they can be difficult to implement; and suffer from problems such as poor mixing, and the difficulty of diagnosing convergence. Here we review three alternatives to MCMC methods: importance sampling, the forward-backward algorithm, and sequential Monte Carlo (SMC). We discuss how to design good proposal densities for importance sampling, show some of the range of models for which the forward-backward algorithm can be applied, and show how resampling ideas from SMC can be used to improve the efficiency of the other two methods. We demonstrate these methods on a range of examples, including estimating the transition density of a diffusion and of a discrete-state continuous-time Markov chain; inferring structure in population genetics; and segmenting genetic divergence data.