Uncertainty theory vs. stochastic models in option pricing: a comparative study on risk and hedging
image: The left side represents the theoretical framework; the top middle contains a labeled box with a circumscribed circle displaying the call and put option prices (c, p), as well as the delta and vega risks (δ, ν), measured in a generic currency ¤. Near the center, the double symbol □ represents the functions that process input information with the comparison criteria to provide the outputs, represented by dashed arrows. A particular output is depicted on the right.
Credit: Carlos Alexander Grajales (University of Antioquia, Colombia) Santiago Medina Hurtado (National University of Colombia, Colombia) Samuel Mongrut (Universidad del Pacifico, Peru)
Background and Motivation
Human financial decision-making often operates under conditions of incomplete information or profound uncertainty. While the classical stochastic approach (exemplified by the Black-Scholes model) relies on historical data and probability distributions, it may falter in turbulent markets, emerging sectors with scant data, or when future disruptions render the past unreliable. Uncertainty theory, pioneered by Liu, offers an alternative by mathematically modelling human belief degrees based on expert judgment. This study directly compares these two foundational frameworks for pricing European options and measuring their key risk sensitivities (delta and vega), addressing a critical gap in understanding how derivative valuation and risk management differ when shifting from a data-driven probabilistic view to an uncertain expert perspective.
Methodology and Scope
The researchers develop formulas to estimate delta (δ) and vega (ν) risks within the uncertain Liu stock model framework. They then conduct a comprehensive numerical experiment comparing prices and risks against the benchmark stochastic Black-Scholes model. The comparison uses a broad four-dimensional grid of parameters: strike price (K), time to maturity (T), expert log-drift (e, reflecting preference/profitability), and log-diffusion (σ, analogous to volatility). Two robust criteria are employed: a relative equality test to identify significant differences and matrix norm metrics (Frobenius, 1-norm, ∞-norm) to quantify the distances between model outputs across thousands of parameter combinations. The study also designs a static delta-hedging experiment to illustrate practical implications.
Key Findings and Contributions
- Significant Divergence: Prices and risks for European call options differ substantially between the uncertain and stochastic models across most parameter scenarios.
- Costly Young OTM Options: Under the uncertainty framework, young (short-maturity) out-of-the-money (OTM) call options are notably more expensive and riskier than their stochastic counterparts.
- Primacy of Expert Preference: The expert log-drift parameter (e), representing belief about profitability, has a more pronounced impact on option premiums and risks than the volatility-like log-diffusion (σ). This highlights the critical role of subjective judgment in uncertain valuation.
- Sensitivity to Maturity and Strikes: The differences between models are most sensitive to changes in the option's time to maturity, with the largest gap occurring for longer-dated options. Differences are also observable across all strike prices.
- Hedging Implications: A static delta-hedging strategy constructed under the uncertain framework, while potentially more expensive to set up, can exhibit smaller portfolio value fluctuations in response to asset price changes, suggesting a possible robust advantage in certain conditions.
Why It Matters
This research provides a crucial, systematic comparison between two competing paradigms in quantitative finance. The findings validate uncertainty theory as a viable and structurally different framework for derivatives pricing, especially relevant for emerging markets, innovative financial products with no trading history, or periods of market stress where historical patterns break down. It demonstrates that incorporating expert views systematically leads to materially different risk assessments and hedging costs, challenging the universality of traditional stochastic methods and offering a formal toolkit for scenarios they inadequately address.
Practical Applications
- For Traders & Risk Managers: In environments with limited data or highly expected turbulence, considering the uncertainty framework can provide a broader, more precautionary view of option costs and risks, particularly for short-dated OTM contracts. It offers a method to quantify and integrate expert views into formal risk metrics (Greeks).
- For Financial Innovation: Designers of new derivatives or participants in nascent markets can use the uncertain approach to price and hedge products where historical calibration is impossible, relying instead on calibrated expert judgment.
- For FinTech & AI Development: The formalised "facts" and methods from this study can serve as foundational principles for developing next-generation AI-driven risk management systems that intelligently incorporate both data and expert belief in uncertain environments.
Discover high-quality academic insights in finance from this article published in China Finance Review International. Click the DOI below to read the full-text!