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Risk-neutral valuation of options under arithmetic Brownian motions

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  • Qiang Liu
  • Shuxin Guo

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.

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  • Qiang Liu & Shuxin Guo, 2024. "Risk-neutral valuation of options under arithmetic Brownian motions," Papers 2405.11329, arXiv.org, revised Nov 2024.
    • Handle: RePEc:arx:papers:2405.11329
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    References listed on IDEAS

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    1. Goldenberg, David H., 1991. "A unified method for pricing options on diffusion processes," Journal of Financial Economics, Elsevier, vol. 29(1), pages 3-34, March.
    2. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
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