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.*

PDF
MPRA_paper_22604.pdf Download (823b) |

## 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 |
---|---|

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

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 |

Item ID: | 22604 |

Depositing User: | C-Rene Dominique |

Date Deposited: | 10. May 2010 12:54 |

Last Modified: | 11. Feb 2013 11:16 |

References: | Bachelier, L. (1900). ‘Théorie de la speculation: Annales de l’École Normale Supérieure.’’ 3e série, 2186. Basu, S. (1977). “Investment Performance of Common Stocks in Relation to Their Price Earnings Ratios: A Test of the Efficiency Market Hypothesis. Journal of Finance, 33: 663-682. Bayraktur, E, et al. (2003). “Estimating the Fractal Dimension of the SP-500 Index Using Wavelet Transform Analysis.” www.princeton.edu/-Sircar/ Public/ Article/bps.pdf. Bernstein, P. (1996). “Against the Gods.” John Wiley & Sons, New York. Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81:637- 659. Casti, J. (1995). “Complexification.” Harper Perennial, New York. Cheredito, P. (2003). Arbitrage in Fractional Brownian Motion Models. Finance and Statistics, 7: 533-553. Comte, F. and Renault, E. (1998). Long-memory in Continuous-Time Stochastic Volatility Models. Mathematical Finance, 8: 291-323. Cox, J. et al. (1979). Option Pricing: A Simplified Approach. Journal of Financial Economics, 7: 229-263. Cutland, N. J., et al. (1995). Stock Price Retrurns and the Joseph Effect: A Fractal Version of the Black-Scholes Model. Progress in Probability, 36: 327-351. Fama, E. and Blume, M. (1966). Filter Rule and Stock Market Trading. A Supplement of the Journal of Business,39:226-241. Falconer, K. (2003). Fractal Geometry. John Wiley & Sons, New York. Grassberger, P. and Procaccia, I. (1983). Measuring The Strangeness of Strange Attractors. Physica, 9D: 189-208. Greene, M. T. and Fielitz, B. D. (1977). Long-Term Dependence in Common Stock Returns. Journal of Financial Economics,4: 339-349. Guasoni, P. (2006). No Arbitrage Under Transaction Costs With Fractional Brownian Motion and Beyond. Mathematical Finance, 16: 569-582. Heston, S. L. (1993). A Close-form Solution for Options With Stochastic Volatility With Applications to Bonds and Currency Options. Review of Financial Studies, 6: 327-343. Hurst, H. E. et al. (1951). Long-term Storage: An Experimental Study. Constable, London. Kendall, M. (1954). The Analysis of Economic Time Series, Part I: Prices. Journal of the Royal Statistical Society, 96: 11-25. Yanhui, L. et al. (1999). Statistical Properties of the Volatility of Price Flustuations. Physical Review E, 60: 1890-1400. Lo, A. W. (1999). A Non-Random Walk Down Wall Street. Princeton University Press, New Jersey. Lyapunov, A. M. (1907). Problème général de stabilité de mouvement. Annuaire de la Faculté des sciences, Université de Toulouse, 9: 203-475. Malkiel, B. G. (1973). A Random Walk Down Wall Street. W. W. Norton, New York. Mandelbrot, B. (1971). When Can Price Be Arbitraged Efficiently? A Limit to the Volatility of the Random Walk and Martingale Models. Review of Economics and Statistics, 53: 225-236. Mandelbrot, B. (1977). Fractal and Scaling in Finance. Springer Verlag, New York. Mandelbrot, B and J. W. van Ness. (1968). Fractional Brownian Motion, Fractional Noises and Applications. SIAM Review,10: 422-437. Merton, R. C. (1969). Life Time Portfolio Selection Under Uncertainty. Review of Economics and Statistics, 51: 247-257. Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal.of Economic and Management Science, 4: 141-183. Medio, A. (1992). Chaotic Dynamics: Theory and Applications to Economics. Cambridge Univ. Press, New York. Moreira, J. G. (1994). On the Fractal Dimension of Self-Affine Profiles. Journal of Physics A, 27: 8079-8089. Osborne, M. (1954). Brownian Motion in the Stock Market. Operation Research., 7: 145-173. Osedelec, V. I. (1968). A Multiplicative Egodic Theorem: Lyapunov Characteristic Numbers for Dynamic systems. Transactions of the Moscow Mathematical Society, 19: 197-231. Peters, E. (1991). A Chaotic Attractor for the SP-500. Financial Analyst Journal, March-April: 55-81. Peters. E. (1994). Fractal Analysis: Applying Chaos to Investment and Economics. John Wiley & Sons, New York. Preciado, James et al. (2008). Proceedings of the Congress on Engineering and Computer Science 2008. WCECS 2008, October 22-24, San Francisco. Shiller, R. I. (1981). Do Stock Prices Move too Much to Be Justified By Subsequent Changes in Dividends. American Economic Review, 71: 421-436. |

URI: | http://mpra.ub.uni-muenchen.de/id/eprint/22604 |