Asymmetric Lévy flight in financial ratios

Abstract

Because financial crises are characterized by dangerous rare events that occur more frequently than those predicted by models with finite variances, we investigate the underlying stochastic process generating these events. In the 1960s Mandelbrot [Mandelbrot B (1963) J Bus 36:394-419] and Fama [Fama EF (1965) J Bus 38:34-105] proposed a symmetric Lévy probability distribution function (PDF) to describe the stochastic properties of commodity changes and price changes. We find that an asymmetric Lévy PDF, L, characterized by infinite variance, models several multiple credit ratios used in financial accounting to quantify a firm's financial health, such as the Altman [Altman EI (1968) J Financ 23:589-609] Z score and the Zmijewski [Zmijewski ME (1984) J Accounting Res 22:59-82] score, and models changes of individual financial ratios, ΔX i . We thus find that Lévy PDFs describe both the static and dynamics of credit ratings. We find that for the majority of ratios, ΔX i scales with the Lévy parameter α ≈ 1, even though only a few of the individual ratios are characterized by a PDF with power-law tails X −1−α i with infinite variance. We also find that α exhibits a striking stability over time. A key element in estimating credit losses is the distribution of credit rating changes, the functional form of which is unknown for alphabetical ratings. For continuous credit ratings, the Altman Z score, we find that PðΔZÞ follows a Lévy PDF with power-law exponent α ≈ 1, consistent with changes of individual financial ratios. Estimating the conditional PðΔZjZÞ versus Z, we demonstrate how this continuous credit rating approach and its dynamics can be used to evaluate credit risk.complex systems | econophysics | rating migrations M ost tests and tools used in statistics assume that any errors in a financial model are Gaussian distributed, and it is a common practice in economics to use a Gaussian distribution to approximate empirical data. Mandelbrot (1) and Fama (2) were among the first to notice that the logarithm of cotton price fluctuations and common stock price fluctuations have fatter tails than those produced by a Gaussian distribution, and they proposed a stable Lévy distribution to model the stochastic properties of the fluctuations. Analyzing high-frequency data, Mantegna and Stanley (3) reported that the stable Lévy distribution accurately models only a broad central region of the probability distribution function (PDF) of stock price changes, whereas Gopikrishnan et al. reported that a power law with an exponent value beyond the Lévy regime is needed to describe the tails (4, 5).The central limit theorem (CLT) implies that the mean of a sufficiently large number of independent random variables, each with finite variance, will approximately follow a normal distribution (6). A generalization of the CLT shows that the mean of a sufficiently large number of independent random variables, each with infinite variance, approximately follows a stable Lévy distribution L α;γ ðxÞ ¼ ð1∕πÞ∫ ∞ 0 dq expð−γq α Þ cosðqxÞ, wher...

Linking agent-based models and stochastic models of financial markets

“…1 shows that the model reproduces the empirical data in both the central region and the tails. In particular, both model and empirical distributions have power-law tails (Pðjr t j > xÞ∼ x −ξ r ; Pðn t > xÞ ∼ x −ξ n ) (13,(29)(30)(31) where the power-law exponents ξ r and ξ n that we obtain for the empirical data by applying the Hill's estimator are in agreement with the ones obtained for empirical data (32).…”

Asymmetric Lévy flights in the presence of absorbing boundaries

“…One striking feature of such processes is that their statistical behavior is dominated by a few rare and very large events, whose occurrence is thus governed by the tail of the distribution. Often the applications of Lévy flights are restricted to the symmetric case when γ = 1, however recently the asymmetric Lévy flights have found applications in search problems [8] and finance [9]. Diffusion in asymmetric disordered potential was recently considered in connection with the ratchet-effect [10].…”

Correlated Disorder in Myelinated Axons Orientational Geometry and Structure

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