# Using FX ATM/RR/BF Volatility to Estimate Smile

Suppose $S$ is some FX rate, EUR/USD say, and $\sigma_{S}(K,T)$ is the implied volatility for some option written on $S$, sourced from the surface $\sigma_{S}(\cdot,\cdot)$ (alternatively, consider the implied volatility surface $(\Delta,T)\mapsto\sigma_{S}(\Delta,T)$, commonly used for FX implied volatility data).

Suppose we have the following term-structure data (maturity tenors $T$ for 1D, 1W, 2W, 3W, 1M, ..., 1Y, ..., 10Y):

1. Forward rates $F(\cdot)$
2. Risk-free zero rates $r_{d}(\cdot)$ and $r_{f}(\cdot)$ for both currencies
3. ATM volatilities $\sigma_{S}(K^{*},\cdot)$ or $\sigma_{S}(\Delta^{*},\cdot)$ (the ATM strike or delta, denoted with the asterisk, is not known)
4. $25\Delta$ "Risk Reversal" $RR(\cdot)$
5. $25\Delta$ "Butterfly" $BF(\cdot)$

Is there anyway to use (1)-(5) to approximate $\sigma_{S}(K,T)$ given the option's $(K,T)$?

(The motivation for asking this question is that I have access to (1)-(5) through an automated data import tool, but not the entire surface, and would like to price some options using the data that can be imported automatically. The valuation is not being done for trading purposes, so an absolutely precise valuation is not needed, only an estimate.)