Testing for Continuous Martingales Using the Crossing Tree

We present statistical tests for the continuous martingale hypothesis; that is, for whether an observed process is a continuous local martingale, or equivalently a continuous time-changed Brownian motion. Our technique is based on the concept of the crossing tree. Simulation experiments are used to assess the power of the tests, which is generally higher than that of recently proposed tests using the estimated quadratic variation (i.e. realized volatility). In particular, the crossing tree shows significantly higher power with shorter data sets. We then show results from applying the methodology to five high-frequency currency exchange rate data sets from 2003. For four of them we show that at small time-scales (less than 15 minutes or so) the continuous martingale hypothesis is rejected, but not so at larger time-scales. For the fifth, the hypothesis is rejected at small time-scales and at some moderate time-scales, but not all.