Learning Agents in Black-Scholes Financial Markets: Consensus Dynamics\n and Volatility Smiles

2017/04/25 by Tushar Vaidya, Vaidya, Tushar, Carlos Murguia +3
Decision Sciences · Economics, Econometrics and Finance · Physics and Astronomy · #Complex Systems and Time Series Analysis #FOS: Computer and information sciences #FOS: Economics and business #Game Theory and Applications #Machine Learning (cs.LG) #Mathematical Finance (q-fin.MF) #Multiagent Systems (cs.MA) #Opinion Dynamics and Social Influence #physics.soc-ph

paper · pdf · doi:10.48550/arxiv.1704.07597

openalex publication_date 2017/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Black-Scholes (BS) is the standard mathematical model for option pricing in\nfinancial markets. Option prices are calculated using an analytical formula\nwhose main inputs are strike (at which price to exercise) and volatility. The\nBS framework assumes that volatility remains constant across all strikes,\nhowever, in practice it varies. How do traders come to learn these parameters?\nWe introduce natural models of learning agents, in which they update their\nbeliefs about the true implied volatility based on the opinions of other\ntraders. We prove convergence of these opinion dynamics using techniques from\ncontrol theory and leader-follower models, thus providing a resolution between\ntheory and market practices. We allow for two different models, one with\nfeedback and one with an unknown leader.\n

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