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I am trying to calculate the implied option premium $C(K)$ and $\Delta$ using the procedure outlined by Castagna and Mercurio in this paper - http://www.fabiomercurio.it/consistentfxsmile.pdf

My question is if I am given a range of option premiums for different maturities, and a swap curve for domestic and foreign rates, should I build the Vanna Volga option premium smile for a range of maturities $T$s and strikes $K$s and then interpolate to find the Vanna Volga option premium from this smile/surface or is it better to interpolate each item of my inputs to find the inputs at the correct maturity $T$ and then calculate the option premium without building a matric for a range of $T$s and $K$s.

Given

  1. risk-free rates for specific Ts from 0, 1wk, 2wk, ..., 2y etc.

  2. $\sigma$ for $ \sigma_{\Delta25p} $, $ \sigma_{\Delta25c} $, $ \sigma_{\Delta_{ATM}} $

Input

$S_0$, $K$, and $T$

Output

$C(K)$ and $\Delta$

So, to clarify I want to kow whether I should interpolate to find a specific risk-free rate and $\sigma$ for a specifc $S_0$, $K$, and $T$ first or to interpolate the smile once I have calculated the above for a range of $T$s and $K$s.

Thanks

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