The Hidden Boundary Zone of the Black-Scholes Formula: A Probabilistic Decomposition

Abstract

In this article, we present an alternative probabilistic decomposition of the well-known Black-Scholes formula for European call option pricing. This framework clarifies the relationship between classical option pricing theory and the Samuelson-Ito equation by introducing an intuitive expectation-based representation. Crucially, this approach exposes a hidden boundary zone within the option’s payoff space—defined as event A3—where the replication system experiences maximum tension. We explicate the exact economic and microstructure components of this zone, allowing investors and financial engineers to visually trace what they are fundamentally paying for under late-stage market uncertainty.