COS Density Estimation Sensitivity to Truncation Ranges
Summary
The document describes a numerical problem in recovering probability densities with the COS method. The author compares truncation intervals built from cumulants through orders two, four, and six, and reports that the resulting densities shift substantially even when the higher-order cumulants are extremely small and the latter two intervals are nearly identical. The coefficient behavior also appears sensitive to the selected interval, with slower decay in the real part of the characteristic function for one choice.
The context is short-maturity Lévy-process densities, which can have sharp peaks and fat tails. The author is considering standard-deviation scaling and raises FFT as an alternative, while noting concern about slow characteristic-function decay and Gibbs effects. The document is a question rather than a resolved method: it offers no validated remedy and cautions that proposed approaches in earlier discussions may not transfer to complex multifactor affine models.
Key ideas
- The COS method approximates a density over a finite interval whose bounds can be selected using cumulants.
- The author reports substantial density changes across cumulant-based intervals, despite nearly matching higher-order intervals.
- Short-maturity Lévy densities may combine sharp peaks with fat tails, complicating truncation and numerical convergence.
- The document raises scaling and FFT as possibilities but does not establish a solution.
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# Probability density from COS method too sensitive to truncation range
# Probability density from COS method too sensitive to truncation range
I have a long-standing confusion around the truncation range of the COS method proposed by Fang and Oosterlee because I find that the results are highly volatile given the different truncation ranges.
To be more specific, given the following three truncation ranges (which I am aware is merely a rule of thumb): $$ [a_{2}, b_{2}]:=\left[\kappa_1-L \sqrt{\kappa_2}, \quad \kappa_1+L \sqrt{\kappa_2}\right], $$ $$ [a_{4}, b_{4}]:=\left[\kappa_1-L \sqrt{\kappa_2+\sqrt{\kappa_4}}, \quad \kappa_1+L \sqrt{\kappa_2+\sqrt{\kappa_4}}\right], $$ and $$ [a_{6}, b_{6}]:=\left[\kappa_1-L \sqrt{\kappa_2+\sqrt{\kappa_4 + \sqrt{\kappa_{6}}}}, \quad \kappa_1+L \sqrt{\kappa_2+\sqrt{\kappa_4+ \sqrt{\kappa_{6}}}}\right], $$ the recovered densities shift around significantly.
And I agree with the authors that the inclusion of high order cumulant ($\kappa_4$ and $\kappa_6$) is valid in theory because the objective density functions of many Lévy processes for short maturity $T$ (in which I use $T=1/252$), exhibit sharp peaks and fat tails. But, in my numerical example, I find that the values of $\kappa_4$ and $\kappa_6$ are extremely small, up to $10^{-21}$ and $10^{-33}$. Hence, the two truncation ranges: $[a_{4}, b_{4}]$ and $[a_{6}, b_{6}]$ are almost identical. So, intuitively speaking, the recovered densities should be similar. In contrast, my numerical results reflect that the recovered density is highly sensitive to the inclusion of high-order cumulants:
And the following graph plots the coefficient vector $\boldsymbol{A}=\left(A_{k}\right)_{k=1}^{N}$, which is calculated as: $$ A_{k}=\frac{2}{b-a}\Re\left\{\Phi\left(\frac{k\pi}{b-a}\right)\exp\left\{-i\left(\frac{ak\pi}{b-a}\right)\right\}\right\}. $$ And one can observe that the real part of the characteristic function has a relatively slow decay if the truncation range $[a_{6}, b_{6}]$ is selected.
Are there any suggestions to remedy this? I am currently working on the standard deviation (i.e. $\sqrt{\kappa_{2}}$)-scaled version of my objective density. Besides, I think FFT is another choice but the short-maturity pdf has significant fat tails. Hence, one should expect a slow decay of the real part of the characteristic function which will render the Gibbs phenomenon as pointed out by Fallahgoul et al. (2022).
P.S. Confusion around the truncation range is not new. There are several discussions on this technical topic such as Floc’h (2020) and Junike and Pankrashkin (2021). But I do not think their solutions are applicable when the model gets complicated (such as multi-factor affine).
I am willing to show the codes if necessary.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.