Tick-by-tick interbank foreign exchange (FX) price series exhibit statistically- significant structures on various time scales. These include negative autocorrelations in tick-by-tick returns and positive autocorrelations (trends) on longer time scales. To account for the observed structures, we propose state space models for financial time series in which the observed price is a noisy version of an unobserved, less-noisy ``True Price'' process.
The True Prices in our models are stochastic processes such as random walks, random trends, and fractional Brownian motions. The observational noise is due to market microstructure. For fractional Brownian motions, we represent the multi-scale correlations using self-similar wavelet decompositions. The state space model parameters are estimated using the Kalman filter or EM algorithms.
Since both the observational noise and the changes in the True Price series have non-gaussian distributions, the Kalman filter and EM algorithms are not able to completely separate the observational noise from the true price components. To improve this separation, we perform an independent component analysis (ICA) using algorithms developed for the blind separation of signals.
Statistical analysis of True Price models supports our assertion that the estimated True Prices are significantly different from the observed prices, and that significant non-random structures exist in the FX markets. Sonification of the observed price and ``true price'' series supports this finding by revealing obvious perceptual differences between the signals.
An extension of the True Price methodology called the ``multi-effect decomposition'' further enables the separation of tick-by-tick price series into market microstructure, short-term news and longer term drift components. We believe that the proposed True Price models may enable discovery of new arbitrage opportunities and construction of better forecasting and trading models.