Learning Agents in Black-Scholes Financial Markets: Consensus Dynamics and Volatility Smiles
This addresses the problem of how traders learn volatility parameters in financial markets, offering a theoretical resolution that is incremental in applying existing control methods to a specific domain.
The paper tackles the discrepancy between the constant volatility assumption in the Black-Scholes model and real-world varying volatility by modeling learning agents who update beliefs based on peer opinions, proving convergence of these dynamics using control theory and leader-follower models to bridge theory and market practices.
Black-Scholes (BS) is the standard mathematical model for 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 models of learning agents, in which they update their beliefs about the true implied volatility based on the opinions of other traders. We prove 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.