4,658 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
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
Self-Excited Multifractal Dynamics
We introduce the self-excited multifractal (SEMF) model, defined such that
the amplitudes of the increments of the process are expressed as exponentials
of a long memory of past increments. The principal novel feature of the model
lies in the self-excitation mechanism combined with exponential nonlinearity,
i.e. the explicit dependence of future values of the process on past ones. The
self- excitation captures the microscopic origin of the emergent endogenous
self-organization properties, such as the energy cascade in turbulent flows,
the triggering of aftershocks by previous earthquakes and the "reflexive"
interactions of financial markets. The SEMF process has all the standard
stylized facts found in financial time series, which are robust to the
specification of the parameters and the shape of the memory kernel:
multifractality, heavy tails of the distribution of increments with
intermediate asymptotics, zero correlation of the signed increments and
long-range correlation of the squared increments, the asymmetry (called
"leverage" effect) of the correlation between increments and absolute value of
the increments and statistical asymmetry under time reversal
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