Optimal phenotypic plasticity in a stochastic environment minimizes the cost/benefit ratio

This paper addresses the question of optimal phenotypic plasticity as a response to environmental fluctuations while optimizing the cost/benefit ratio, where the cost is energetic expense of plasticity, and benefit is fitness. The dispersion matrix \Sigma of the genes' response (H = ln|\Sigma|) is used: (i) in a numerical model as a metric of the phenotypic variance reduction in the course of fitness optimization, then (ii) in an analytical model, in order to optimize parameters under the constraint of limited energy availability. Results lead to speculate that such optimized organisms should maximize their exergy and thus the direct/indirect work they exert on the habitat. It is shown that the optimal cost/benefit ratio belongs to an interval in which differences between individuals should not substantially modify their fitness. Consequently, even in the case of an ideal population, close to the optimal plasticity, a certain level of genetic diversity should be long conserved, and a part, still to be determined, of intra-populations genetic diversity probably stem from environment fluctuations. Species confronted to monotonous factors should be less plastic than vicariant species experiencing heterogeneous environments. Analogies with the MaxEnt algorithm of E.T. Jaynes (1957) are discussed, leading to the conjecture that this method may be applied even in case of multivariate but non multinormal distributions of the responses

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, ``generic'' interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as observed recently by Bonanno et al., naturally emerge. We then propose a simple solvable ``stochastic volatility'' model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided.Comment: 21 pages, 5 figure

Causal cascade in the stock market from the ``infrared'' to the ``ultraviolet''

Modelling accurately financial price variations is an essential step underlying portfolio allocation optimization, derivative pricing and hedging, fund management and trading. The observed complex price fluctuations guide and constraint our theoretical understanding of agent interactions and of the organization of the market. The gaussian paradigm of independent normally distributed price increments has long been known to be incorrect with many attempts to improve it. Econometric nonlinear autoregressive models with conditional heteroskedasticity (ARCH) and their generalizations capture only imperfectly the volatility correlations and the fat tails of the probability distribution function (pdf) of price variations. Moreover, as far as changes in time scales are concerned, the so-called ``aggregation'' properties of these models are not easy to control. More recently, the leptokurticity of the full pdf was described by a truncated ``additive'' L\'evy flight model (TLF). Alternatively, Ghashghaie et al. proposed an analogy between price dynamics and hydrodynamic turbulence. In this letter, we use wavelets to decompose the volatility of intraday (S&P500) return data across scales. We show that when investigating two-points correlation functions of the volatility logarithms across different time scales, one reveals the existence of a causal information cascade from large scales (i.e. small frequencies, hence to vocable ``infrared'') to fine scales (``ultraviolet''). We quantify and visualize the information flux across scales. We provide a possible interpretation of our findings in terms of market dynamics.Comment: 9 pages, 3 figure

A multivariate multifractal model for return fluctuations

In this paper we briefly review the recently inrtroduced Multifractal Random Walk (MRW) that is able to reproduce most of recent empirical findings concerning financial time-series : no correlation between price variations, long-range volatility correlations and multifractal statistics. We then focus on its extension to a multivariate context in order to model portfolio behavior. Empirical estimations on real data suggest that this approach can be pertinent to account for the nature of both linear and non-linear correlation between stock returns at all time scales.Comment: To be published in the Proceeding of the APFA2 conference (Liege, Belgium, July 2000) in the journal Quantitative Financ

Volatility fingerprints of large shocks: Endogeneous versus exogeneous

Finance is about how the continuous stream of news gets incorporated into prices. But not all news have the same impact. Can one distinguish the effects of the Sept. 11, 2001 attack or of the coup against Gorbachev on Aug., 19, 1991 from financial crashes such as Oct. 1987 as well as smaller volatility bursts? Using a parsimonious autoregressive process with long-range memory defined on the logarithm of the volatility, we predict strikingly different response functions of the price volatility to great external shocks compared to what we term endogeneous shocks, i.e., which result from the cooperative accumulation of many small shocks. These predictions are remarkably well-confirmed empirically on a hierarchy of volatility shocks. Our theory allows us to classify two classes of events (endogeneous and exogeneous) with specific signatures and characteristic precursors for the endogeneous class. It also explains the origin of endogeneous shocks as the coherent accumulations of tiny bad news, and thus unify all previous explanations of large crashes including Oct. 1987.Comment: Latex document, 12 pages, 2 figure

Linear processes in high-dimension: phase space and critical properties

In this work we investigate the generic properties of a stochastic linear model in the regime of high-dimensionality. We consider in particular the Vector AutoRegressive model (VAR) and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable and an unstable phase. We find that along the transition region separating the two regimes, the correlations of the process decay slowly, and we characterize the conditions under which these slow correlations are expected to become power-laws. We check our findings with numerical simulations showing remarkable agreement with our predictions. We finally argue that real systems with a strong degree of self-interaction are naturally characterized by this type of slow relaxation of the correlations.Comment: 40 pages, 5 figure