Pólya-Based Approximation for the ATM-Forward Implied Volatility

We introduce a closed form approximation for the implied volatility of ATM-forward options. The relative error of this approximation is uniformly bounded for all option maturities and implied volatilities. The approximation is extremely precise, having relative error less than 10−6 for all options with integrated volatility less than 1.9, such as options with maturity less than three years and implied volatility less than 100%. Moreover, the approximate implied volatilities fall within the implied volatility bid-ask spread for all the liquid options, such as options with volatility less than 200% and maturity less than nine years