On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance

We study the structure and properties of an infinite-activity CGMY Lévy process X with given skewness S and kurtosis K of X_1 , without a Brownian component, but allowing a drift component. The jump part of such a process is specified by the Lévy density which is Ce^-Mx/x^1+Y for x>0 and Ce^-G|x|/|x|^1+Y for x<0 . A main finding is that the quantity R=S^2/K plays a major role, and that the class of CGMY processes can be parametrised by the mean 𝔼[X_1] , the variance Var[X_1] , S , K and Y , where Y varies in [0,Y_max(R)) with Y_max(R)=(2-3R)/(1-R) . Limit theorems for X are given in various settings, with particular attention to X approaching a Brownian motion with drift, corresponding to the Black–Scholes model; for this, sufficient conditions in a general Lévy process setup are that K→ 0 or, in the spectrally positive case, that S→ 0 . Implications for moment fitting of log-return data are discussed. The paper also exploits the structure of spectrally positive CGMY processes as exponential tiltings (Esscher transforms) of stable processes, with the purpose of providing simple formulas for the log-return density f(x) , short derivations of its asymptotic form, and quick algorithms for simulation and maximum likelihood estimation.