Maximum likelihood estimation of stable Paretian distribution applied to index and option data

Abstract. Ample evidence exists documenting the fat-tailed character of returns in finan-cial markets (Rachev and Mittnik, 2000). Several papers attempt to model these fat tailed distributions as power laws of the form Pr(k) = k −α (α) , where k is a positive integer measuring asset returns. Pr(k) is the probability of actually observing k, α is the power law exponent, and (α) is the Riemann zeta function defined as k=1 k −α. The method most employed in determining the power law exponent is graphical analysis of the log of the ranked data followed by regression. In this paper, we point out the flaws of this method of discovering power laws, and argue for a more direct method of discovery using maximum likelihood estimation over a bounded field. After deriving our model, we test it on two high frequency time series: AMEX-OIL, and European-style exercise options on FSTES 100. Relative to the standard OLS power law fitting procedure, we find a much more accurate fit to the data us-ing the maximum likelihood method, estimating α exponents of approximately 3 − 5, which we verify using Kolmogorov-Smirnov tests