Dynamic Asset Pricing in a Unified Bachelier-Black-Scholes-Merton Model

TL;DR

This paper introduces a unified asset pricing model combining Bachelier and Black-Scholes-Merton frameworks, allowing for negative prices and riskless rates, and deriving various derivative prices within this comprehensive approach.

Contribution

The paper develops a novel unified model that encompasses both Bachelier and Black-Scholes-Merton models, extending asset pricing theory to more general market conditions.

Findings

Unified model supports negative prices and riskless rates.

Option pricing varies with the choice of riskless asset used.

Model reduces to classical Black-Scholes-Merton and Bachelier limits.

Abstract

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.