Lévy distribution | EPFL Graph Search

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile. It is a special case of the inverse-gamma distribution. It is a stable distribution. The probability density function of the Lévy distribution over the domain is where is the location parameter and is the scale parameter. The cumulative distribution function is where is the complementary error function and is the Laplace Function (CDF of the Standard Normal Distribution). The shift parameter has the effect of shifting the curve to the right by an amount , and changing the support to the interval [, ). Like all stable distributions, the Levy distribution has a standard form f(x;0,1) which has the following property: where y is defined as The characteristic function of the Lévy distribution is given by Note that the characteristic function can also be written in the same form used for the stable distribution with and : Assuming , the nth moment of the unshifted Lévy distribution is formally defined by: which diverges for all so that the integer moments of the Lévy distribution do not exist (only some fractional moments). The moment generating function would be formally defined by: however this diverges for and is therefore not defined on an interval around zero, so the moment generating function is not defined per se. Like all stable distributions except the normal distribution, the wing of the probability density function exhibits heavy tail behavior falling off according to a power law: as which shows that Lévy is not just heavy-tailed but also fat-tailed. This is illustrated in the diagram below, in which the probability density functions for various values of c and are plotted on a log–log plot. The standard Lévy distribution satisfies the condition of being stable where are independent standard Lévy-variables with .

Les mathématiques financières robustes, le maritime et les FFA

Jérôme Reboulleau

Powerful mathematical tools have been developed for trading in stocks and bonds, but other markets that are equally important for the globalized world have to some extent been neglected. We decided to study the shipping market as an new area of development in mathematical finance. The market in shipping derivatives (FFA and FOSVA) has only been developed after 2000 and now exhibits impressive growth. Financial actors have entered the field, but it is still largely undiscovered by institutional investors. The first part of the work was to identify the characteristics of the market in shipping, i.e. the segmentation and the volatility. Because the shipping business is old-fashioned, even the leading actors on the world stage (ship owners and banks) are using macro-economic models to forecast the rates. If the macro-economic models are logical and make sense, they fail to predict. For example, the factor port congestion has been much cited during the last few years, but it is clearly very difficult to control and is simply an indicator of traffic. From our own experience it appears that most ship owners are in fact market driven and rather bad at anticipating trends. Due to their ability to capture large moves, we chose to consider Lévy processes for the underlying price process. Compared with the macro-economic approach, the main advantage is the uniform and systematic structure this imposed on the models. We get in each case a favorable result for our technology and a gain in forecasting accuracy of around 10% depending on the maturity. The global distribution is more effectively modelled and the tails of the distribution are particularly well represented. This model can be used to forecast the market but also to evaluate the risk, for example, by computing the VaR. An important limitation is the non-robustness in the estimation of the Lévy processes. The use of robust estimators reinforces the information obtained from the observed data. Because maximum likelihood estimation is not easy to compute with complex processes, we only consider some very general robust score functions to manage the technical problems. Two new class of robust estimators are suggested. These are based on the work of F. Hampel ([29]) and P. Huber ([30]) using influence functions. The main idea is to bound the maximum likelihood score function. By doing this a bias is created in the parameters estimation, which can be corrected by using a modification of the following type and as proposed by F. Hampel. The procedure for finding a robust estimating equation is thus decomposed into two consecutive steps : Subtract the bias correction and then Bound the score function. In the case of complex Lévy processes, the bias correction is difficult to compute and generally unknown. We have developed a pragmatic solution by inverting the Hampel's procedure. Bound the score function and then Correct for the bias. The price is a loss of the theoretical properties of our estimators, besides the procedure converges to maximum likelihood estimate. A second solution to for achieving robust estimation is presented. It considers the limiting case when the upper and lower bounds tend to zero and leads to B-robust estimators. Because of the complexity of the Lévy distributions, this leads to identification problems.

EPFL2008

Les mathématiques financières robustes, le maritime et les FFA

Jérôme Reboulleau

EPFL2008