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

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

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

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

Multifractal point processes and the spatial distribution of wildfires in French Mediterranean regions

We introduce a simple and wide class of multifractal spatial point patterns as Cox processes which intensity is multifractal, i.e., the class of Poisson processes with a stochastic intensity corresponding to a random multifractal measure. We then propose a maximum likelihood approach by means of a standard Expectation-Maximization procedure in order to estimate the distribution of these intensities at all scales. This provides, as validated on various numerical examples, a simple framework to estimate the scaling laws and therefore the multifractal properties for this class of spatial point processes. The wildfire distribution gathered in the Prom\'eth\'ee French Mediterranean wildfire database is investigated within this approach that notably allows us to compute the statistical moments associated with the spatial distribution of annual likelihood of fire event occurence. We show that for each order $q$, these moments display a well defined scaling behavior with a non-linear spectrum of scaling exponents $\zeta_q$. From our study, it thus appears that the spatial distribution of the widlfire ignition annual risk can be described by a non-trivial, multifractal singularity spectrum and that this risk cannot be reduced to providing a number of events per $km^2$. Our analysis is confirmed by a direct spatial correlation estimation of the intensity logarithms whose the peculiar slowly decreasing shape corresponds to the hallmark of multifractal cascades. The multifractal features appear to be constant over time and similar over the three regions that are studied.Comment: 41 pages, 14 figure

A multifractal random walk

We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). To our knowledge, it is the first multifractal processes with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascade-like multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by few parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.Comment: 5 pages, 4 figures, uses RevTe

Uncovering latent singularities from multifractal scaling laws in mixed asymptotic regime. Application to turbulence

In this paper we revisit an idea originally proposed by Mandelbrot about the possibility to observe ``negative dimensions'' in random multifractals. For that purpose, we define a new way to study scaling where the observation scale $\tau$ and the total sample length $L$ are respectively going to zero and to infinity. This ``mixed'' asymptotic regime is parametrized by an exponent $\chi$ that corresponds to Mandelbrot ``supersampling exponent''. In order to study the scaling exponents in the mixed regime, we use a formalism introduced in the context of the physics of disordered systems relying upon traveling wave solutions of some non-linear iteration equation. Within our approach, we show that for random multiplicative cascade models, the parameter $\chi$ can be interpreted as a negative dimension and, as anticipated by Mandelbrot, allows one to uncover the ``hidden'' negative part of the singularity spectrum, corresponding to ``latent'' singularities. We illustrate our purpose on synthetic cascade models. When applied to turbulence data, this formalism allows us to distinguish two popular phenomenological models of dissipation intermittency: We show that the mixed scaling exponents agree with a log-normal model and not with log-Poisson statistics.Comment: 4 pages, 3 figure