Hierarchical models for financial markets and turbulence - SOSA CORREA, William Oswaldo (2018)
In this thesis we present a study about the modeling of multiscale fluctuation phenomena and its applications to different problems in econophysics and turbulence. The thesis was organized in three parts according to the different problems considered. In the first part, we present an empirical study of the Brazilian option market in light of three option pricing models, namely the Black-Scholes model, the exponential model, and a model based on a power law distribution, the so-called q-Gaussian distribution or Tsallis distribution. It is found that the q-Gaussian performs better than BlackScholes in about one-third of the option chains analyzed. But among these cases, the exponential model performs better than the q-Gaussian in 75% of the time. The superiority of the exponential model over the q-Gaussian model is particularly impressive for options close to the expiration date. In the second part, we study a general class of hierarchical models for option pricing with stochastic volatility. We adopt the idea of an information cascade from long to short time scales, aiming to implement a hierarchical stochastic volatility model whose dynamics is described by a system of coupled stochastic differential equations. Assuming that the time scales of the different processes in the hierarchy are well separated, the stationary probability distribution for the volatility is obtained analiticaly in terms of a Meijer G-function. The option price is then computed as the average of the Black-Scholes formula over the volatility distribution, resulting in an explicit formula for the price in terms of a bivariate Meijer G-function. We also analyze the behavior of the theoretical price with the parameters of the model and we briefly compare it to empirical data from the Brazilian options market. In the third part, we study a stochastic model for the distribution of velocity increments in turbulent flows. As a basic hypothesis, we assume that the velocity increments distribution conditioned on a given energy transfer rate is a normal distribution whose variance is proportional to the energy transfer rate and whose mean depends linearly on the variance. The dynamics of the energy flux among the different scales of the hierarchy is described by a hierarchical stochastic process similar to that used in the second part of this thesis for the volatility. Therefore, the stationary distribution of the energy transfer rate is also expressed in terms of a Meijer G-function. The marginal probability distribution for the velocity increments is obtained as a statistical composition of the conditional distribution (Gaussian) with the distribution of the energy transfer rate (a G-function), which results in an asymmetric distribution written in terms of a bivariate Meijer G-function. Our model describes very well the asymmetry observed in empirical velocity increments distributions both from experimental data and numerical simulations of the Navier-Stokes equation.