Analytic Approximation for Bachelier Option Prices and Applications

It is well-known that, in the Bachelier model, when asset prices and volatilities are uncorrelated, the at-the-money implied volatility coincides with the fair value of the volatility swap. Using this identity as a starting point and applying classical Itô calculus and Taylor expansions, we write the price for out-of the-money (OTM) and in-the-money (ITM) options as an expansion with respect to the moneyness, where the coefficients are related to the negative (non-integer) powers of the future mean volatility. As an a application, we use it as a control variate to reduce the variance of Monte Carlo option prices in the correlated case.