Interpretation of parameters in the CGMY model

Question

I would like to understand role of parameters $C,G,M,Y$ in CGMY model, especially $G$ and $M$. The Lévy measure is $$\nu(x)=C\frac{e^{-Mx}}{x^{1+Y}}1_{x>0}+C\frac{e^{-G|x|}}{|x|^{1+Y}}1_{x<0} $$In one text I found this interpretation:

$G$ and $M$ model the decay rate of large positive and negative jumps, respectively. We can deduce that that for large values of $G$, large positive jumps become less probable, thereby increasing the occurence of small positive jumps.

Is this the correct interpretation? In my opinion it should be

$G$ and $M$ model the decay rate of large negative and positive jumps, respectively. We can deduce that that for large values of $G$, large negative jumps become less probable, thereby increasing the occurence of small negative jumps.

Can anyone explain the role of each parameter?

Answer 1

Have a look at page 311 in the original paper from Carr, Geman, Madan and Yor (2002). The paramters are for the names of the authors. They explain the role of each parameter there. Note that $C>0$, $G\geq0$, $M\geq0$ and $Y<2$.

These parameters play an important role in capturing various aspects of the stochastic process under study. The parameter $C$ may be viewed as a measure of the overall level of activity. Keeping the other parameters constant and integrating over all moves exceeding a small level, we see that the aggregrate activity level may be calibrated through movements in $C$. For example, if one were to construct a model with a stochastic aggregate activity rate, then one could model $C$ as an independent positive process, possibly following a square root law of its own. In the special case when $G=M$, the Lévy measure is symmetric, and, in this case, Madan et al. (1998) show that the parameter $C$ provides control over the kurtosis of the distribution of $X(t)$.

This point seems important to you:

The parameters $G$ and $M$, respectively, control the rate of exponential decay on the right and left of the Lévy density, leading to skewed distributions when they are unequal. For $G<M$, the left tail of the distribution for $X(t)$ is heavier than the right tail, which is consistent with the risk-neutral distribution typically implied from option prices. Thus, when $G$ and $M$ are implied from the risk-neutral distribution, their difference calibrates the price of a fall relative to a rise, while their sum measures the price of a large move relative to a small one. In contrast, in the statistical distribution, the difference between $G$ and $M$ determines the relative frequency of drops relative to rises, while their sum measures the frequency of large moves relative to small ones. The exponential factor in the numerator of the Lévy density leads to the finiteness of all moments for the process $X(t)$. As we typically construct a process at the return level, it is reasonable to enforce finiteness of the moments at this level.

Note that $C$ and $Y$ don't change if you switch between equivalent measures.