Simulation Methods for Nonlinear and Non-Gaussian Models in Finance, Premio SIE

Financial variables, such as asset returns in international stock and bond markets or interest rates in the liquidity market, often exhibit a heterogeneous time evolution, with a unconditional density characterised by heavy tails, skewness, multimodality and time changing volatility. Through an empirical study, all these features appear clearly in some financial indexes sampled with monthly frequency and become more evident when data are collected with a higher frequency (i.e. weekly, daily or intra-day frequencies). Gaussian distribution and linear dynamic assumptions reveal unsatisfactory in many financial applications like asset pricing, risk measurement and management. Nonlinear and non-Gaussian models have been introduced in finance in order to come to more attractive results. Many stochastic models are now available as alternatives to the linear and Gaussian ones. But all of them are generally difficult to handle and represent challenging problems in applied mathematics. Some recent works (see for example Doucet, Freitas and Gordon [7], Robert and Casella [9] and Del Moral [6]) highlight the ability of the Monte Carlo simulation methods in solving optimisation and integration problems, which arise in treating complex probabilistic models and suggest moreover a Bayesian approach to optimal decision and inference making. Within the simulation based inference framework the Bayesian approach has been widely applied in many recent studies, due to the natural way the Monte Carlo approximation can enter into the inference procedure. The Bayesian framework accounts for prior information about the parameters and allows to treat complex models, such as mixtures of distributions, stochastic volatility and stochastic trend models. For an introduction to the basic and more advanced simulation methods we refer the interested reader to Robert and Casella [9], Doucet, Freitas and Gordon [7] and Liu [8].