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I am unable to reproduce the example of FX polynomial smile interpolation on page 59 of the book FX Option Pricing by Iain Clark shown below. Consider just the ATM volatility as a specific case. I calculate the following using the parameter values (c0, c1 and c2) given in section 3.9.1:

enter image description here

I can only get agreement for the ATM volatility if I subtract 0.50 from the value of $\delta(x)$ in equation 3.22. This would centre the quadratic around the forward where the moneyness is one. It also ensures that $f(0)=c_0$ so that $\sigma_X(F_{0,T})=\sigma_{ATM}$ if the ATM is forward.

Can anyone confirm that I am correct to subtract 1/2.

PS. Note that there is an erratum (from the book website) that I have fixed where the denominator in (3.22) is $\sigma_0$ rather than $\delta_0$.

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  • $\begingroup$ are you still following this post? I am also struggling with with this example and I am not able to reproduce the values of c_0, c_1 and c_2. I would like to share what I tried so far. Just let me know if you keep track of this question. :) Thanks! $\endgroup$
    – Candidate
    Dec 21, 2020 at 14:52
  • $\begingroup$ Hi - I have now implemented this in my FinancePy library and get almost exactly the same results. Here is the calling function.github.com/domokane/FinancePy/blob/master/tests/… $\endgroup$
    – Dom
    Dec 22, 2020 at 14:28
  • $\begingroup$ thank you for your reply! I will have a look at your code. What I was trying to do was to simply use the results obtained in the book - take the values of $c_0$, $c_1$, $c_2$ and plug them into the interpolated smile curve function and re-compute the values of the different sigmas (from Table 3.5). This exercise was unsuccessful with the "polynomial in delta", but I managed to obtain the figures regarding the SABR. I am using excel, unfortunately the "Goal Seek" and "Solver" can not solve for $c_0$, $c_1$, $c_2$ (taking into account the restrictions over the smile curve function) or at least I $\endgroup$
    – Candidate
    Dec 28, 2020 at 21:40
  • $\begingroup$ I was trying to replicate the example as well, I think I am facing the same issue, I saw the github link you have posted, but it looks like not available for access any more, would you mind to renew the link if you still have it on your git? $\endgroup$
    – jocas
    May 13, 2022 at 2:15
  • $\begingroup$ @jocas Try github.com/domokane/FinancePy/blob/master/tests_golden/… $\endgroup$
    – Dom
    May 14, 2022 at 12:50

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