Financial Models with Lévy Processes and Volatility Clustering

Reward-risk momentum strategies using classical tempered stable distribution

“…Additionally, we assume that the innovations of the model are generated from classical tempered stable (CTS) distribution (Rosinski (2007)) in order to model asymmetry and fattails in empirical distributions. The ARMA-GARCH-CTS model is proposed by Kim et al (2010a), Kim et al (2010b), Kim et al (2011) and there exist several applications of the model to finance (Tsuchida et al (2012); Beck et al (2013)). This paper follows the original literature.…”

“…For the CTS distribution, there exists the closed form expression for CVaR (1−η) (ǫ t+1 ) in Kim et al (2011). For a given CVaR value, it is straightforward to compute STAR-ratio and R-ratio.…”

“…After then the residuals of the ARMA-GARCH model are calculated based on the estimated ARMA-GARCH parameters. Finally, the parameters of the CTS distribution can be found from the residuals by using MLE (Rachev et al (2011)). The probability density function of the CTS distribution is obtained from the characteristic function, eq.…”

“…Similar to the tail index α, a fatter downside (upside) tail is characterized by smaller λ − (λ + ). According to Kim et al (2011), parameter estimation for the ARMA(1,1)-GARCH(1,1)-CTS model is as follows. First of all, the parameters of the ARMA-GARCH model are estimated from the maximum likelihood estimation (MLE) under an assumption of Student-t distributed residuals.…”

“…Expected returns and standard deviations obtained from the risk model are used for calculating Sharpe ratio. The parameters are also plugged to the CVaR formula for the ARMA-GARCH-CTS model given in Kim et al (2011):…”