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## From Minority Game to Black & Scholes pricing

Fri, 18 May 2012 03:15:55 GMT

In this paper we study the continuum time dynamics of a stock in a market where agents behavior is modeled by a Minority Game with number of strategies for each agent S=2 and "fake" market histories. The dynamics derived is a generalized geometric Brownian motion; from the Black&Scholes formula the calibration of the Mi 660 nority Game, by means of the game parameter $\sigma^{2}$, on the European options on DAX Index market is performed. An "$(\alpha,\sigma^{2})$ -matrix" containing, given options' moneyness and maturities, values of the parameters $\alpha$ and $\sigma^{2}$ that make the theoretical option price agree with the market price is constructed. We conclude that the asymmetric phase of the Minority Game with $\alpha$ close to $\alpha_c$ is coherent with options implied volatility market.