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

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.