Further results on the modelling of complex fractals in finance, scaling observation and optimal portfolio selection

Authors:
Guy Jumarie
Affiliations:
Department of Mathematics, University of Quebec at Montreal, P.O. Box 8888, Downtown Station, Montreal, Qc, H3C 3P8, Canada
Venue:
Systems Analysis Modelling Simulation
Year:
2002

Citing 3
Cited 2

Relative information: theories and applications

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

Systems Analysis Modelling Simulation
Fractals and Scaling In Finance: Discontinuity, Concentration, Risk

Fractals and Scaling In Finance: Discontinuity, Concentration, Risk

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Abstract

It has been proposed to use complex-valued fractional Brownian motion to describe fractals in finance, and mainly to derive a Black-Scholes equation of order n in which the parameter n takes account of the investment risk. Here, this approach is improved in two ways. Firstly, by using properties of complex white noises, one shows how one can by-pass setting the problem in the complex plane, and one derives a fractal stochastic differential equation for stock prices. Secondly, in order to prove that complex white noise is quite relevant, one points out that it can be derived in quite a natural way, as the result of an observation process which converts a 1-D system into a 2-D one. As a new application, one examines the incidence of fractal noises on optimal portfolio selection policy. The problem is firstly solved by using stochastic dynamic programming of order n, and then the same solution is obtained by using signed probability density.