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

Author

Listed:
  • Dominique, C-Rene
  • Rivera-Solis, Luis Eduardo
  • Des Rosiers, Francois

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’.

Suggested Citation

  • Dominique, C-Rene & Rivera-Solis, Luis Eduardo & Des Rosiers, Francois, 2010. "Determining The Value-at-risk In The Shadow Of The Power Law: The Case Of The SP-500 Index," MPRA Paper 22604, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:22604
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    References listed on IDEAS

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    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Shiller, Robert J, 1981. "Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?," American Economic Review, American Economic Association, vol. 71(3), pages 421-436, June.
    5. Greene, Myron T. & Fielitz, Bruce D., 1977. "Long-term dependence in common stock returns," Journal of Financial Economics, Elsevier, vol. 4(3), pages 339-349, May.
    6. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    Cited by:

    1. Dominique, C-René & Rivera-Solis, Luis Eduardo, 2011. "Mixed fractional Brownian motion, short and long-term Dependence and economic conditions: the case of the S&P-500 Index," MPRA Paper 34860, University Library of Munich, Germany.

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    More about this item

    Keywords

    Market Collapse; Fractional Brownian Motion; Fractal Attractors; Maximum Hausdorff Dimension of Markets and Affine Profiles; Hurst Exponent; Power Spectrum Exponent; Value at Risk;
    All these keywords.

    JEL classification:

    • C90 - Mathematical and Quantitative Methods - - Design of Experiments - - - General
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)

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