What is the “true price”? state space models for high frequency FX data

In previous work, we have found statistically significant structures in tick-by-tick interbank foreign exchange (FX) price series on various time scales. These structures include negative autocorrelations in successive 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 with short-term, long-term, or multi-scale memory structures. The processes we consider include random walks, random trends, and fractional Brownian motions. For fractional Brownian motion processes, we represent the multi-scale correlational structures using self-similar wavelet decompositions. We estimate the state space model parameters using the Kalman filter and 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 a neural-network-based independent component analysis (ICA) using algorithms developed for the blind separation of signals. Statistical analysis of our true price models supports our assertion that the estimated true prices are significantly different from the observed prices, and that significant non-random structures may exist in the FX markets