Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets

Area of Science:

• Quantitative Finance
• Financial Market Analysis
• Stochastic Processes

Background:

• Fat-tailed distributions are frequently observed in financial market fluctuations.
• Previous analyses often assume stationary increments in stochastic processes, a common but potentially flawed assumption.
• Sliding interval techniques are widely used but may introduce analytical artifacts.

Purpose of the Study:

• To investigate the validity of the stationary increments assumption for the Euro-Dollar exchange rate.
• To analyze intraday increments and identify deviations from standard assumptions.
• To introduce empirical methods for evaluating dynamical scaling indices and functions.

Main Methods:

• Analysis of intraday increments of the Euro-Dollar exchange rate.
• Identification of time intervals exhibiting power-law behavior in the standard deviation of increments.
• Modeling stochastic dynamics using diffusion processes with time- and rate-dependent diffusion coefficients.
• Empirical evaluation of dynamical scaling indices and scaling functions.

Main Results:

• The assumption of stationary increments is explicitly shown to be invalid for the Euro-Dollar exchange rate.
• Specific intraday time intervals demonstrate power-law behavior in the standard deviation of increments.
• The empirically evaluated dynamical scaling index is generally significantly smaller than the commonly reported 0.5.
• Apparent fat-tailed distributions and scaling indices near 0.5 can arise as artifacts of sliding interval analysis.

Conclusions:

• The Euro-Dollar exchange rate exhibits non-stationary increments, challenging standard financial market models.
• Diffusion processes with dynamic diffusion coefficients accurately describe intraday stochastic behavior.
• Sliding interval analysis can create misleading results, including spurious fat tails and scaling indices.