MULTIFRACTAL FLUCTUATIONS IN FINANCE

Abstract

We consider the structure functions S^(q)(τ), i.e. the moments of order q of the increments X(t + τ)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S^(q)(τ)∝τ^ζ(q)). We demonstrate that the nonlinearity of the observed scaling exponent ζ(q) is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity of ζ(q) corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponent q_D > 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.