We introduce a multivariate Hawkes process that accounts for the dynamics of market prices through the impact of market order arrivals at microstructural level. Our model is a point process mainly characterized by 4 kernels associated with respectively the trade arrival self-excitation, the price changes mean reversion the impact of trade arrivals on price variations and the feedback of price changes on trading activity. It allows one to account for both stylized facts of market prices microstructure
(including random time arrival of price moves, discrete price grid, high frequency mean reversion, correlation functions behavior at various time scales) and the stylized facts of market impact (mainly the concave-square-root-like/relaxation characteristic shape of the market impact of a meta-order). Moreover, it allows one to estimate the entire market impact profile from anonymous market data. We show that these kernels can be empirically estimated from the empirical conditional mean intensities. We provide numerical examples, application to real data and comparisons to former approaches.
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In this paper we consider the problem of the existence of some large correlation (integral) scale in random cascade models. We propose a new model that possesses multifractal properties without involving any integral scale. This model relies on a non stationary log-normal process which proper- ties, over any finite time interval, are very close to continuous cascade models. These latter models are notably well known to reproduce faithfully the main stylized fact of financial time series but the integral scale where the cascade is initiated is hard to interpret. Moreover the reported empirical values of this large scale turn out to be closely correlated to the overall length of the sample. As illustrated by the example of Dow-Jones index, this feature is precisely predicted by our model.
We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time in- terval [0,T] in the limit T → ∞. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh \Delta over [0,T] up to some further time shift τ. The behaviour of this functional depends on the relative size of \Delta and τ with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in a micro- scopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce important empirical stylised fact such as the Epps effect and the lead-lag effect. Moreover, our approach enable to track these effects across scales in rigorous mathematical terms.
We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method.
We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.
We present the construction of a continuous time stochastic process which has moments that satisfy an exact scaling relation, including odd order moments. It is based on a natural extension of the MRW construction. This allows us to propose a continuous time model for the price of a financial asset that reflects most major stylized facts observed on real data, including asymmetry and multifractal scaling.
We introduce a new stochastic model for the variation of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on linear self and mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e., strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect ({\it i.e.} the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales.
More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures BUND 10Y and BOBL 5Y in several situations.
By studying all the trades and best bids/asks of ultra high frequency snapshots recorded from the order books of a basket of 10 futures assets, we bring qualitative empirical evidence that the impact of a single trade
depends on the intertrade time lags. We find that when the trading rate becomes faster, the return variance per trade or the impact, as measured by the price variation in the direction of the trade, strongly increases. We provide evidence that these properties persist at coarser time scales. We also show that the spread value is an increasing function of the activity. This suggests that order books are more likely empty when the trading rate is high.
Hawkes processes are used for modeling tick-by-tick variations of a single or of a pair of asset prices. For each asset, two counting processes (with stochastic intensities) are associated respectively with the positive and negative jumps of the price. We show that, by coupling these two intensities, one can reproduce high-frequency mean reversion structure that is characteristic of the microstructure noise. Moreover, in the case of two assets, by coupling the stochastic intensities corresponding to the positive (resp. negative) jumps of each asset, we are able to reproduce the Epps effect, i.e., the decorrelation of the increments at microscopic scales.
At large scale our model becomes diffusive and converge towards a standard Brownian motion. Analytical closed-form formulae for the mean signature plot, the diffusive correlation matrix and the cross-asset correlation function at any time-scale are given. Empirical results are shown on futures Euro-Bund and Euro-Bobl high frequency data.
In this paper, we make a short overview of multifractal models of asset returns. All the proposed models rely upon the notion of random multiplicative cascades. We focus in more details on the simplest of such models namely the log-normal Multifractal Random Walk. This model can be seen as a stochastic volatility model where the (log-) volatility has a peculiar long-range correlated memory. We briefly address calibration issues of such models and their applications to volatility and VaR forecasting.
Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of ”mixed” partitions functions i.e., the estimator of the cumulant generating function of the cascade generator distribution, depends on some “mixed asymptotic” exponent χ respectively above and beyond two critical value. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. These results are shown to remain valid within a general wavelet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreover, within the mixed asymptotic framework, we establish a “box-counting” multifractal formalism that can be seen as a rigorous formulation of Mandelbrot’s negative dimension theory. Numerical illustrations of our purpose on specific examples are also
provided.
Log-normal continuous random cascades form a class of multifractal processes that has already been successfully used in various fields. Several statistical issues related to this model are studied. We first make a quick but extensive review of their main properties and show that most of these properties can be analytically studied. We then develop an approximation theory of these processes in the limit of small intermittency, i.e., when the degree of multifractality is small. This allows us to prove that the probability distributions associated with these processes possess some very simple aggregation properties accross time-scales. Such a control of the process properties at different time-scales, allows us to address the problem of parameter estimation. We show that one has to distinguish two different asymptotic regimes: the first one, referred to as the ”low frequency regime”, corresponds to taking a sample whose overall size increases whereas the second one, referred to as the ”high frequency regime”, corresponds to sampling the process at an increasing sampling rate. We show that, the first regime leads to convergent estimators whereas, in the high frequency regime, the situation is much more intricate: only the intermittency coefficient can be estimated using a consistent estimator. However, we show that, in practical situations, one candetect the nature of the asymptotic regime (low frequency versus high frequency) and consequently decide whether the estimations of the other parameters are reliable or not. We finally illustrate how both our results on parameter estimation and on aggregation properties, allow one to successfully use these models for modelization and prediction of financial time series.
In this note, we present results on the behavior for the partition function of multiplicative cascades in the case where the total time of observation is large, compared to the scale of decay for the correlation of the cascade process.
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 τ and the total sample length L are respectively going to zero and to infinity. This “mixed” asymptotic regime is parametrized by an exponent χ 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 χ 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.
In this paper, we make a short overview of continuous time
multifractal processes recently introduced to model asset return
fluctuations. We show that these models account in a very parcimonious
manner for most of ``stylized facts'' of financial time series.
We review in more details the simplest of such models namely
the log-normal Multifractal Random Walk. It can simply be considered
as a stochastic volatility model where the (log-) volatility memory has
a peculiar ''logarithmic'' shape. This model possesses some
appealing stability properties as respect to time aggregation.
We describe how one can
estimate it using a GMM method and we present some applications to
volatility and VaR forecasting.
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Removing noise from audio signals requires a non-diagonal processing of time-frequency coefficients to avoid producing ``musical noise''. State of the art algorithms perform a parameterized filtering of spectrogram coefficients with empirically fixed parameters. A block thresholding estimation procedure is introduced, which adjusts all parameters adaptively to signal property by minimizing a Stein estimation of the risk. Numerical experiments demonstrate the performance and robustness of this procedure through objective and subjective evaluations.
We investigate a new audio denoising algorithm. Complex wavelets protect phase of signals and are thus preferred in audio signal processing to real wavelets. The block attenuation eliminates the residual noise artifacts in reconstructed signals and provides a good approximation of the attenuation with oracle. A connection between the block attenuation and the decision-directed \textit{a priori} SNR estimator of Ephraim and Malah is studied. Finally we introduce an adaptive block technique based on the dyadic CART algorithm. The experiments show that not only the proposed method does eliminate the residual noise artifacts, but it also preserves transients of signals better than short-time Fourier based methods do.
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We discuss a possible scenario explaining in what respect the
observed fat tails of asset returns or volatility fluctuations
can be related to volatility long-range correlations. Our approach is based
on recently introduced multifractal models for asset returns that account for
the volatility correlations
through a multiplicative random cascade. Within the framework of these models,
it can be shown that the sample size required for a correct estimation of
the behavior of extreme return fluctuations is generally huge and outside
the range of accessible size of data.
Consequently, in many cases, the extreme tail probability appears as a power-law,
with a rather small (underestimated) tail exponent.
We point out that increasing the amount of data by using smaller and smaller (intraday) scales,
does not contribute to reduce the bias and, as observed empirically, the tail exponent turns
out to be rather stable across scales.
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In this paper we discuss the problem of the estimation of
extreme event occurrence probability for data drawn from some
multifractal process. We also study
the heavy (power-law) tail behavior of probability density function
associated with such data.
We show that because of strong correlations, standard extreme value approach
is not valid and classical tail exponent estimators should be interpreted
cautiously. Extreme statistics associated with multifractal
random processes turn out to be characterized by non self-averaging properties.
Our considerations rely upon some analogy between random
multiplicative cascades and the physics of disordered systems
and also on recent mathematical results about the so-called
multifractal formalism.
Applied to financial time series, our
findings allow us to propose an unified framemork
that accounts for the observed
multiscaling properties of return fluctuations, the volatility
clustering phenomenon and the observed ``inverse cubic law'' of the
return pdf tails.
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We define a large class of multifractal random measures and processes with
arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes
generalize within a unified framework both the recently defined log-normal Multifractal
Random Walk processes (MRW) (Bacry etal.) and the log-Poisson ``product of cynlindrical
pulses" (Barral and Mandelbrot). Their construction involves some ``continuous
stochastic multiplication'' (Schmitt and Marsan) from coarse to fine scales.
They are obtained as limit processes when the finest scale goes to zero. We
prove the existence of these limits and we study their main statistical properties
including non degeneracy, convergence of the moments and multifractal scaling.
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We introduce a dictionary of elementary waveforms, called harmonic atoms
that extends the Gabor dictionary and fits well the natural harmonic structures
of audio signals. By modifying the ``standard'' matching pursuit, we define
a new pursuit along with a fast algorithm, namely the Fast Harmonic Matching
Pursuit, to approximate N-dimensional audio signals with a linear combination
of M harmonic atoms. Our algorithm has a computational complexity of
O(MKN), where K is the number of partials in a given harmonic
atom. The decomposition method is demonstrated on musical recordings, and we
describe a simple note detection algorithm that shows how one could use a harmonic
matching pursuit to detect notes even in difficult situations, e.g., very different
note durations, lots of reverberation, and overlapping notes.
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We define a large class of continuous time multifractal random measures and
processes with arbitrary log-infinitely divisible exact or asymptotic scaling
law. These processes generalize within a unified framework both the recently
defined log-normal Multifractal Random Walk (MRW) (Bacry et.al) and the log-Poisson
"product of cynlindrical pulses" (Barral and Mandelbrot). Our construction is
based on some "continuous stochastic multiplication" from coarse to fine scales
that can be seen as a continuous interpolation of discrete multiplicative cascades.
We prove the stochastic convergence of the defined processes and study their
main statistical properties. The question of genericity (universality) of limit
multifractal processes is addressed within this new framework. We finally provide
some methods for numerical simulations and discuss some specific examples.
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