DETERMINING THE VALUE AT RISK IN THE SHADOW OF THE POWER LAW - THE CASE OF THE SP-500 INDEX

Oct 13, 2026

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The paper examines why standard financial market models fail to predict extreme events such as October 1987 and August 2007. It proposes that combining a generalized fractional Brownian motion with its amplitude spectra reveals the attractor surface governing price evolution. This framework characterizes the SP-500 index by a high long-term Hurst exponent, producing “black noise” with a power spectrum proportional to f^(−β), where β>2. The model then determines event frequencies and builds a frequency–value-at-risk relationship for practical risk assessment.

LanguageEnglish

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Pages22

Reading Time8 min

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I-INTRODUCTION

DETERMINING THE VALUE AT RISK IN THE SHADOW OF THE POWER LAW: THE CASE OF THE SP-500 INDEX C-René Dominique, Luis Eduardo Rivera-Soli, and François Des Rosiers* ABSTRACT: In extant financial market models, including the Black-Scholes‟ contruct, the dramatic events of October 1987 and August 2007 are totally unexpected, because these models are based on the assumptions of „independent price fluctuations‟ and the existence of some „fixed-point equilibrium‟. This paper argues that the convolution of a generalized fractional Brownian motion (into an array in frequency or time domain) and their corresponding amplitude spectra describes the surface of the attractor driving the evolution of prices. This more realistic approach shows that the SP-500 Index is characterized by a high long term Hurst exponent and hence by a „black noise‟ with a power spectrum proportional to f-􀁅 (􀁅 > 2) . In that set up, the above dramatic events are expected and their frequencies are determined. The paper also constructs an exhaustive frequency-variation relationship which can be used as practical guide to assess the „value at risk‟. Keywords: Market Collapse, Fractional Brownian Motion, Fractal Attractors, Maximum Hausdorff Dimension of Markets and Affine Profiles, Hurst Exponent, Power Spectrum Exponent, Value at Risk. JEL Classification System: C90, G10 . I-INTRODUCTION The process known as Brownian motion (B(t)), first observed by Robert Brown (1773-1858), the botanist, has found applications in many other areas, mainly due to its scale and time invariance properties. For example, it has inspired Karl Weierstrass (1815-1897) to develop a whole new class of functions that bear his name today. B (t) was next introduced to mathematical physics by Albert Einstein (1879-1953) and Marian Smoluchowski (1872-1917), and refined by another physicist Norbert Wiener (1894-1964), giving what is known today as a „Wiener Process‟. B (t) is also applied in medicine, chemistry, telecommunication, thermodynamics, etc. Its generalization is the foundation of the field of tomography where it is successfully used to model a variety of natural phenomena such as attractors‟ surfaces, terrains, coastlines, clouds, etc. Bachelier (1900) was perhaps the first to import the regular B (t) into mathematical finance. That idea resurfaced in Chicago in the early 1950s where financial theorists began to use it to model a variety of stochastic processes. However, its importation in extenso into mathematical finance has not always been an enlightening experience. To begin with, Bachelier was so convinced that commodity prices evolve according to a B (t) that he concluded, rather boldly, that “the mathematical expectation of a speculator is zero.” That belief was resuscitated in the 1950sto support the construction of stochastic models based on continuous and discrete-time random walk (Kendall, 1953; Osborne, 1959; and more recently, Malkiel, 1973, among others) . That line of research subsequently gave rise to the notion of Efficient Market Hypothesis (EMH) (Fama and Blume, 1966) and to more sophisticated stochastic partial *Formerly Titular Professor of Applied Economics, Laval University, [rdom1@Netzero.net](rdom1@Netzero.net) Professor of Finance, Dowling College, [RiveraL@dowling.edu](RiveraL@dowling.edu) Professor of Finance, Insurance and Real Estate, Laval University, [francois.desrosiers@fsa.ulaval.ca](francois.desrosiers@fsa.ulaval.ca). differential equations describing optimal investment or stock and derivative pricing strategies (Merton, 1969, 1973; Black and Scholes, 1973; Cox et al., 1979) . Despite numerous criticisms (Basu, 1977; Shiller, 1981; Lo, 1999), however, these constructs (the later vintage in particular) remain the foundation of financial analyses until the cumulated effects of „black Tuesday‟ in October 1987 and the collapse of the credit market in August 2007 sounded their death knell. Many factors underline this dramatic failur

FAQs

Why do conventional financial market models struggle with extreme events like October 1987 and August 2007?

They rely on assumptions such as independent price fluctuations and fixed-point equilibrium, which are argued to be incompatible with the dynamics driving real market evolution during dramatic events.

What alternative modeling approach does the paper propose for market dynamics?

It uses the convolution of a generalized fractional Brownian motion with amplitude spectra to describe the attractor surface that drives price evolution, rather than relying on standard equilibrium-based assumptions.

How does the paper relate the SP-500 index to long-term dependence and risk measurement?

It claims the SP-500 has a high long-term Hurst exponent, yielding “black noise” with a power spectrum proportional to f^(−β) (β>2), and it constructs a frequency–variation relationship to guide assessment of value at risk.

Oct 13, 2026

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