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:

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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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