Nonlinear Time Series: Models and Simulation
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Most financial time series data exhibit nonlinear features which cannot be captured by the linear models seen in the previous two chapters. In this last chapter, we present the elements of a theory of nonlinear time series adapted to financial applications. We review a set of standard econometric models which were first introduced in the discrete time setting. They include the famous, ARCH, GARCH, ... models, but we also discuss stochastic volatility models and we emphasize the differences between these concepts which are too often confused. However, because of the growing influence of the theoretical developments of continuous time finance in the everyday practice, we spend quite a significant part of the chapter analyzing the time series models derived from the discretization of continuous time stochastic differential equations. This new point of view can bring a fresh perspective. Indeed, the classical calculus based on differential equations can be used as a framework for time evolution modeling. Its stochastic extension is adapted to the requirements of modeling of uncertainty, and powerful intuition from centuries of analyses of physical and mechanical systems can be brought to bear. We examine its implications at the level of simulation. The last part of the chapter is devoted to a new set of algorithms for the filtering of nonlinear state space systems. We depart from the time honored tradition of the extended Kalman filter, and we work instead with discrete approximations called particle filters. This modern approach is consistent with our strong bias in favor of Monte Carlo simulations. We illustrate the versatility of these filtering algorithms with the example of price volatility tracking.
KeywordsInterest Rate Conditional Variance Stochastic Volatility Implied Volatility GARCH Model
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