Measurement of Financial Risk Persistence

Abstract:     
This paper discusses various ways of measuring the persistence or Long Memory (LM) of financial market risk in both its time and frequency domains. For the measurement of the risk, irregularity or randomness of these series, we can compute a set of critical Lipschitz-Holder exponents, in particular, the Hurst Exponent and the Levy Stability Alpha, and relate them to the Mandelbrot-Hoskings' fractional difference operators, as occur in the Fractional Brownian Motion model (which is our benchmark). The main contribution of this paper is to provide a compaison table of the various critical exponents available in various scientific disciplines to measure the LM persistence of time series. It also discusses why Markov and (G)ARCH models cannot capture this LM, long term dependence or risk persistence, because these models have finite lag lengths, while the empirically observed long memory risk phenomenon is an infinite lag length phenomenon. Currently, there are three techniques of nonstationary time series analysis to measure time-varying financial risk: Range/Scale analysis, windowed Fourier analysis, and wavelet MRA. This paper relates these powerful analytic techniques to classical Box-Jenkins-type time series analysis and to Pearson's spectral frequency analysis, which both rely on the uncorroboated assumption of stationarity and ergodicity.

Number of Pages in PDF File: 37

Keywords: Persistence, long memory, dependence, time series, frequency, critical exponents, fractional Brownian motion, (G)ARCH, risk measurement

JEL Classification: C15, C23, C53, G10

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