Stock market crashes like those in October 1987 and October 1997, the turbulent period around the Asian Crisis in 1998 through 1999 or the burst of the “dotcom bubble” together with the severely volatile period after September 11, 2001, the market meltdown in 2008, and finally the May 6, 2010 Flash Crash are constant reminder to financial engineers and risk managers of how often extreme events actually happen in real-world financial markets. These events have led to increased efforts to improve the flexibility and statistical reliability of existing models to capture the dynamics of economic/financial variables.
In this presentation, we will discuss general frameworks (1) to measure risk, (2) to construct optimal portfolios, and (3) to price options. The economic ideas underlying the model come from three stylized facts about real-world financial markets. First, observed financial return series are asymmetric and heavy tailed, where the tails are important because bad news are tail events. The normal distribution is symmetric and has too light tails to match market data, but generally infinitely divisible distributions introduce heavier tails and skewness. Second, there is volatility clustering in time series (i.e., calm periods followed by highly volatile periods and vice versa). Finally, a dependence structure for risk factors is non-linear and asymmetric. Hence linear and symmetric correlation coefficients cannot describe the dependence structure.
In searching for an acceptable model to describe these three stylized facts, we present, in the first part, some parametric distributions with asymmetric and heavy tailed properties, including the -stable and a few subclasses of tempered stable distributions. In the second part, we present a GARCH model with infinitely divisible distributed innovation and subclasses of that GARCH model that incorporates volatility clustering together with excess leptokurtosis and asymmetry for the residual distribution.
This is a joint talk with Young Shin (Aaron) Kim, Frank Fabozzi and Boryana Racheva-Yotova.
Applied Mathematics and Statistics, Stony Brook University
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