Asset Pricing Under Deep Uncertainty: Beyond the Black-Scholes Paradigm
Abstract
We revisit foundational assumptions of options pricing theory in the context of Knightian uncertainty, proposing a robust pricing framework that accommodates model ambiguity and produces tighter empirical predictions in tail-risk scenarios.
Introduction
The Black-Scholes-Merton framework transformed financial markets by providing an analytically tractable solution to the options pricing problem. Yet its core assumptions—continuous trading, constant volatility, log-normally distributed returns—have been empirically rejected in virtually every careful empirical test conducted over the past five decades...
Theoretical Framework
We distinguish between risk, where probabilities are known, and uncertainty, where the probability distribution itself is unknown. Knight's original distinction has been formalized by various authors; our contribution is to integrate this framework directly into a no-arbitrage pricing model...
The Robust Pricing Model
Let S_t denote the underlying asset price at time t. We model the dynamics as dS_t = μ(t,S_t)dt + σ(t,S_t)dW_t where the drift μ and volatility σ are now elements of an uncertainty set Θ rather than fixed parameters...
Empirical Validation
Using options data on S&P 500 constituents over 2015–2023, we calibrate both the standard BSM model and our robust alternative. The robust model reduces out-of-sample pricing error by 23% for deep out-of-the-money puts, precisely where tail-risk concerns are most acute...