Evaluating the Gamma Function in the CGMY Characteristic Function

Summary

The document addresses a numerical implementation issue in the CGMY model’s characteristic function. The questioner encounters an error because the expression includes gamma functions evaluated at negative values and asks whether the formula is incorrect. The answer explains that gamma is extended to negative non-integer arguments using its recurrence relation, so negative values alone do not make the function undefined.

The important exception is a negative integer argument, where the gamma function has a pole and is undefined; the response says the CGMY parameter values are consequently excluded at those points. This distinction helps diagnose an implementation error and identify problematic parameter inputs. The explanation is brief and does not discuss numerical stability near poles, alternative limiting formulations, or the full parameter constraints for a valid CGMY process, so it should not be treated as a complete implementation guide.

Key ideas

• The gamma function can be evaluated at negative non-integer arguments through its recurrence relation.
• The gamma function is undefined at negative integer arguments.
• CGMY parameter choices that make the gamma argument a negative integer require special care.
• The response does not cover numerical stability near these singular points.

Tags

# Characteristic function of CGMY model


# Characteristic function of CGMY model












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?

## Answer by Forgottenscience (score 1)

https://quant.stackexchange.com/a/42769

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.