Risk-neutral valuation of options under arithmetic Brownian motions

Abstract

On April 22, 2020, the CME Group switched to Bachelier pricing for a group of oil futures options. The Bachelier model, or more generally the arithmetic Brownian motion (ABM), is not so widely used in finance, though. This paper provides the first comprehensive survey of options pricing under ABM. Using the risk-neutral valuation, we derive formulas for European options for three underlying types, namely an underlying that does not pay dividends, an underlying that pays a continuous dividend yield, and futures. Further, we derive Black-Scholes-Merton-like partial differential equations, which can in principle be utilized to price American options numerically via finite difference.