# Two questions about COS method

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

1. 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 $$K$$.

2. 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.

• See this paper for a re-formulation in terms of the characteristic function of $X_t = \ln \left( S_t / F_0(t) \right)$ which then also gives you strike-independent intervals. Sep 3 '20 at 20:02
• Ok but what interval $[a,b]$ and $N$ should I use for calibration? Sep 4 '20 at 9:58
• This is discussed in the original paper already and works analogously for the modification. Use a number $x$ of modified standard deviations $\sqrt{c_2 + \sqrt{\vert c_4 \vert}}$ where $c_2$ and $c_4$ are the 2nd and 4th cumulants and e.g. x = 10. Sep 4 '20 at 11:39
• I understand, so I have to set a new interval each time, thanks. There is new book of Osterlee quantfinancebook.com, do you heard about it maybe? Is it worth to buy? Sep 4 '20 at 11:51
• No you don't since in the re-formulation the cumulants are not contract-dependent. Don't have opinion on the book - sorry. Sep 4 '20 at 12:11