Hey I have some problem with COS method. Here is the paper of Fang and Oosterlee https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.466.267&rep=rep1&type=pdf
Why in table 1.1 they write characteristic functions for $\ln(S_t/K)$ instead of $\ln(S_t/S_0)$ and where is this $K$? Is it hidden in $\mu$ term? And why cumulants are also calculated for $\ln(S_t/K)$? Then the interval $[a,b]$ which depends on the cumulants would be different for different values of $K$.
How to use this method for calibration procedure? What interval $[a,b]$ should I use for different models and what value of $N$? Since in calibration we are searching for model parameter we can't use cumulant to determine this interval.