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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.

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    $\begingroup$ 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. $\endgroup$ Sep 3, 2020 at 20:02
  • $\begingroup$ Ok but what interval $[a,b]$ and $N$ should I use for calibration? $\endgroup$
    – Math122
    Sep 4, 2020 at 9:58
  • $\begingroup$ 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. $\endgroup$ Sep 4, 2020 at 11:39
  • $\begingroup$ 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? $\endgroup$
    – Math122
    Sep 4, 2020 at 11:51
  • $\begingroup$ No you don't since in the re-formulation the cumulants are not contract-dependent. Don't have opinion on the book - sorry. $\endgroup$ Sep 4, 2020 at 12:11

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