  We present a unified, market-complete model that integrates both the
Bachelier and Black-Scholes-Merton frameworks for asset pricing. The model
allows for the study, within a unified framework, of asset pricing in a natural
world that experiences the possibility of negative security prices or riskless
rates. In contrast to classical Black-Scholes-Merton, we show that option
pricing in the unified model displays a difference depending on whether the
replicating, self-financing portfolio uses riskless bonds or a single riskless
bank account. We derive option price formulas and extend our analysis to the
term structure of interest rates by deriving the pricing of zero-coupon bonds,
forward contracts, and futures contracts. We identify a necessary condition for
the unified model to support a perpetual derivative. Discrete binomial pricing
under the unified model is also developed. In every scenario analyzed, we show
that the unified model simplifies to the standard Black-Scholes-Merton pricing
under specific limits and provides pricing in the Bachelier model limit. We
note that the Bachelier limit within the unified model allows for positive
riskless rates. The unified model prompts us to speculate on the possibility of
a mixed multiplicative and additive deflator model for risk-neutral option
pricing.
