An Infinite-Dimensional Insider Trading Game

Abstract

We generalize the seminal framework of Kyle (1985) to a many-asset setting, bridging the gap between informed-trading theory and modern trading practices. Specifically, we formulate an infinite-dimensional Bayesian trading game in which the informed trader's private information may concern arbitrary aspects of the cross-sectional payoff structure across a continuum of traded assets. In this general setting, we obtain a parsimonious equilibrium characterized by a single scalar fixed point, yielding closed-form characterizations of equilibrium trading strategy, price impact within and across markets, and the informational efficiency of equilibrium prices.

Figures (5)

• Figure 1: Vertical Spreads (Example \ref{['example: Kyle with options']}) Panels (a,c) plot the signal-contingent payoff densities $\eta(\,\cdot\,,s_i)$ (solid, shaded) against the prior (dashed). Panels (b,d) plot the corresponding equilibrium insider strategies $W^*(\,\cdot\,,s_i)$: the strategies flip sign across $s_1$ and $s_2$ and are well-approximated by the standard bull/bear vertical call spread payoff.
• Figure 2: $\;$ Straddle/Butterfly (Example \ref{['example: vol straddle']}) The left column shows the insider's private signal (filled densities) against the market maker's prior (dashed). The right column shows the insider portfolio $W^*$ conditional on the corresponding signals. Price Impact Three states are indicated---$x$, $y$, $z$. Security $y$ has zero price impact on all securities, $\Lambda_{y,w} = 0$ for all $w$, because its payoff has zero variation across signals, $\eta(y, s_1) = \eta(y, s_2)$. Therefore, the informed demand for $y$ must be zero---$W^*(y, s_1) = W^*(y, s_2) = 0$ as shown in Figures \ref{['fig: Binary_Signal_AD_demand_s2_observed, vol']} and \ref{['fig: Binary_Signal_AD_demand_s1_observed, vol']}. $\Lambda_{x,z} > 0$ because $x$ and $z$ have positively correlated (in fact, identical) payoffs.
• Figure 3: $\;$ Ratio Spreads (Example \ref{['example: skewness']}). The left column shows the insider's private signal (filled densities) against the market maker's prior (dashed). The right column shows the insider portfolio $W^*$ conditional on the corresponding signals. Panel (b) is a risk reversal ($s_1$) and panel (d) is a put spread ($s_2$).
• Figure 4: Sensitivity Analysis - Market Maker's Posterior (finite-$S$ illustration) ($x$-axis - $\{ s_1, \cdots, s_I \}$, $y$-axis - probability) For the finite-signal case $S=\{s_1,\dots,s_I\}$ with uniform prior, these graphs show the market maker's expected posterior probabilities on $S$ conditional on the insider observing signal $s_1$.
• Figure 5: Sensitivity Analysis: Information Efficiency (finite-$S$ illustration). In the finite-signal case $S=\{s_1,\dots,s_I\}$ with uniform prior, let $q^{(i)}=(q^{(i)}_1,\dots,q^{(i)}_I)$ denote the market maker's equilibrium posterior weights conditional on $S=s_i$, so that $\mathrm{IE}(s_i)=\mathbb{E}[q_i^{(i)}]$. The figure plots $\mathrm{IE}(s_i)$ as a function of $I$ (by symmetry, independent of $i$); the expectation is taken under the equilibrium law of the posterior induced by the symmetric equilibrium with intensity $\alpha^*=\alpha^*(I)$.

Theorems & Definitions (47)

• Remark 2.1
• Definition 1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Definition 2
• Remark 4.1