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

## TL;DR

This paper models how traders learn implied volatility in Black-Scholes markets through opinion dynamics, proving convergence and bridging the gap between theoretical assumptions and market realities.

## Contribution

It introduces novel learning agent models for volatility estimation and proves their convergence using control theory, addressing a key gap in financial market modeling.

## Key findings

- Opinion dynamics converge under specified models
- Models explain the emergence of volatility smiles
- Bridges theory and market practice in volatility estimation

## Abstract

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.