Abstract
Black–Scholes (BS) is a remarkable quotation model for European 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 agent-based models, in which traders update their beliefs about the true implied volatility based on the opinions of other agents. We prove exponentially fast 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.
Highlights
Statistical Econophysics relies on data, fitting certain power laws to existing asset prices at various time scales [1,2]
The focus is on the macroscopic aggregation of interactions in the form of available data. While this is an important area of research, agent-based Econophysics offers the opportunity to study the microscopic interactions in more detail, where agents are heterogeneous
We model this feedback by introducing an extra driving term into the opinion dynamics (3.1)
Summary
Statistical Econophysics relies on data, fitting certain power laws to existing asset prices at various time scales [1,2]. In statistical Econophysics, zero-intelligence agents have random interactions. The central object of study is historical price data. The viewpoint is that interacting zero-intelligence traders’ actions are already incorporated into price fluctuations. The focus is on the macroscopic aggregation of interactions in the form of available data. While this is an important area of research, agent-based Econophysics offers the opportunity to study the microscopic interactions in more detail, where agents are heterogeneous. Our objective is to offer a cogent and clear motivation for agent-based Econophysics in the context of option volatilities, whereby learning and interaction are made explicit. We aim to take a more nuanced view of agentbased Econophysics as espoused by Chakraborti et al [5]
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