Extreme Risk and Fractal Regularity in Finance

Abstract

As the Great Financial Crisis reminds us, extreme movements in the level and volatility of asset prices are key features of financial markets. These phenomena are difficult to quantify using traditional approaches that specify extreme risk as a singular rare event detached from ordinary dynamics. Multifractal analysis, whose use in finance has considerably expanded over the past fifteen years, reveals that price series observed at different time horizons exhibit several major forms of scale-invariance. Building on these regularities, researchers have developed a new class of multifractal processes that permit the extrapolation from high-frequency to low-frequency events and generate accurate forecasts of asset volatility. The new models provide a structured framework for studying the likely size and price impact of events that are more extreme than the ones historically observed. Fractal modeling uses invariance principles to parsimoniously specify complex objects at multiple scales. It has proven to be of major importance in mathemat- ics and the natural sciences, as this issue illustrates. Fractals also offer enormous benefits for the field of finance, in particular for modeling the price of traded se- curities, for computing the risk of financial portfolios, for managing the exposure of institutions, or for pricing derivative securities. These benefits should become more apparent as the adoption of fractal methods by the financial industry contin- ues to gain ground. The fields of finance and economics also play a singular role in the intellectual history of fractals. Benoˆ it Mandelbrot first discovered evidence of fractal behavior in financial returns, household income and household wealth in the late 1950's and early 1960's, and subsequently found similar patterns in coast- lines, earthquakes and other natural phenomena. These observations prompted the