Risk and complex fractals in finance: application to a Black - Scholes equation of ordern

  • Authors:

  • Guy Jumarie

  • Affiliations:

  • Department of Mathematics, University of Quebec at Montré/al, P.O. Box 8888, Downtown Station Montreal, Qc/ H3C 3P8, Canada

  • Venue:
  • Systems Analysis Modelling Simulation
  • Year:
  • 2002

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Abstract

The short term risks in finance cannot be suitably analyzed by using the classical risk theory, and a way to circumvent this pitfall is to use fractional processes. The higher the risk is the higher the order of the fractal is. Recently, a model of complex-valued fractional Brownian motion of order n has been suggested in the literature, and the purpose herein is to examine which kind of results it can provide in mathematics of finance. After a short background to this model, one comments on fractals in finance, and then one suggests a general model of complex fractals for stock market. Various state variables can be identified with the parameters so involved in this model, and mainly, one can introduce a new concept of stock market time different from the standard absolute physical time. As an application, a Black-Scholes equation of order n is derived.