How close are the option pricing formulas of Bachelier and Black-Merton-Scholes?

TL;DR

This paper compares Bachelier's and Black-Merton-Scholes's option pricing formulas, showing they align closely both theoretically and empirically, and explains why Bachelier's model provides good short-term approximations.

Contribution

It demonstrates the close agreement between the two models and explains the effectiveness of Bachelier's approach using chaos expansion methods.

Findings

Prices of both models coincide very well

Bachelier's model provides good short-time approximations

Historical data supports the theoretical comparison

Abstract

We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we explain -- by simple methods from chaos expansion -- why Bachelier's model yields good short-time approximations of prices and volatilities.