Binary Option Market Manipulation by Influencing Belief Dynamics

TL;DR

The paper addresses how inequality among traders affects price dynamics in a binary option market by developing an information-geometric, semi-Hamiltonian model of trader beliefs and price evolution. Beliefs evolve on a Fisher-information manifold with dynamics governed by a Hamiltonian, while price dynamics provide a natural damping mechanism, yielding a structured decomposition into centre, stable, and slow manifolds in the symmetric case. Key contributions include (i) characterizing the centre manifold dimension $2N-2$ (with a 2D stable and 1D slow manifold) for identical traders, (ii) showing that symmetry breaking can decrease the centre-manifold dimension and promote convergence via an enlarged stable manifold, and (iii) exploring external information, inter-agent communication, and power inequalities that can trigger bubbles and pump-and-dump dynamics, along with an optimal price-manipulation path along information-geodesics. These results illuminate how information and power asymmetries shape market stability and offer a framework for detecting or mitigating bubbles in binary-option-like markets, including a geodesic-based manipulation strategy as a theoretical benchmark.

Abstract

Using techniques from information geometry, we construct a semi-Hamiltonian system modelling trader beliefs in a binary asset market and study the impact of inequality or asymmetry in beliefs, information, and power on price dynamics. We show that in a market with no inequality and $N$ completely symmetric traders, the resulting dynamics evolve on a $2N + 1$ dimensional manifold consisting of a $2N-2$ dimensional centre manifold, a $2$ dimensional stable manifold and a $1$ dimensional slow manifold. Introducing asymmetry into the traders has the potential to decrease the dimension of the centre manifold, which we prove using a parameter analysis. Using the belief model, we also study the impact of inter-agent communication, exogenous information and asymmetric purchasing power on price dynamics, showing that market bubbles can emerge when powerful traders produce outsize influence in the market, thus impacting other traders' beliefs as well as the price. This process is exacerbated when back-channel communication is permitted. The impact of areas of high curvature in belief space is also discussed.

Figures (14)

• Figure 1: (Left) A small market simulation consisting of five identical agents with random starting beliefs and no momentum ($\gamma_i(0) = 0$ for all $i$). The dashed line represents the starting price $p_0 = 0.5$ and the blue line is the oscillating price. (Right) A second example with the same initial momenta and starting price, but different initial conditions, showing the impact of initial conditions on agent belief and spot-price. For both plots, $m_i = k_i = Q_i = 1$.
• Figure 2: An example of the centre manifold that can occur in these dynamics using a two agent market.
• Figure 3: (Top Row) For $p_0 = \tfrac{1}{2}$, the density plot for $\left\langle p\right\rangle$ as a function $[l,u]$ shows convergence to $\tfrac{1}{2}$. Likewise, the density plot for $\mu$ shows this value is close $0$, suggesting convergence of the mean belief. The density plot for $\left\langle p - \left\langle p\right\rangle\right\rangle$ shows this value is close to zero for all starting conditions, suggesting the price converges. However, the density plot for $v$ is non-zero, suggesting the dynamics converge to the centre manifold and oscillate around their mean belief. (Bottom row) Similar results are shown for the starting price $p_0 = \tfrac{3}{4}$.
• Figure 4: (Left) Illustration of the geometry pushing the price toward $p^* = \tfrac{1}{2}$. (Right) Inconsistent price and initial conditions, converging to the basin of attraction around $p^*$.
• Figure 5: An alternate basin of attraction for two agents.