Non-Gaussian processes of Ornstein-Uhlenbeck type, or OU processes for short, offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory. Keywords: Background driving L'evy process; Econometrics; L'evy density; L'evy process; Option pricing; OU process; Particle filter; Stochastic volatility; Subordination; Superposition. Authors' note: This paper supersedes our previously circulated but unpublished papers "Aggregation and model ...

In this chapter we propose using compound Poisson processes to model trade-by-trade #nancial data. Our main focus will be on developing speci#c types of Cox processes in order to accurately depict the trading process. We study the problem of signal extracting the intensity of the trading process. We #nish by studying the implication for price changes over pre-speci#ed intervals of times, such as 30 seconds, 20 minutes or a day and assessing the empirical plausibility of OU based models for the intensity of the trading process.. Some keywords: Cox process; Durations; Kalman #lter; Intensity; Ornstein# Uhlenbeck processes; Particle #lter; Trade-by-trade dynamics. 1 Introduction 1.1 The data and model Most modern theoretical and empirical #nance is based on continuous time models with continuous sample paths or, in other words, di#usion processes which are driven by Wiener processes. Prominent recent references include Du#e #1992# and Ait-Sahalia #1996#, while the most well known example ...

informs doi 10.1287/mnsc.1060.0520

Comments are welcome In this paper we study the impact of market microstructure effects on the properties of realized variance using a pure jump process for high frequency security prices. Closed form expressions for the bias and mean squared error of realized variance are derived under alternative sampling schemes. Importantly, we show that business time sampling is generally superior to the common practice of calendar time sampling in that it leads to a reduction in mean squared error. Using IBM transaction data we estimate the model parameters and de-termine the optimal sampling frequency for each day in the data set. The empirical results reveal a downward trend in optimal sampling frequency over the last 4 years with considerable day-to-day variation that is closely related to changes in market liquidity.

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In this paper we extend Rydberg-Shephard’s activity, direction and size decomposition of trade-by-trade price movements to the multivariate case. We illustrate our ideas using a bivariate modelling problem — modelling the evolution of the prices of Ford and GM shares. Throughout we use the continuous record of trades made in the first five months of 1997 on the New York Stock Exchange (NYSE).

In this paper we introduce a new class of models, called OSV, by combining an ordinal response model and the idea of stochastic volatility. Corresponding time series occur in high-frequency finance when the stocks are traded on a coarse grid. For parameter estimation we develop an efficient Grouped Move Multigrid Monte Carlo (GM-MGMC) sampler. This sampler is based on a scale transformation group, whose elements operate on the random samples of a certain conditional distribution. Also volatility estimates are provided. For illustration, we apply our new model class to price changes of the IBM stock. Dependencies on covariates are quantified and compared with theoretical results for such processes.

We provide a structural approach to identify instantaneous causality effects between durations and stock price volatility. So far, in the literature, instantaneous causality effects have either been excluded or cannot be identified separately from Granger type causality effects. By giving explicit moment conditions for observed returns over (random) duration intervals, we are able to identify an instantaneous causality effect. The documented causality effect has significant impact on inference for tick-by-tick data. We find that instantaneous volatility forecasts for, e.g., IBM stock returns must be decreased by as much as 40 % when not having seen the next quote change before its (conditionally) median time. Also, instantaneous volatilities are found to be much higher than indicated by standard volatility assessment procedures using tick-by-tick data. For IBM, a naive assessment of spot volatility based on observed returns between quote changes would only account for 60 % of the actual volatility. For less liquidly traded stocks at NYSE this effect is even stronger.

Option pricing in the jumpdiffusion model with a random jump amplitude: A complete market approach

Ordinal valued time series can be found in many different areas, for example in analysis of stock prices where the transaction price changes often occur in discrete increments as sixteenths of a dollar. We consider these price changes as discrete random variables which are assumed to be generated by a latent process which represents the underlying true price change process and which incorporates both exogenous variables and autoregressive components. A standard Gibbs sampling algorithm has been developed to estimate the parameters of the model. However this algorithm exhibits bad convergence properties. To get a more efficient sampling method we utilize a special transformation group on the sample space which allows to develop a Grouped Move Multigrid Monte Carlo Gibbs sampler. A simulation study is given to demonstrate the substantial improvement by this new algorithm. Finally we apply our model to the data of the IBM stock on Nov 13, 2000, and estimate the influence of the duration between transactions, the volume, and the bid-offer-spread both to model fit and prediction.