We derive a phenomenological model of the underlying microscopic Langevin equation of the nonlinear Fokker-Planck equation, which is used to describe anomalous correlated diffusion. The resulting distributiondependent stochastic equation is then analyzed and properties such as long-time scaling and the Hurst exponent are calculated both analytically and from simulations. Results of this microscopic theory are compared with those of fractional Brownian motion.

Options are financial instruments that depend on the underlying stock. We explain their non-Gaussian fluctuations using the nonextensive thermodynamics parameter q. A generalized form of the Black-Scholes (BS) partial differential equation and some closed-form solutions are obtained. The standard BS equation (q=1) which is used by economists to calculate option prices requires multiple values of the stock volatility (known as the volatility smile). Using q=1.5 which well models the empirical distribution of returns, we get a good description of option prices using a single volatility.

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann-Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accomodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy SBG = −k i pi ln pi, and typically provides exponential laws for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form Sq = k(1 − i p q i )/(q − 1) (with S1 = SBG), and typically provides, for the same type of description, (asymptotic) power laws.Connections between dynamics and thermodynamics are far from being completely elucidated. Frequently, statistical mechanics is presented as a self-contained body, which could dispense dynamics from its formulation. This is an unfounded assumption (see, for instance, [1] and references therein). Questions still remain open even for one of its most well established equilibrium concepts, namely the Boltzmann-Gibbs (BG) factor e −Ei/kT , where E i is the energy associated with the ith microscopic state of a conservative Hamiltonian system, k is Boltzmann constant, and T the absolute temperature. For example, no theorem exists stating the necessary and sufficient conditions for the use of this celebrated and ubiquitous factor to be justified. In the mathematician F. Takens' words [2]:The values of p i are determined by the following dogma: if the energy of the system in the i th state is E i and if the temperature of the system is T then: One possible reason for this essential point having been poorly emphasized is that when dealing with short-range interacting systems, BG thermodynamical equilibrium may be formulated without much referring to the underlying dynamics of its constituents. One rarely finds in textbooks much more than a quick mention to ergodicity. A full analysis of the microscopic dynamical requirements for ergodicity to be ensured is still lacking, in spite of the pioneering studies of N. Krylov [3]. In his words:In Another possibly concomitant reason no doubt is the enormous success, since more than one century, of BG statistical mechanics for very many systems....

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Quantitative Finance, Vol. 2, No. 6, December 2002, 415-431