Learning agents in Black-Scholes financial markets

Area of Science:

• Quantitative Finance
• Computational Economics
• Financial Market Modeling

Background:

• The Black-Scholes (BS) model is foundational for European option pricing.
• BS assumes constant volatility, which contradicts observed market behavior where volatility varies with strike price.
• The mechanism by which traders ascertain implied volatility in practice remains an open question.

Purpose of the Study:

• To introduce agent-based models that simulate how traders learn implied volatility.
• To reconcile theoretical option pricing models with empirical market observations.
• To analyze the convergence properties of traders' belief updating mechanisms.

Main Methods:

• Development of natural agent-based models for simulating trader behavior.
• Application of control theory and leader-follower model techniques.
• Analysis of opinion dynamics under different model assumptions (feedback, unknown leader).

Main Results:

• Demonstrated exponentially fast convergence in traders' opinion dynamics regarding implied volatility.
• Provided a theoretical framework that bridges the gap between the BS model's assumptions and market realities.
• Validated the models' ability to capture realistic learning processes in financial markets.

Conclusions:

• Agent-based modeling offers a robust approach to understanding implied volatility learning.
• The proposed models provide a resolution to the volatility smile/skew phenomenon observed in option markets.
• Fast convergence of opinions suggests efficient information aggregation among traders.