Risk-neutral valuation of options under arithmetic Brownian motions

TL;DR

This paper provides a comprehensive survey of options pricing under arithmetic Brownian motion, deriving formulas for European options and PDEs for American options, expanding the use of Bachelier models in finance.

Contribution

It introduces the first extensive review of ABM-based option pricing, including formulas and PDEs for various underlying types, enhancing the applicability of Bachelier models.

Findings

Derived formulas for European options under ABM

Established PDEs for American options pricing

Expanded understanding of Bachelier model applications

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.