The use of «multifractal processes» for risk calculation in finance allows us to use the concept of «scale invariance» and its implications developed in fluid mechanics…
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Studying the world’s sensitivity brings never lasting questioning which often tortures our rational and causal oriented mind. The observation and studying of seemingly very strange situations leads us to build theories that can later be used in other fields. This is indeed the case of mutifractal processes which originate mainly from the study of Kolmogorov fluid dynamics within highly developed turbulent environments. These environments are such that any obstacle which is external to the fluid injects sufficient energy for the fluid’s trajectory to become completely chaotic. Fluids are home to various sized whirlpools. Their study has shown that large whirlpools transform into smaller ones until they reach scales where viscosity become efficient and scatters their energy into heat. At differing scales of observation, the whirlpools display similar behaviour. Fluid behaviour can be described through “multiplying cascades” or by “multifractal processes”.
The use of multifractal processes for risk calculation in finance allows us to make use of the concept of “scale invariance” and its implications developed in fluid mechanics. The price of a security follows a fractal process if by enlarging the image representing this price with given diluting coefficients, we come back to a price with a similar appearance from a statistical point of view. It shall be described as multifractal if the diluting coefficients must be progressively adapted during successive zooms.
Multifractal VAR looks like an interesting alternative to commonly used modelsVincent Boisbourdain
We explain sensitivities, stress tests and VAR in this section. Sensitivities provide a relatively detailed vision and stress tests enable the evaluation of the impact which past or hypothetical crises would have nowadays. VAR provides a reasonable estimate of the maximum loss that can be incurred by a financial organism within a given time frame. Its value comes from providing a synthetic measure which will most notably be used in determining the mandatory capital reserves required in facing market risk. The most common methods used in estimating VAR are historical, parametric and Monte-Carlo. Historical VAR is based on the assumption that future market movements will be identical to past ones. Parametric VAR is built on the basis that weight representing the market follows a simple parametric law. Finally, Monte Carlo VAR is obtained by randomly simulating a set of future scenarios with loss estimation corresponding to the targeted quartile.
Evaluating probability distribution is crucial for VAR calculation. The thesis of Kozhemyak [1] shows how to use multifractal processes in order to calculate VAR with the help of a “semi parametric” method.
Financial market evolution mainly borrows two characteristics from turbulent fluids. On one hand, financial phenomena are more brutal than they initially seem and on the other hand the probability laws that define them are not stable over time. These are well known facts and financial mathematicians try to replicate them by using the “leap model” or “stochastic volatility process”. A simpler or more generic approach could be attempted with multifractal models. The latter are quite recent but are starting to attain a certain degree of maturity. At the beginning, they were founded on a particular representation of time whereas nowadays security price volatility models have appeared making them more attractive to financiers. The MRW model standing for “Multi Fractal Walks” introduced by Bacry, Delour and Muzy in 2001 [2] can be mentioned. The statistical features of the prices such as “return decorrelation” and “long term volatility auto correlation” are described by this model. We also find that the more brutal phenomena are proportionally more important when markets are observed within thinner time scales, a feature better known as Kurtosis. This model has been perfected by Bouchaud and Pochard [3] in order to take into account differing occurrence probabilities during a fall and a rise in the price of a security price. This allows us to obtain asymmetrical distributions. The whole set of statistical features which are taken into account by multifractal models then allows for a best estimation of probability density and afterwards the VAR.
During these ongoing crisis times, the current VAR models have displayed their limits. With the exception of the historical model, they are often based on Gaussian distribution hypotheses. Risk estimation through these methods which suppose that a financial market evolves like a fluid with a laminar flow does not reflect reality where distributions can be asymmetric and leptokurtic at the same time. If we take the fluid illustration once more, the market follows a turbulent regime. Relatively calm phases can be followed by whirlpools. The distributions must be estimated with a model that adapts itself naturally to turbulences. In order to increase further the reactivity of the models, it is possible to carry out high frequency measures and extrapolate the results obtained by lower frequencies by using multifractal models which allow through construction to establish the link between different times scales.
Therefore, multifractal VAR looks like an interesting alternative to the usual models. It allows for a better anticipation of the most dangerous events (through the inclusion of asymmetric and leptokurtic distributions) by displaying a self adaptation quality and great reactivity.
Vincent Boisbourdain , November 2009
Article also available in : English | français
[1] "Modélisation de séries financières à l’aide de processus invariants d’échelle. Application à la prédiction du risque" Thèse, (2006) Alexey Kozhemyak
[2] "Modelling financial time series using Multifractal Random Walks" E. Bacry, J. Delour, J.F Muzy (2001)
[3] "The skewed multifractal random walk with applications to option smiles" B. Pochard et J.P Bouchaud (2002)
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