From Minority Game to Black & Scholes pricing

TL;DR

This paper models stock market dynamics using Minority Game frameworks, deriving a generalized geometric Brownian motion and calibrating it with Black-Scholes, revealing insights into implied volatility behaviors.

Contribution

It introduces a continuum time model based on Minority Games, linking agent behavior to market dynamics and option pricing.

Findings

Generalized geometric Brownian motion derived from Minority Game dynamics

Calibration aligns model with implied volatility market data

Asymmetric phase near critical parameters matches observed market volatility

Abstract

In this paper we study the continuum time dynamics of a stock in a market where agents behavior is modeled by a Minority Game and a Grand Canonical Minority Game. The dynamics derived is a generalized geometric Brownian motion; from the Black & Scholes formula the calibration of both the Minority Game and the Grand Canonical Minority Game, by means of their characteristic parameters, is performed. We conclude that for both games the asymmetric phase with characteristic parameters close to critical ones is coherent with options implied volatility market.