Determining The Value-at-risk In The Shadow Of The Power Law: The Case Of The SP-500 Index

Dominique, C-Rene and Rivera-Solis, Luis Eduardo and Des Rosiers, Francois (2010): Determining The Value-at-risk In The Shadow Of The Power Law: The Case Of The SP-500 Index.

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-b (b > 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’.

• Item Type: MPRA Paper
• Language: English
• Keywords: Market Collapse; Fractional Brownian Motion; Fractal Attractors; Maximum Hausdorff Dimension of Markets and Affine Profiles; Hurst Exponent; Power Spectrum Exponent; Value at Risk
• Subjects: C - Mathematical and Quantitative Methods > C9 - Design of Experiments > C90 - General; G - Financial Economics > G1 - General Financial Markets > G10 - General

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