Abstract. In the present work we investigate the multiscale nature of the correlations for high frequency data (1 minute) in different futures markets over a period of two years, starting on the 1 st of January 2003 and ending on the 31st of December 2004. In particular, by using the concept of local Hurst exponent, we point out how the behaviour of this parameter, usually considered as a benchmark for persistency/antipersistency recognition in time series, is largely time-scale dependent in the market context. These findings are a direct consequence of the intrinsic complexity of a system where trading strategies are scale-adaptive. Moreover, our analysis points out different regimes in the dynamical behaviour of the market indices under consideration.

This paper develops a matrix-variate adaptive Markov chain Monte Carlo (MCMC) methodology for Bayesian Cointegrated Vector Auto Regressions (CVAR). We replace the popular approach to sampling Bayesian CVAR models, involving griddy Gibbs, with an automated efficient alternative, based on the Adaptive Metropolis algorithm of Roberts and Rosenthal, (2009). Developing the adaptive MCMC framework for Bayesian CVAR models allows for efficient estimation of posterior parameters in significantly higher dimensional CVAR series than previously possible with existing griddy Gibbs samplers. For a n-dimensional CVAR series, the matrix-variate posterior is in dimension 3n 2 + n, with significant correlation present between the blocks of matrix random variables. Hence, utilizing a griddy Gibbs sampler for large n becomes computationally impractical as it involves approximating an n × n full conditional posterior using a spline over a high dimensional n × n grid. The adaptive MCMC approach is demonstrated to be ideally suited to learning on-line a proposal to reflect the posterior correlation structure, therefore improving the computational efficiency of the sampler.We also treat the rank of the CVAR model as a random variable and perform joint inference on the rank and model parameters. This is achieved with a Bayesian posterior distribution defined over both the rank and the CVAR model parameters, and inference is made via Bayes Factor analysis of rank.Practically the adaptive sampler also aids in the development of automated Bayesian cointegration models for algorithmic trading systems considering instruments made up of several assets, such as currency baskets. Previously the literature on financial applications of CVAR trading models typically only considers pairs trading (n=2) due to the computational cost of the griddy Gibbs. We are able to extend under our adaptive framework to n >> 2 and demonstrate an example with n = 10, resulting in a posterior distribution with parameters up to dimension 310. By also considering the rank as a random quantity we can ensure our resulting trading models are able to adjust to potentially time varying market conditions in a coherent statistical framework.

We consider a statistical model for pairs of traded assets, based on a Cointegrated Vector Auto Regression (CVAR) Model. We extend standard CVAR models to incorporate estimation of model parameters in the presence of price series level shifts which are not accurately modeled in the standard Gaussian error correction model (ECM) framework. This involves developing a novel matrixvariate Bayesian CVAR mixture model, comprised of Gaussian errors intra-day and α-stable errors inter-day in the ECM framework. To achieve this we derive conjugate posterior models for the Scale Mixtures of Normals (SMiN CVAR) representation of α-stable inter-day innovations. These results are generalized to asymmetric intractable models for the innovation noise at inter-day boundaries allowing for skewed α-stable models via Approximate Bayesian computation.Our proposed model and sampling methodology is general, incorporating the current CVAR literature on Gaussian models, whilst allowing for price series level shifts to occur either at random estimated time points or known a priori time points. We focus analysis on regularly observed non-Gaussian level shifts that can have significant effect on estimation performance in statistical models failing to account for such level shifts, such as at the close and open times of markets.We illustrate our model and the corresponding estimation procedures we develop on both synthetic and real data. The real data analysis investigates Australian dollar, Canadian dollar, five and ten year notes (bonds) and NASDAQ price series.In two studies we demonstrate the suitability of statistically modeling the heavy tailed noise processes for inter-day price shifts via an α-stable model. Then we fit the novel Bayesian matrix variate CVAR model developed, which incorporates

In the present work we demonstrate the application of different physical methods to high-frequency or tick-by-tick financial time series data. In particular, we calculate the Hurst exponent and inverse statistics for the price time series taken from a range of futures indices. Additionally, we show that in a limit order book the relaxation times of an imbalanced book state with more demand or supply can be described by stretched exponential laws analogous to those seen in many physical systems.Comment: 14 Pages and 10 figures. Proceeding to the SPIE conference, 4 - 7 December 2007 Australian National Univ. Canberra, ACT, Australi

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