I have a basic question about the CGMY model which has characteristic function

$$ \Gamma(-Y_p)\left((M-iu)^{Y_p}-M^{Y_p}\right)+\frac{C_n}{C_p}\Gamma(-Y_n)\left((G+iu)^{Y_n}-G^{Y_n}\right) $$

whith $Y_p<2$ and $Y_n<2$. However, when I implement pricing with this characteristic function I get thrown an error as the gamma function is not defined for negative values. So is there an error in the classical characteristif function of the CGMY model?


1 Answer 1


Y in the CGMY model is not defined for negative integer values due to divergence of the gamma function at those values, and implicitly the characteristic function. However, in the case of negative non-integer $x$ we extend the gamma function in the sense that whenever $x \in (-\infty,0) \setminus \mathbb{Z_{-}}$, we define the value of $\Gamma(x)$ via the trivial identity $\Gamma(x) = \frac{\Gamma(x+1)}{x}$. You can see some discussion on the values of $Y$ and their interpretation in [1].

[1]: Fiorani, F. (2004). Option pricing under the variance gamma process.


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