TY - JOUR
T1 - On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance
AU - Asmussen, Søren
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022/7
Y1 - 2022/7
N2 - We study the structure and properties of an infinite-activity CGMY Lévy process X with given skewness S and kurtosis K of X1, 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/ x1+Y for x> 0 and Ce −G|x|/ | x| 1+Y for x< 0. A main finding is that the quantity R= S2/ K plays a major role, and that the class of CGMY processes can be parametrised by the mean E[X1] , the variance Var [X1] , S, K and Y, where Y varies in [0 , Ymax(R)) with Ymax(R) = (2 − 3 R) / (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.
AB - We study the structure and properties of an infinite-activity CGMY Lévy process X with given skewness S and kurtosis K of X1, 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/ x1+Y for x> 0 and Ce −G|x|/ | x| 1+Y for x< 0. A main finding is that the quantity R= S2/ K plays a major role, and that the class of CGMY processes can be parametrised by the mean E[X1] , the variance Var [X1] , S, K and Y, where Y varies in [0 , Ymax(R)) with Ymax(R) = (2 − 3 R) / (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.
KW - Cumulant
KW - Exponentially tilted stable distribution
KW - Functional limit theorem
KW - Log-return distribution
KW - Moment method
KW - Wasserstein distance
UR - http://www.scopus.com/inward/record.url?scp=85132989496&partnerID=8YFLogxK
U2 - 10.1007/s00780-022-00482-x
DO - 10.1007/s00780-022-00482-x
M3 - Journal article
AN - SCOPUS:85132989496
SN - 0949-2984
VL - 26
SP - 383
EP - 416
JO - Finance and Stochastics
JF - Finance and Stochastics
IS - 3
ER -