We discuss recent evidence that B. Mandelbrot's proposal to model market fluctuations as a Lévy stable process is adequate for short enough time scales, crossing over to a Brownian walk for larger time scales. We show how the reasoning of Black and Scholes should be extended to price and hedge...

We define and study a rather complex market model, inspired from the Santa Fe artificial market and the Minority Game. Agents have different strategies among which they can choose, according to their relative profitability, with the possibility of not participating to the market. The price is...

We solve exactly a simple model of trend following strategy, and obtain the analytical shape of the profit per trade distribution. This distribution is non trivial and has an option like, asymmetric structure. The degree of asymmetry depends continuously on the parameters of the strategy and on...

The low temperature physics of disordered systems is governed by the statistics of extremely low energy states. It is thus rather important to discuss the possible universality classes for extreme value statistics. We compare the usual probabilistic classification to the results of the replica...

We discuss several models in order to shed light on the origin of power-law distributions and power-law correlations in financial time series. From an empirical point of view, the exponents describing the tails of the price increments distribution and the decay of the volatility correlations are...

We study, both analytically and numerically, an ARCH-like, multiscale model of volatility, which assumes that the volatility is governed by the observed past price changes on different time scales. With a power-law distribution of time horizons, we obtain a model that captures most stylized...

We reconsider the problem of option pricing using historical probability distributions. We first discuss how the risk-minimisation scheme proposed recently is an adequate starting point under the realistic assumption that price increments are uncorrelated (but not necessarily independent) and of...

We show how one can actually take advantage of the strongly non-Gaussian nature of the fluctuations of financial assets to simplify the calculation of the Value-at-Risk of complex non linear portfolios. The resulting equations are not hard to solve numerically, and should allow fast VaR and...

It is commonly believed that the correlations between stock returns increase in high volatility periods. We investigate how much of these correlations can be explained within a simple non-Gaussian one-factor description with time independent correlations. Using surrogate data with the true...