Black-Scholes-Merton Option Pricing Revisited: Did we Find a Fatal Flaw?

TL;DR

This paper critically reexamines the foundational assumptions of the Black-Scholes-Merton option pricing model, revealing a timing mistake in Merton's model that undermines key theoretical results in financial mathematics.

Contribution

It identifies a fundamental timing error in Merton's continuous-time model, challenging the validity of the Black-Scholes formula and related models.

Findings

Discovered a timing mistake in Merton's model

Invalidated key results in option pricing literature

Showed that Black-Scholes formula relies on implicit assumptions

Abstract

The option pricing formula of Black and Scholes (1973) hinges on the continuous-time self-financing condition, which is a special case of the continuous-time budget equation of Merton (1971). The self-financing condition is believed to formalize the economic concept of portfolio rebalancing without inflows or outflows of external funds, but was never formally derived in continuous time. Moreover, and even more problematically, we discover a timing mistake in the model of Merton (1971) and show that his self-financing condition is misspecified both in discrete and continuous time. Our results invalidate seminal contributions to the literature, including the budget equation of Merton (1971), the option pricing formula of Black and Scholes (1973), the continuous trading model of Harrison and Pliska (1981), and the binomial option pricing model of Cox, Ross and Rubinstein (1979). We also show that Black and Scholes (1973) and alternative derivations of their formula implicitly assumed the replication result.