A Robust Non-linear Multivariate Kalman Filter for Arbitrage Identification in High Frequency Data

We present a methodology for modelling real world high frequency financial data. The methodology copes with the erratic arrival of data and is robust to additive outliers in the data set. Arbitrage pricing relationships are formulated into a linear state space representation. Arbitrage opportunities violate these pricing relationships and are analogous to multivariate additive outliers. Robust identification/filtering of arbitrage opportunities in the data is accomplished by Kalman filtering. The state space model used to describe the pricing relationships is general enough to handle both linear and non-linear models. The recursive Kalman equations are adapted to filter tick data, cope with the erratic arrival of observations and produce estimates of all the arbitrage prices on every time step. We demonstrate the methodology with a robust neural network filter applied to foreign exchange triangular arbitrage. Tick data from three markets is used: $/DM, £/$, £/DM 1993-1995. The filter produces estimates of the arbitrage price for all exchange rates on every second, increasing both the speed and efficiency of arbitrage identification.