Is Kyle’s equilibrium model stable?

: In the dynamic discrete-time trading setting of Kyle (1985), we prove that Kyle’s equilibrium model is stable when there are one or two trading times. For three or more trading times, we prove that Kyle’s equilibrium is not stable. These theoretical results are proven to hold irrespectively of all Kyle’s input parameters.

1 Introduction Kyle (1985) is a cornerstone model in today's market microstructure theory.Its relevance is long established; see, e.g., the textbook discussions in Back (2017).We consider the discrete-time formulation where an informed trader, noise traders, and market makers dynamically trade the stock at N ∈ N time points.After observing the aggregate order flow, the market makers set the stock price to clear the stock market.Kyle (1985) proves existence of a unique linear equilibrium, and we study its stability properties.We prove that the number of trading time points N ∈ N determines all stability properties of Kyle's equilibrium.Specifically, irrespectively of all other input parameters, we prove that Kyle's equilibrium is stable for N ∈ {1, 2} and not stable for N ≥ 3. Hadamard (1902) deems a model well-posed if existence, uniqueness, and stability hold.Kyle (1985) gives existence and uniqueness of a linear equilibrium, and we use the convergence of policy iterations to determine if Kyle's linear equilibrium is stable. 1 We start the policy iterations from a marginal perturbation away from the insider's equilibrium orders.Then, we iteratively create a sequence of insider orders by considering the market makers' response to the insider's perturbed orders and the subsequent response by the insider and so on.Kyle's equilibrium is deemed stable if this iteratively constructed sequence of insider orders converges to Kyle's equilibrium orders whenever the initial orders are only marginally different from Kyle's equilibrium orders.This definition of fixed-point stability in terms of iterations of a marginal perturbation away from the fixed point itself is standard in numerical analysis, see, e.g., Definition 1.3 in the textbook Süli and Mayers (2003). 2  Defining stability in terms of the convergence of policy iterations is natural in the context of a financial market equilibrium because policy iterations can be viewed as the best responses of rational agents given the current state of the market.Thus, a 1 Using policy iterations to iteratively calculate optimizers is well established and is intimately related to the Bellman equation in optimal control theory, see, e.g., Theorem 3 in Chapter I.11 in Bellman (1957).
2 There are several related notions of stability.For example, stability of the fixed-point operator itself is defined in Definition 7.1 in Berinde (2007), and stability for dynamical systems (such as ODE solutions) is defined in Definition 6.1 in Betounes (2010).Stability of games is defined in Kohlberg and Mertens (1986).Robustness of games is defined Stauber (2006).
stable equilibrium has the property that if the agents find themselves in the equilibrium's vicinity, their actions draw the economy closer to equilibrium 3 .
To the best of our knowledge stability of Kyle's dynamic equilibrium model has not been studied in the literature.The closest study to our paper is Boulatov and Bernhardt (2015), who proves a robustness property for Kyle's equilibrium when N = 1.We study stability of Kyle's discrete-time model in full generality.In addition to being of theoretical interest, understanding stability properties of Kyle's model is relevant because algorithmic market makers have become an important part of asset pricing; see, e.g., Colliard, Foucault, and Lovo (2022).
We prove two theoretical results.First, when N ∈ {1, 2}, we prove that Kyle's equilibrium is always stable.Our proof establishes that the policy iterations are locally contracting near Kyle's equilibrium.Even stronger, for N = 1, we show that Kyle's equilibrium is a super-attractive fixed point in the sense that local convergence is strictly faster than linear.Second, when N ≥ 3, we prove that Kyle's equilibrium is always not stable.To provide some intuition for our results, we illustrate numerically the nonstability of Kyle's equilibrium when N = 3.We numerically compute the limit of the policy iterations when the insider's measurement of noise-trader variance slightly differ from the correct noise-trader variance.Our numerics are based on a 10th digit difference.Then, we numerically illustrate that the policy iterations converge, however, not to Kyle's equilibrium.This happens because there is an eigenvalue of the Jacobian of the policy iteration operator evaluated at Kyle's equilibrium strictly big- 3 This notion of stability can be found in financial economics literature.In DeMarzo, Kaniel, and Kremer (2008), a stable equilibrium leads to a price bubble, which means that small shocks to the agents' beliefs may result in departures from optimal risk sharing associated with typically non-stable equilibria.Biais, Foucault, and Moinas (2015) focus on stable equilibria in their study of firms investing in fast trading technologies.
ger than one (in absolute value), which implies non-stability.However, there are fixed points for which all corresponding eigenvalues are strictly less than one (in absolute value).Whenever the insider's variance estimate differs from the correct variance, the policy iterations converge numerically to such a fixed point.
While the main part of the paper is about policy iterations in the insider's control (i.e., informed stock orders), we consider an alternative in Appendix A, where we iterate the market makers' control (i.e., the pricing rule).This variation leads to the same stability conclusions in that Kyle's equilibrium is stable for N ∈ {1, 2} and not stable for N ≥ 3.
All proofs are in Appendix B. Our proofs rely on a new characterization of Kyle's equilibrium in terms of a one-dimensional fully autonomous recursion, which is independent of all model inputs.
Throughout the text, we use the symbol to transpose vectors.For example, x = (x 1 , ..., x N ) denotes a column vector in R N .For numbers, we use ... to indicate that we have excluded remaining decimals.For example, we have π = 3.14159....

Kyle's discrete-time model
This section briefly recalls the discrete-time model in Kyle (1985) with N ∈ N trading times.The noise traders' orders ∆u n at trading time n ∈ {1, ..., N } are Gaussian random variables with mean zero and variance σ 2 u ∆, where ∆ > 0 is the time step.The stock's liquidating value is denoted by ṽ, which is assumed Gaussian with mean zero and variance Σ 0 := V[ṽ] > 0. These exogenous random variables (ṽ, ∆u 1 , ..., ∆u N ) are assumed mutually independent.
At time n = 1, the insider submits orders ∆x 1 to the market makers.The orders ∆x 1 are required to be measurable with respect to σ(ṽ).At later times n ∈ {2, ..., N }, the insider submits orders ∆x n , which are required to be measurable with respect to σ(ṽ, ∆u 1 , ..., ∆u n−1 ).The aggregate orders are defined as ∆y n := ∆u n + ∆x n , n ∈ {1, ..., N }. (2.1) For a given pricing rule p n = p n (∆y 1 , ..., ∆y n ), the insider seeks orders (∆x 1 , ..., ∆x n ) that maximize her expected profit given by 2) The market makers set prices p n in the following sense.At time n ∈ {1, ..., N }, the market makers observe the aggregate orders ∆y n from (2.1) before setting the stock price as (2. 3) The next result (due to Kyle, 1985) gives existence of a linear Kyle equilibrium in the sense that items 2. and 3. in Theorem 2.1 hold.
In what follows, we will refer to β as the insider's equilibrium trading intensity as β determines how aggressively the insider trades when the market price differs from her own valuation.

Policy iterations and stability
To iteratively create a sequence of insider orders, we start with some vector The initial pricing rule is defined by where (∆x 1 , ..., ∆x N ) denote arbitrary insider orders.When she faces the pricing rule (3.2), the insider's optimal orders that maximize (2.2) are similar to (2.6), and given by In (3.3), the next policy iteration β (1) := (β N ) is computed by α N := 0 and Given the pricing rule (3.2), the orders (3.3) maximize (2.2) provided that the secondorder condition α (1) n < 1 holds.However, because Kyle's equilibrium coefficients from Theorem 2.1 satisfy αn λn < 1, a continuity argument gives that α n .We write the above policy iteration step compactly as The domain of T is given by 2. For N = 2, the function T in (3.5) is given by The domain of T is given by

♦
Based on β (1) from (3.5), we use forward recursion to iteratively construct the sequence β (2) , β (3) , ....More specifically, given the m'th policy iteration β (m) ∈ R N , the next policy iteration is defined as (3.9) We use the following definition of stability, which is based on Definition 1.3 in Süli and Mayers (2003).Definition 3.2.Kyle's equilibrium is locally stable with respect to policy iterations for the insider if there exists > 0 such that all initial policies β (0) ∈ dom(T ) with 0 where the sequence β (m) is defined recursively by (3.9).♦ In Definition 3.2, the term locally refers to the smallness condition | β (0) − β| < .Definition 5.2 below allows for more general policy iterations.
To gain some intuition of local stability, Figure 1 depicts a fictitious operator T : R → R with two fixed points.The dashed line is the 45-degree line and the intersections of the two lines correspond to T 's two fixed points.The fixed point to the left is not stable because policy iterations starting from a vicinity of this point moves away from this fixed point.This is called a repellent fixed point.At the left fixed point, the graph of T intersects the 45-degree line from below at this point indicating that T 's derivative is larger than 1.On the other hand, the fixed point to the right is stable and T 's derivative at this point is smaller than 1.This is called an attractive fixed point.In the subsequent sections, we prove that the derivative of the policy iteration operator T in (3.5) evaluated at β has a norm less than one when N ∈ {1, 2}, and a norm larger than 1 when N ≥ 3.In view of the above discussion, these norms indicate that Kyle's equilibrium is stable if and only if N ∈ {1, 2}.

T( )
Figure 1: Two fixed points of a fictitious operator T : R → R. The left fixed point is not stable and the right fixed point is stable.

One or two trading times
Let T be defined in (3.7) for N = 1 or (3.8) for N = 2.In the next result, ∇T ( β) ∈ R N ×N denotes the Jacobian matrix of T 's derivatives evaluated at β ∈ dom(T ).

Three or more trading times
Before rigorously proving that the policy iterations are not stable, we consider a numerical illustration.

Numerics
The recursive formulas in (2.4) in Theorem 2.1 show that Kyle's equilibrium β depends continuously on the input parameters ∆ > 0, σ u > 0, and Σ 0 := V[ṽ] > 0. This implies that for any > 0, we can choose δ > 0 such that the equilibrium β (0) corresponding to a marginal perturbation of the noise-trader variance σ 2 u ± δ satisfies | β − β (0) | < , where β corresponds to the the equilibrium trading intensity of the insider when the noise-trader variance equals σ 2 u .To illustrate the non-stability of Kyle's equilibrium when N = 3 and σ u := ∆ := V[ṽ] := 1, we consider a 10th digit perturbation δ := 1 10 10 of the noise-trader variance.Kyle's equilibrium corresponding to the noise-trader variance σ 2 u + δ = 1.0000000001 is given by (5.2) To illustrate the non-stability, we start the policy iterations from (5.1) in a model with σ u := ∆ := V[ṽ] := 1. Numerically, the non-linear policy iterating scheme (3.9) starting from β (0) converges to To gain some intuition for why 2 reports two sets of eigenvalues.2 shows that ||∇T ( β)|| > 1, which indicates that Kyle's equilibrium is not stable.On the other hand, the eigenvalues for ∇T ( β (∞) ) in the first row in Table 2 are smaller than one and therefore do not contradict local stability.To build an analogy with Figure 1, β (resp.β (∞) ) can be associated with the non-stable left fixed point (resp.the stable right fixed point) at which the operator has a derivative bigger (resp.less) than 1.Mathematically speaking, for N = 3, Kyle's equilibrium β is a repelling fixed point for the policy iteration operator T whereas the fixed point β (∞) is an attractive fixed point for T .

Theory
To rigorously disprove local stability when N ≥ 3, we iterate only in the third-to-last variable β N −2 whereas all other coefficients are set equal to Kyle's equilibrium values.Then, we show that the resulting policy iterations diverge, and, consequently, there is no > 0 such that (3.10) holds.To this end, we let T be from (3.5) and define the scalar function T : R → R as T 's (N − 2)'th coordinate where ( β1 , ..., βN ) are Kyle's equilibrium coefficients from Theorem 2.1 and β Theorem 5.1.Let N ≥ 3.For ∆ > 0, σ u > 0, and Σ 0 > 0, we have: 1.For any starting value β (0) ) 2. Kyle's equilibrium is not locally stable with respect to policy iterations for the insider in the sense of Definition 3.2.
Theorem 5.1 implies that policy iterations based on more general starting policies are also not locally stable in the following sense.We say stock holdings x = (x 0 , x 1 , ...., x N ) ∈ L 2 if the random variables ∆x n := and is said to be consistent with T if T • ( x) = T ( x) for all x ∈ dom(T ).Definition 5.2.Let T • be a consistent extension of T .Kyle's equilibrium is locally stable with respect to generalized policy iterations for the insider if there exists > 0 such that all starting policies x (0) ∈ dom(T where The following by-product is an immediate consequence of Theorem 5.1.
Corollary 5.3.Let N ≥ 3 and let T • be a consistent extension of T .Then, for ∆ > 0, σ u > 0, and Σ 0 > 0, Kyle's equilibrium is not locally stable with respect to generalized policy iterations for the insider in the sense of Definition 5.2.
We conclude this section by considering the related definition of an unstable equilibrium.This definition can be found in, e.g., Definition 1.3 in Süli and Mayers (2003) and differs from non locally stable equilibria (i.e., unstable and non-stable are different mathematical concepts).Definition 5.4.If β (0) = β is the only starting policy for which (3.10) holds, we say that Kyle's equilibrium is unstable with respect to policy iterations for the insider.♦ By comparing Definitions 3.2 and 5.4, we see that an unstable fixed point is always also not stable.However, Exercise 1.2 in Süli and Mayers (2003) shows that a nonstable fixed point can fail to be unstable.For N ≥ 3, our next and last theoretical result shows that while Kyle's equilibrium is not stable, it is also not unstable.
In our proof of Theorem 5.5, we marginally perturb the last equilibrium coordinate by setting β Then, we show the corresponding policy iterations converge to β.Alternatively, the iterations also converge to β when we set β (0) N −1 := βN−1 + δ and β (0) n := βn for n ∈ {1, ..., N − 2, N }.However, no matter how small a perturbation, as soon as we perturb one of the first N − 2 coordinates of β, the policy iterations do not converge to β.

Conclusion
Based on a standard notion of stability used widely in both numerical analysis and financial economics, we proved that the dynamic equilibrium model of informed trading in Kyle (1985) is stable when N ∈ {1, 2} and not stable when N ≥ 3. To investigate further the severity of non-stability, we proved that Kyle's equilibrium is not unstable when N ≥ 3. We numerically illustrated that policy iterations can converge to fixed points, which are not equilibria.Proof of Theorem 4.1.1.We consider the parametrization where γ is a point between β and β.The second equality in (B.5) follows from T ( β) = 0.4 Let ∈ (0, 1) be such that (i) < β and (ii) for β = (β 1 , ..., β N ) ∈ R N .As functions of β N −2 ∈ R alone, f is a polynomial of degree 7 and g is a polynomial of degree 6.
The chain rule gives the derivative g(z) .
equilibrium β corresponding to the noise-trader variance σ 2 u = 1 differs slightly from β (0) and is given by β