# Carr-madan vs COS method vs other methods

Hey during calibration we have to calculate option prices very fast. The most popular method was developed by Carr-Madan, but COS method also is very popular. The problem is for example with Variance Gamma with Stochastic arrival model and other models of this type, where we don't know the cumulants in closed form. How to calculate these cumulants in such a situation, or what other algorithm do you recommend for quick calculation of option prices?

• My workhorse is COS with an adaptive integration span and number of points. I fall back to automatic differentiation when I cannot compute the derivatives of the cumulant generating function analytically. Sep 28 '20 at 6:40
• What are the values ​​for $N$and $L$ you most often use? Sep 28 '20 at 12:52
• My default is $N = 256$ and $L = 10$ but I chose the values adaptively during every pricer invocation to account for the moneyness of the option ($L$) and how fast the characteristic function converges ($N$). Sep 28 '20 at 14:21
• How do you price options for different strike prices? I.e. in interval $[a,b]$ we have cumulants of $\log\frac{S_T}{K}$ so for different $K$ we have different interval. We get the same interval for every $K$ if we delete term $\log\frac{S_0}{K}$ from 1st cumulant. Is it okay or we shouldnt do it? Sep 29 '20 at 10:38
• @LocalVolatility what is more authors of COS Method write only about situation when $a<0<b$ but there can be also situation when $a<b<0$ and $0<a<b$ and then coefficient $V_k$ have another form. Do you include these two cases in your valuation or do you focus only on a situation when $a<0<b$? Sep 29 '20 at 17:40