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Black-Scholes (BS) is the standard mathematical model for option pricing infinancial markets. Option prices are calculated using an analytical formulawhose main inputs are strike (at which price to exercise) and volatility. TheBS framework assumes that volatility remains constant across all strikes,however, in practice it varies. How do traders come to learn these parameters?We introduce natural models of learning agents, in which they update theirbeliefs about the true implied volatility based on the opinions of othertraders. We prove convergence of these opinion dynamics using techniques fromcontrol theory and leader-follower models, thus providing a resolution betweentheory and market practices. We allow for two different models, one withfeedback and one with an unknown leader.

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