Analytic approximation for Bachelier option prices and applications

TL;DR

This paper develops an analytic approximation for Bachelier option prices using Taylor expansions, enabling variance reduction in Monte Carlo simulations, especially in correlated cases.

Contribution

It introduces a Taylor expansion-based method to approximate Bachelier option prices and applies it as a control variate for variance reduction in Monte Carlo methods.

Findings

Derived explicit expansion formulas for OTM and ITM options.

Demonstrated variance reduction in Monte Carlo simulations using the approximation.

Connected the coefficients to negative powers of future mean volatility.

Abstract

It is well-known that, in the Bachelier model, when asset prices and volatilities are uncorrelated, the implied volatility coincides with the fair value of the volatility swap. In this paper, via classical It\^o 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.