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

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  • Tushar Vaidya
  • Carlos Murguia
  • Georgios Piliouras

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.

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  • Tushar Vaidya & Carlos Murguia & Georgios Piliouras, 2017. "Learning Agents in Black-Scholes Financial Markets: Consensus Dynamics and Volatility Smiles," Papers 1704.07597, arXiv.org, revised Jul 2020.
    • Handle: RePEc:arx:papers:1704.07597
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    References listed on IDEAS

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    1. Carr, Peter & Madan, Dilip B., 2005. "A note on sufficient conditions for no arbitrage," Finance Research Letters, Elsevier, vol. 2(3), pages 125-130, September.
    2. Jacob Abernethy & Rafael M. Frongillo & Andre Wibisono, 2012. "Minimax Option Pricing Meets Black-Scholes in the Limit," Papers 1202.2585, arXiv.org.
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