The relationship between return fractality and bipower variation

Abstract

This paper presents an intuitively simple asset pricing model designed to predict stock returns and volatilities, when stock prices may follow a fractal walk rather than a random walk. The model utilizes similarity ratio of the return fractals as the basis for forecasting. We argue that a collection of past returns such as moving average statistics can be 'expanded' to generate future returns through the similarity ratios by trading time multiples. We also argue that stock returns with fractal dimensions in excess of 1.5, the norm for market efficiency, may be prone to frequent jumps and discontinuity. The paper builds an econometrically testable model to estimate fractal dimensions to offer an alternative way to forecast return volatilities without having to estimate separately the bipower variation and the jumps in existing studies. We argue that fractional Brownian motion may be a more suitable description than the standard Wiener process when describing stock return behaviors. The paper also demonstrates the application aspect of our asset pricing model to high frequency algorithmic trading.