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This is a big industry, but here are some alternatives(as usual, the best choice depends on purpose and desired accuracy): Fit a quadratic in delta space: $\sigma_{\Delta}=a + b \left( \Delta - \Delta_{ATM} \right) + c \left( \Delta - \Delta_{ATM} \right)^2$. When you have fitted this equation, you can input delta, and the function will return ...