Quantitative Analyses on Non-Linearities in Financial Markets

"The brief market plunge was just a small indicator of how complex and chaotic, in the formal sense, these systems have become. Our nancial system is so complicated and so interactive [...]. What happened in the stock market is just a little example of how things can cascade or how technology can interact with market panic" (Ben Bernanke, IHT, May 17, 2010) One of the most important issues in economics is modeling and fore- casting the uctuations that characterize both nancial and real mar- kets, such as interest rates, commodities and stock prices, output growth, unemployment, or exchange rate. There are mainly two op- posite views concerning these economic uctuations. According to the rst one, which was the predominant thought in the 1930s, the economic system is mainly linear and stable, only randomly hit by exogenous shocks. Ragnar Frisch, Eugen Slutsky and Jan Tinbergen, to cite a few, are important exponents of this view, and they demon- strated that the uctuations observed in the real business cycle may be produced in a stable linear system subject to an external sequence of random shocks. This view has been criticized starting from the 1940s and the 1950s, since it was not able to provide a strong eco- nomic explanation of observed uctuations. Richard Goodwin,John Hicks and Nicholas Kaldor introduced a nonlinear view of the econ- omy, showing that even in absence of external shocks, uctuations might arise. The economists then suggested an alternative within the exogenous approach, at rst by using the stochastic real busi- ness cycle models (Finn E. Kidland and Edward C. Prescott, 1982) and, more recently, by the adoption of the New Keynesian Dynamic Stochastic General Equilibrium (DSGE) models, very adopted from the most important institutions and central banks. These models, however, have also been criticized for the assumption of the rational- ity of agents' behaviour, since rational expectations have been found to be systematically wrong in the business cycle. Expectations are of fundamental importance in economics and nance, since the agents' decisions about the future depends upon their expectations and their beliefs. It is in fact very unlikely that agents are perfect foresighters with rational expectations in a complex world, characterized by an irregular pattern of prices and quantities dealt in nancial markets, in which sophisticated nancial instruments are widespread. In the rst chapter of this dissertation, I will face the machine learn- ing technique, which is a nonlinear tool used for a better tting, fore- casting and clustering of dierent nancial time series and existing information in nancial markets. In particular, I will present a collec- tion of three dierent applications of these techniques, adapted from three dierent joint works: "Yield curve estimation under extreme conditions: do RBF net- works perform better?, joint with Pier Giuseppe Giribone, Marco Neelli, Marina Resta, published Anna Esposito, Marcos Faundez- Zanuy, Carlo Francesco Morabito, Eros Pasero Edrs, Multidisci- plinary Approaches to Neural Computing/Vol. 69/ WIRN 2017 and Chapter 22 in book "Neural Advances in Processing Non- linear Dynamic Signals", Springer; Interest rates term structure models and their impact on actuarial forecasting, joint with Pier Giuseppe Giribone and Marina Resta, presented at XVIII Quantitative Finance Workshop, University of Roma 3, January 2018; Applications of Kohonen Maps in financial markets: design of an automatic system for the detection of pricing anomalies, joint with Pier Giuseppe Giribone and published on Risk Management Magazine, 3-2017. In the second chapter, I will present the study A nancial market model with conrmation bias, in which nonlinearity is present as a result of the formation of heterogeneous expectations. This work is joint with Fabio Tramontana and it has been presented during the X MDEF (Dynamic Models in Economics and Finance) Workshop at University of Urbino Carlo Bo. Finally, the third chapter is a rielaboration of another joint paper, "The eects of negative nominal risk rates on the pricing of American Calls: some theoretical and numerical insights", with Pier Giuseppe Giribone and Marina Resta, published on Modern Economy 8(7), July 2017, pp 878-887. The problem of quantifying the value of early ex- ercise in an option written on equity is a complex mathematical issue that deals with continuous optimal control. In order to solve the con- tinuous dynamic optimization problem that involves high non linearity in the state variables, we have adopted a discretization scheme based on a stochastic trinomial tree. This methodology reveals a higher reliability and exibility than the traditional approaches based on approximated quasi-closed formulas in a context where financial markets are characterized by strong anomalies such as negative interest rates.