Learning agents in black-scholes financial markets

Black-Scholes (BS) is a remarkable quotation model for European option
pricing in financial markets. Option prices are calculated using an
analytical formula whose main inputs are strike (at which price to
exercise) and volatility. The BS 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 agent-based
models, in which traders update their beliefs about the true implied
volatility based on the opinions of other traders. We prove exponentially
fast convergence of these opinion dynamics using techniques from control
theory and leader-follower models, thus providing a resolution between
theory and market practices. We allow for two different models, one with
feedback and one with an unknown leader.