The usual returns fat-tailed distributions show power law decay in the tails. A subsequent question thus pertains to the forecast of the numerical value (of the exponent) describing the fat tail(s). The irrational fractional Brownian Motion (IfBM) modifies the Geometric Brownian Motion (GBM) by including an extra functional form which contains two parameters namely c and K.

Parameter c identifies the onset of the fat tails and K provides a measure for the weight of fat tails. Optimal estimation of the parameters through simulations allows the IfBM to model the leptokurtic return distribution of financial asset prices. In this study, optimal c and K parameters are fitted to consecutive daily two-year period returns of S&P500 index from 1956 to 2016, generating 30-time series estimations of c and K. Through an econometric model specification analysis, the empirical kurtosis of returns distribution is modelled as multiplicative function of c and K. Subsequently a vector auto regression (VAR) analysis on c and K advances the understanding of modelling and forecasting Kurtosis of return distributions. Furthermore, the power law of fat tails in the returns distribution is investigated using this approach.

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