Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function ("bad" probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring "adiabatic" conditions on the volatility smile

Turbulence and Multiscaling in the Randomly Forced Navier Stokes Equation

We present an extensive pseudospectral study of the randomly forced Navier-Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance $\sim k^{4-d-y}$, where $k$ is the wavevector and the dimension $d = 3$. We present the first evidence for multiscaling of velocity structure functions in this model for $y \geq 4$. We extract the multiscaling exponent ratios $\zeta_p/\zeta_2$ by using extended self similarity (ESS), examine their dependence on $y$, and show that, if $y = 4$, they are in agreement with those obtained for the deterministically forced Navier-Stokes equation ($3d$NSE). We also show that well-defined vortex filaments, which appear clearly in studies of the $3d$NSE, are absent in the RFNSE.Comment: 4 pages (revtex), 6 figures (postscript

Chaotic Cascades with Kolmogorov 1941 Scaling

We define a (chaotic) deterministic variant of random multiplicative cascade models of turbulence. It preserves the hierarchical tree structure, thanks to the addition of infinitesimal noise. The zero-noise limit can be handled by Perron-Frobenius theory, just as the zero-diffusivity limit for the fast dynamo problem. Random multiplicative models do not possess Kolmogorov 1941 (K41) scaling because of a large-deviations effect. Our numerical studies indicate that deterministic multiplicative models can be chaotic and still have exact K41 scaling. A mechanism is suggested for avoiding large deviations, which is present in maps with a neutrally unstable fixed point.Comment: 14 pages, plain LaTex, 6 figures available upon request as hard copy (no local report #

On two-dimensionalization of three-dimensional turbulence in shell models

Applying a modified version of the Gledzer-Ohkitani-Yamada (GOY) shell model, the signatures of so-called two-dimensionalization effect of three-dimensional incompressible, homogeneous, isotropic fully developed unforced turbulence have been studied and reproduced. Within the framework of shell models we have obtained the following results: (i) progressive steepening of the energy spectrum with increased strength of the rotation, and, (ii) depletion in the energy flux of the forward forward cascade, sometimes leading to an inverse cascade. The presence of extended self-similarity and self-similar PDFs for longitudinal velocity differences are also presented for the rotating 3D turbulence case

Heavy-Tailed Distribution of Cyber-Risks

With the development of the Internet, new kinds of massive epidemics, distributed attacks, virtual conflicts and criminality have emerged. We present a study of some striking statistical properties of cyber-risks that quantify the distribution and time evolution of information risks on the Internet, to understand their mechanisms, and create opportunities to mitigate, control, predict and insure them at a global scale. First, we report an exceptionnaly stable power-law tail distribution of personal identity losses per event, ${\rm Pr}({\rm ID loss} \geq V) \sim 1/V^b$, with $b =0.7 \pm 0.1$. This result is robust against a surprising strong non-stationary growth of ID losses culminating in July 2006 followed by a more stationary phase. Moreover, this distribution is identical for different types and sizes of targeted organizations. Since $b<1$, the cumulative number of all losses over all events up to time $t$ increases faster-than-linear with time according to $\mathbf{\simeq t^{1/b}}$, suggesting that privacy, characterized by personal identities, is necessarily becoming more and more insecure. We also show the existence of a size effect, such that the largest possible ID losses per event grow faster-than-linearly as $\sim S^{1.3}$ with the organization size $S$. The small value $b \simeq 0.7$ of the power law distribution of ID losses is explained by the interplay between Zipf's law and the size effect. We also infer that compromised entities exhibit basically the same probability to incur a small or large loss.Comment: 9 pages, 3 figure

Dragon-kings: mechanisms, statistical methods and empirical evidence

This introductory article presents the special Discussion and Debate volume "From black swans to dragon-kings, is there life beyond power laws?" published in Eur. Phys. J. Special Topics in May 2012. We summarize and put in perspective the contributions into three main themes: (i) mechanisms for dragon-kings, (ii) detection of dragon-kings and statistical tests and (iii) empirical evidence in a large variety of natural and social systems. Overall, we are pleased to witness significant advances both in the introduction and clarification of underlying mechanisms and in the development of novel efficient tests that demonstrate clear evidence for the presence of dragon-kings in many systems. However, this positive view should be balanced by the fact that this remains a very delicate and difficult field, if only due to the scarcity of data as well as the extraordinary important implications with respect to hazard assessment, risk control and predictability.Comment: 20 page

Universal behavior of extreme value statistics for selected observables of dynamical systems

The main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely, up to now, on the Gnedenko approach. In this framework, extremes are basically identified with the block maxima of the time series of the chosen observable, in the limit of infinitely long blocks. It has been proved that, assuming suitable mixing conditions for the underlying dynamical systems, the extremes of a specific class of observables are distributed according to the so called Generalized Extreme Value (GEV) distribution. Direct calculations show that in the case of quasi-periodic dynamics the block maxima are not distributed according to the GEV distribution. In this paper we show that, in order to obtain a universal behaviour of the extremes, the requirement of a mixing dynamics can be relaxed if the Pareto approach is used, based upon considering the exceedances over a given threshold. Requiring that the invariant measure locally scales with a well defined exponent - the local dimension -, we show that the limiting distribution for the exceedances of the observables previously studied with the Gnedenko approach is a Generalized Pareto distribution where the parameters depends only on the local dimensions and the value of the threshold. This result allows to extend the extreme value theory for dynamical systems to the case of regular motions. We also provide connections with the results obtained with the Gnedenko approach. In order to provide further support to our findings, we present the results of numerical experiments carried out considering the well-known Chirikov standard map.Comment: 7 pages, 1 figur

Trauma Hemorrhagic Shock-Induced Lung Injury Involves a Gut-Lymph-Induced TLR4 Pathway in Mice

Injurious non-microbial factors released from the stressed gut during shocked states contribute to the development of acute lung injury (ALI) and multiple organ dysfunction syndrome (MODS). Since Toll-like receptors (TLR) act as sensors of tissue injury as well as microbial invasion and TLR4 signaling occurs in both sepsis and noninfectious models of ischemia/reperfusion (I/R) injury, we hypothesized that factors in the intestinal mesenteric lymph after trauma hemorrhagic shock (T/HS) mediate gut-induced lung injury via TLR4 activation.The concept that factors in T/HS lymph exiting the gut recreates ALI is evidenced by our findings that the infusion of porcine lymph, collected from animals subjected to global T/HS injury, into naïve wildtype (WT) mice induced lung injury. Using C3H/HeJ mice that harbor a TLR4 mutation, we found that TLR4 activation was necessary for the development of T/HS porcine lymph-induced lung injury as determined by Evan's blue dye (EBD) lung permeability and myeloperoxidase (MPO) levels as well as the induction of the injurious pulmonary iNOS response. TRIF and Myd88 deficiency fully and partially attenuated T/HS lymph-induced increases in lung permeability respectively. Additional studies in TLR2 deficient mice showed that TLR2 activation was not involved in the pathology of T/HS lymph-induced lung injury. Lastly, the lymph samples were devoid of bacteria, endotoxin and bacterial DNA and passage of lymph through an endotoxin removal column did not abrogate the ability of T/HS lymph to cause lung injury in naïve mice.Our findings suggest that non-microbial factors in the intestinal mesenteric lymph after T/HS are capable of recreating T/HS-induced lung injury via TLR4 activation