Measuring financial risk and portfolio reversion with time changed tempered stable Lévy processes

Abstract

Given that underlying assets in financial markets exhibit stylized facts such as leptokurtosis, asymmetry, clustering properties and heteroskedasticity effect, this paper applies the stochastic volatility models driven by tempered stable Lévy processes to construct time changed tempered stable Lévy processes (TSSV) for financial risk measurement and portfolio reversion. The TSSV model framework permits infinite activity jump behaviors of returns dynamics and time varying volatility consistently observed in financial markets by introducing time changing volatility into tempered stable processes which specially refer to normal tempered stable (NTS) distribution as well as classical tempered stable (CTS) distribution, capturing leptokurtosis, fat tailedness and asymmetry features of returns in addition to volatility clustering effect in stochastic volatility. Through employing the analytical characteristic function and fast Fourier transform (FFT) technique, the closed form formulas for probability density function (PDF) of returns, value at risk (VaR) and conditional value at risk (CVaR) can be derived. Finally, in order to forecast extreme events and volatile market, we perform empirical researches on Hangseng index to measure risks and construct portfolio based on risk adjusted reward risk stock selection criteria employing TSSV models, with the stochastic volatility normal tempered stable (NTSSV) model producing superior performances relative to others.

Hedging and pricing early-exercise options with complex fourier series expansion

“…Numerical methods based on fast Fourier transform (FFT) are traditionally very efficient at pricing options due to the existence of the 5 characteristic functions of asset price dynamics. Famous papers (e.g., Carr and Madan, 1999;Itkin, 2005;Lipton, 2002;Gong and Zhuang, 2017) provide good techniques using FFT to price European vanilla options under Lévy processes. The success of these papers has extended the use of FFT or a combination of it and other transformation methods, e.g., the Hilbert transform or Gaussian transform, to pricing exotic options (e.g., Broadie and Yamamoto, 2005;Feng and Linetsky, 2008;Jackson et al, 2008;Lord et al, 2008;Wong and Guan, 2011;Zeng and Kwok, 2014).…”

Mathematical modeling for a new portfolio selection problem in bubble condition, using a new risk measure

“…Yan [6] illustrated deviations and asymptotic behavior of convex and coherent entropic risk measures. Gong and Zhuang [7] presented measuring financial risk and portfolio reversion with time changed. They used stochastic volatility by tempered stable Lévy processes to construct time changed tempered stable Lévy processes.…”