236 research outputs found
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
On signal-noise decomposition of timeseries using the continuous wavelet transform: Application to sunspot index
We show that the continuous wavelet transform can provide a unique
decomposition of a timeseries in to 'signal-like' and 'noise-like' components:
From the overall wavelet spectrum two mutually independent skeleton spectra
can be extracted, allowing the separate detection and monitoring in even
non-stationary timeseries of the evolution of (a) both stable but also
transient, evolving periodicities, such as the output of low dimensional
dynamical systems and (b) scale-invariant structures, such as discontinuities,
self-similar structures or noise. An indicative application to the
monthly-averaged sunspot index reveals, apart from the well-known 11-year
periodicity, 3 of its harmonics, the 2-year periodicity (quasi-biennial
oscillation, QBO) and several more (some of which detected previously in
various solar, earth-solar connection and climate indices), here proposed being
just harmonics of the QBO, in all supporting the double-cycle solar magnetic
dynamo model (Benevolenskaya, 1998, 2000). The scale maximal spectrum reveals
the presence of 1/f fluctuations with timescales up to 1 year in the sunspot
number, indicating that the solar magnetic configurations involved in the
transient solar activity phenomena with those characteristic timescales are in
a self-organized-critical state (SOC), as previously proposed for the solar
flare occurence (Lu and Hamilton, 1991).Comment: 22 pages, 2 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
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