Learning agents in Black–Scholes financial markets

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.

Averaging plus learning models and their asymptotics

“…To our knowledge, the setting of these types of consensus models to trading is new. We use the framework of [3,22] as the base case for our models, generalizing these models in the form…”

“…Agents (algorithms) move simultaneously: cancellations are the norm in today's fast markets. In practical terms, sequential learning models do not seem appropriate [3,4]. Interaction is important in the emergence of consensus.…”

“…To our knowledge, the setting of these types of consensus models to trading is new. We use the framework of [3,22] as the base case for our models, generalizing these models in the form xti=fi,tfalse(Xt−1false)⏟averaging+gi,tfalse(xt−1ifalse)⏟learning,where Xt is the vector of previous beliefs of n agents at time t and fi,t, gi,t determine the dynamic for the ith agent. We treat the case where f is linear and g could be allowed to be nonlinear.…”

“…In §2, we introduce the main definitions and notations. Then, in §3, we discuss the base model, an extension of the result from [22]. In §4, we introduce the models with noise.…”

A bibliometric analysis and basic model introduction of opinion dynamics

“…Much more interesting research was developed by Baum and Kernell in 2001 to detect the opinion dynamics of presidential support based on Franklin Roosevelt's public support [38]. In the support of econophysics and socio-economics, there were also pieces of research on the economic area which involve the product price [40,41], personal wealth [42], tax evasion which are proceeding papers, while the others are articles. The top 3 most cited documents with more than 1,000 citations are review, article, and proceeding paper respectively.…”