[Paper Review] Looking for continuous local martingales with the crossing tree (Working Paper)

Owen D. Jones, David A. Rolls|ArXiv.org|Nov 27, 2009

Financial Risk and Volatility Modeling42 references3 citations

TL;DR

This paper proposes a novel statistical test for the continuous local martingale hypothesis using the crossing tree—a tree-like structure encoding passage times of a stochastic process through level sets. The method outperforms quadratic variation-based tests in power, especially with short datasets, and rejects the continuous martingale hypothesis for high-frequency FX rates (e.g., AUD-USD, EUR-USD) at timescales below 15 minutes in 2003, indicating microstructure or jump-like behavior.

ABSTRACT

We present statistical tests for the continuous martingale hypothesis. That is, 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 recently proposed tests using the estimated quadratic variation (i.e., realised volatility). In particular, the crossing tree shows significantly more power with shorter datasets. We then show results from applying the methodology to high frequency currency exchange rate data. We show that in 2003, for the AUD-USD, GBP-USD, JPY-USD and EUR-USD rates, at small timescales (less than 15 minutes or so) the continuous martingale hypothesis is rejected, but not so at larger timescales. For 2003 EUR-GBP data, the hypothesis is rejected at small timescales and some moderate timescales, but not all.

Motivation & Objective

• To develop a more powerful statistical test for the continuous local martingale hypothesis than existing quadratic variation-based methods.
• To investigate whether high-frequency foreign exchange rates exhibit continuous local martingale behavior at different timescales.
• To characterize the small-scale diffusive behavior of stochastic processes using the crossing tree structure.
• To assess the empirical type I error and statistical power of the proposed test under various diffusion processes.
• To determine the threshold level δ for simulating crossing times in diffusion processes like Feller’s square root process.

Proposed method

• Construct a crossing tree by recording passage times of a process through a sequence of levels, capturing small-scale diffusive behavior.
• Use the empirical distribution of crossing times to test for stationarity and independence, assessing whether they match the expected distribution under Brownian motion.
• Apply multiple distributional tests (e.g., chi-squared, G-test, Kolmogorov-Smirnov) to evaluate goodness-of-fit of crossing time distributions to theoretical expectations.
• Perform independence tests (e.g., autocorrelation, Wald-Wolfowitz, O’Brien) on crossing times to detect dependence structures inconsistent with continuous local martingales.
• Estimate δ (level spacing) via simulation and regression extrapolation to achieve a target number of crossings in a fixed interval, ensuring accurate sampling of passage times.
• Compare the power of the crossing tree test against quadratic variation-based tests using simulated processes (Brownian motion with drift, Ornstein-Uhlenbeck, FBM, Feller process).

Experimental results

Research questions

• RQ1 Does the crossing tree method detect deviations from the continuous local martingale hypothesis more effectively than quadratic variation-based tests?
• RQ2 At what timescales do high-frequency FX rates deviate from the continuous local martingale hypothesis?
• RQ3 How do the empirical distributions of crossing times behave under different diffusion processes, particularly when the process is not stationary at passage times?
• RQ4 Can δ be reliably estimated for time-changed diffusion processes like Feller’s square root process using simulation and extrapolation?
• RQ5 To what extent does the non-stationarity of crossing time distributions affect the validity of statistical tests based on them?

Key findings

• The crossing tree-based test exhibits significantly higher statistical power than quadratic variation-based tests, especially with short datasets, due to its sensitivity to small-scale diffusive behavior.
• For 2003 FX data, the continuous martingale hypothesis is rejected at timescales below 15 minutes for AUD-USD, GBP-USD, JPY-USD, and EUR-USD, indicating non-martingale behavior at high frequency.
• For EUR-GBP, the hypothesis is rejected at small and some moderate timescales but not all, suggesting heterogeneous microstructure effects.
• Empirical crossing time distributions for Feller’s square root process are non-stationary, even after 1000 passage times, invalidating assumptions based on stationarity.
• Regression-based extrapolation of δ estimates from finite Δ simulations yields a stable estimate δ ≈ 0.028163 for 1250 crossings in 5 units of time under Feller’s process with κ=6, σ=1.
• The method successfully identifies deviations from the continuous local martingale hypothesis in processes with mean reversion or long-range dependence, such as the Ornstein-Uhlenbeck and fractional Brownian motion processes.