# Multiscale Volatility Analysis for Noisy High-Frequency Data

## Hurst exponent and time-varying intraday volatility

The study of multiscale properties and scaling laws has long been a topic in finance. One of the key questions is: how does the distribution of returns behave at different timescales?

The standard Brownian motion (or random walk) model intrinsically determines the distribution of returns at all time scales due to the independent increments and other model properties. The generalization to **fractional Brownian motion (fBm) **by Mandelbrot and Van Ness (1968) leads to a much larger class of stochastic processes, allowing for new scaling properties and a long-memory process.

In our new paper, we develop new statistical methods to estimate and analyze the multiscale volatility in asset prices using intraday high-frequency prices.

We analyze a price model with microstructure noise and present a new method to estimate it using real-world high-frequency price data.

The observed price is assumed to be the sum of latent fractional Brownian motion and an independent microstructure noise.

Price = Fractional Brownian Motion + Noise