Arithmetic Brownian Call Formula

This question is about pricing a European call option when the underlying follows arithmetic Brownian motion instead of the usual geometric Brownian motion. The setup is an at-the-money option with zero interest rates and no dividends, a classic Bachelier-style framework sometimes seen in interviews focused on fixed-income or low-rate environments, and in roles where understanding nonstandard dynamics is important. The candidate is asked to derive a closed-form expression for the option price under these assumptions, starting from the stochastic differential equation and working through to a compact formula.

It leans on understanding risk-neutral valuation, the normal distribution of terminal prices under arithmetic Brownian motion, and the translation of an option payoff into an expectation over a Gaussian variable. The derivation requires comfort with conditioning on a normal variable, evaluating integrals of truncated normal distributions, and manipulating standard normal cumulative and density functions. Interviewers are watching for fluency in switching from the SDE to the marginal distribution, correct use of the risk-neutral measure with zero drift, and clean, error-free calculus and algebra leading to a standard Bachelier-type expression.

What it tests

When pricing options under arithmetic Brownian motion, the key is recognizing that the terminal value of the underlying asset is normally distributed, not lognormally as in geometric Brownian motion. This means the payoff of an option can be expressed as an expectation over a normal distribution, leading to integrals that are analytically tractable. The absence of drift under the risk-neutral measure (when the risk-free rate is zero) further simplifies the distribution: the expected value remains at the current price, and only the variance grows with time. The price of an at-the-money option thus depends only on the volatility, the time to expiry, and the properties of the normal distribution, not on the level of the underlying asset. This structure is why the Bachelier formula emerges so simply for at-the-money options.