Many of the concepts in theoretical and empirical finance developed over the past decades - including the classical portfolio theory, the Black-Scholes-Merton option pricing model or the RiskMetrics variance-covariance approach to VaR - rest upon the assumption that asset returns follow a normal distribution. But this assumption is not justified by empirical data! Rather, the empirical observations exhibit excess kurtosis, more colloquially known as fat tails or heavy tails. This chapter is intended as a guide to heavy-tailed models. We first describe the historically oldest heavy-tailed model - the stable laws. Next, we briefly characterize their recent lighter-tailed generalizations, the socalled truncated and tempered stable distributions. Then we study the class of generalized hyperbolic laws, which - like tempered stable distributions - can be classified somewhere between infinite variance stable laws and the Gaussian distribution. Finally, we provide numerical examples.
heavy-tailed distribution stable distribution tempered stable distribution generalized hyperbolic distribution asset return random number generation parameter estimation