5. Illiquidity of Equity

Jun He May 31, 2026

公司金融Corporate Finance 流动性Liquidity 信息不对称Asymmetric Information 分离均衡Separating Equilibrium 证券设计Security Design 内幕交易Insider Trading Kyle模型Kyle Model 学习笔记Study Note

Note

股权非流动性：股东无法以任意数量出售股权而不引致价格下跌。三个模型从不同视角解释：§5.1 Leland-Pyle (1977) 部分出售——风险厌恶 CARA 代理人私知资产质量 $\mu$，在分离均衡中通过出售比例 $\alpha$ 揭示质量，价格 $p(\alpha)=\underline\mu+r\sigma^2(\alpha-\ln\alpha-1)$ (5.5)，$p'(\alpha)<0$（卖得越多=类型越差），用于 IPO 与 IPO 折价（Rock 1986 赢者诅咒）。§5.2 DeMarzo-Duffie (1999) 证券设计——代理人有持有成本 $\delta<1$，分离均衡价 $P(q)=\mathbb E[\tilde X\mid Z_0]q^{\delta-1}$ (5.9)；在单调证券 + 一致最坏情形假设下，债务 $\min\{\tilde X,d\}$ 最优（最大化代理人利润），即分级 (tranching) 对发行人有利。§5.3 Kyle (1985) 内幕交易 + 做市商——三方（知情者/噪声交易者/做市商），线性均衡 $P(y)=\mu+\lambda y$、$X(v)=\alpha+\beta v$，$\lambda\beta=\tfrac12$ (5.15)，做市商 Kalman 滤波定价 (5.16)，后验方差 $\sigma_1^2=\tfrac12\sigma_0^2$ (5.18)——交易把一半私有信息泄露给市场，且与噪声方差无关（噪声越大知情者越激进、$\beta$ 越高）。

Note

Illiquidity of equity: shareholders cannot sell equity at any amount they like without incurring a price drop. Three models explain it from different perspectives: §5.1 Leland-Pyle (1977) selling in fraction — a risk-averse CARA agent privately knows asset quality $\mu$ and, in a separating equilibrium, reveals it through the selling fraction $\alpha$, with price $p(\alpha)=\underline\mu+r\sigma^2(\alpha-\ln\alpha-1)$ (5.5), $p'(\alpha)<0$ (selling more = worse type), used for IPOs and IPO under-pricing (Rock 1986 winner's curse). §5.2 DeMarzo-Duffie (1999) security design — the agent has a holding cost $\delta<1$, separating-equilibrium price $P(q)=\mathbb E[\tilde X\mid Z_0]q^{\delta-1}$ (5.9); under monotone security + uniform worst-case assumptions, debt $\min\{\tilde X,d\}$ is optimal (maximizes the agent's profit), i.e. tranching is good for the issuer. §5.3 Kyle (1985) insider trading + market maker — three parties (informed / noise traders / market maker), linear equilibrium $P(y)=\mu+\lambda y$, $X(v)=\alpha+\beta v$, $\lambda\beta=\tfrac12$ (5.15), market-maker Kalman-filter pricing (5.16), posterior variance $\sigma_1^2=\tfrac12\sigma_0^2$ (5.18) — trading discloses half of the private information to the market, independent of noise variance (more noise → more aggressive insider, higher $\beta$).

5.1 Equity Selling in Fraction: Leland and Pyle (1977)

5.1.1 Setup

两期：$t=0$ 资产交易，$t=1$ 收益实现，无贴现。
风险厌恶代理人拥有一资产，效用 $u(\cdot)$ 严格凹。本节用 CARA（指数）效用 $u(x)=-e^{-rx}$（$r>0$）。
资产在 $t=1$ 产生收益 $\tilde x\sim\mathcal N(\mu,\sigma^2)$。代理人有信号 $\theta$（按 $i$ 索引不同私有信息），信号越高资产越好；对信号 $\theta_i$ 的代理人，资产期望收益为 $\mathbb E[\tilde x\mid\theta_i]$。
代理人可将资产的 $\alpha\in[0,1]$ 比例卖给风险中性市场投资者。
因代理人风险厌恶、投资者风险中性，一阶最优应是代理人把全部股权卖给市场。然而由于信息不对称，在混同均衡 (pooling) 中市场投资者只愿为整个资产支付 $\mathbb E[\tilde x]$，对有好信号的代理人而言太低。

5.1.2 The Model

聚焦分离均衡 (separating equilibrium)（不同质量以不同比例出售，完全揭示质量）。
市场投资者在观察出售比例 $\alpha$ 后，将资产质量的先验更新为后验 $g(\theta\mid\alpha)$。
分离均衡中，投资者通过观察 $\alpha$ 可确切识别 $\theta$，并据此定价 $p(\alpha)=\mathbb E_\theta[\tilde x\mid\alpha]$（竞争市场，投资者回本）。
代理人 IC：真实信号为 $\theta_i$ 的代理人若卖 $\alpha_j$ 使市场相信他是 $\theta_j$，则期望（终端财富）为 $V(\alpha_j\mid\theta_i)=\mathbb E[(1-\alpha_j)\tilde x+\alpha_j p(\alpha_j)\mid\theta_i]$；分离均衡 IC 要求真实类型 $\theta_i$ 选择的 $\alpha_i$ 最优地揭示其类型。

5.1.3 Solve the problem

设信号即收益分布 $\theta=\mu$，假设 $\mu\in[\underline\mu,\infty)$。分离均衡中 $\alpha^\star(\mu)$ 是一一映射，市场定价 $p(\alpha)$ 也是一一映射。先验地知道 $p'(\alpha)<0$：若 $p'(\alpha)>0$，任何质量都想尽量多卖以获更高价，结果所有类型 $\alpha=1$，无法支撑分离均衡，矛盾。给定 $p(\alpha)$，代理人求解 (5.1)：

Two periods: $t=0$ trading of the asset, $t=1$ payoff realized, no discounting.
A risk-averse agent owns an asset, with strictly concave utility $u(\cdot)$. We use CARA (exponential) utility $u(x)=-e^{-rx}$ ($r>0$).
The asset generates payoff $\tilde x\sim\mathcal N(\mu,\sigma^2)$ at $t=1$. The agent has a signal $\theta$ (indexed by $i$ for different private information), higher signal means a better asset; to the agent with signal $\theta_i$, the asset's expected return is $\mathbb E[\tilde x\mid\theta_i]$.
The agent can sell a fraction $\alpha\in[0,1]$ of the asset to risk-neutral market investors.
Since the agent is risk-averse and investors risk-neutral, the first-best is for the agent to sell all equity to the market. However, due to information asymmetry, in a pooling equilibrium the market investor is only willing to pay $\mathbb E[\tilde x]$ for the entire asset, which is too low for the agent with a good signal.

5.1.2 The Model

Focus on the separating equilibrium (different quality sold at a different fraction, fully revealing quality).
Market investors update their prior of asset quality to a posterior $g(\theta\mid\alpha)$ after observing the selling fraction $\alpha$.
In the separating equilibrium, investors can correctly tell $\theta$ with certainty from observing $\alpha$, and price the asset by $p(\alpha)=\mathbb E_\theta[\tilde x\mid\alpha]$ (competitive market, investors break even).
The agent's IC: an agent with true signal $\theta_i$ who sells $\alpha_j$ so the market believes he is $\theta_j$ has expected (terminal wealth) $V(\alpha_j\mid\theta_i)=\mathbb E[(1-\alpha_j)\tilde x+\alpha_j p(\alpha_j)\mid\theta_i]$; the separating-equilibrium IC requires the $\alpha_i$ chosen by true type $\theta_i$ to optimally reveal its type.

5.1.3 Solve the problem

Let the signal be the payoff distribution $\theta=\mu$, assume $\mu\in[\underline\mu,\infty)$. In the separating equilibrium $\alpha^\star(\mu)$ is a one-to-one mapping, and the market price $p(\alpha)$ is also one-to-one. We know a priori that $p'(\alpha)<0$: if $p'(\alpha)>0$, any quality would love to sell as many shares as possible for a higher price, ending in $\alpha=1$ for all types, which cannot support a separating equilibrium — a contradiction. Given $p(\alpha)$, the agent solves (5.1):

$$\max_{\alpha\in[0,1]}\mathbb E\!\left[u\big((1-\alpha)\tilde x+\alpha p(\alpha)\big)\mid\mu\right]\ \Longleftrightarrow\ \max_{\alpha\in[0,1]}-\mathbb E\!\left[e^{\overbrace{-r((1-\alpha)\tilde x+\alpha p(\alpha))}^{\equiv Z}}\mid\mu\right]\tag{5.1}$$

其中 $Z\sim\mathcal N\!\left(-r[(1-\alpha)\mu+\alpha p(\alpha)],\,(1-\alpha)^2r^2\sigma^2\right)$。

Note

求解 / Solution（(5.1)→(5.2)→(5.5)：CARA 化简、定价方案）用正态分布 MGF（He 2019a Prop 27.2）化简 (5.1)： $$\max_\alpha -e^{-r[(1-\alpha)\mu+\alpha p(\alpha)]+\frac12(1-\alpha)^2r^2\sigma^2}\ \Longleftrightarrow\ \max_\alpha (1-\alpha)\mu+\alpha p(\alpha)-\tfrac12(1-\alpha)^2 r\sigma^2,$$ 一阶条件 (5.2)： $$-\mu+\alpha p'(\alpha)+p(\alpha)+(1-\alpha)r\sigma^2=0.$$ 四项含义：$-\mu$ 多卖一点的均值损失；$\alpha p'(\alpha)$ 价格压力损失；$p(\alpha)$ 出售收入收益；$(1-\alpha)r\sigma^2$ 风险敞口下降的收益。竞争均衡中理性投资者出价 (5.3) $p(\alpha^\star(\mu))=\mu$。代入 (5.2)： $$\alpha p'(\alpha)+(1-\alpha)r\sigma^2=0\ \Rightarrow\ p'(\alpha)=\Big(1-\tfrac1\alpha\Big)r\sigma^2\ \Rightarrow\ p(\alpha)=C+r\sigma^2(\alpha-\ln\alpha)\tag{5.4}$$ 用初始条件 $\alpha^\star(\underline\mu)=1$（分离均衡中最差类型价格已最差、故卖出整个资产）定 $C$：$p(1)=C+r\sigma^2\Rightarrow C=\underline\mu-r\sigma^2$。故分离均衡价格方案 (5.5)： $$p(\alpha)=\underline\mu+r\sigma^2(\alpha-\ln\alpha-1).\tag{5.5}$$ $\blacksquare$

(5.5) 是 Pareto 最优分离均衡（最低类型 $\underline\mu$ 卖出整个资产，Pareto 占优所有 $\alpha^\star(\underline\mu)<1$ 的分离均衡）。且 $p'(\alpha)=r\sigma^2(1-\tfrac1\alpha)<0$：卖得越多说明类型越差、市场给越低价——与逆向选择一致。

Tip

Remark 5.1 分离均衡价格方案 (5.5) 不含类型分布的任何信息，直观上因为投资者能通过观察每个 $\mu$ 的 $\alpha^\star(\mu)$ 精确识别私有类型，故不关心 $\mu$ 的分布。

5.1.4 Comparative Statics

由 (5.5)（均衡中 $p(\alpha)=\mu$），定义隐函数 $F=\alpha-\ln\alpha-1-\dfrac{\mu-\underline\mu}{r\sigma^2}$。对 $\sigma^2$ 与 $\alpha$ 微分：

where $Z\sim\mathcal N\!\left(-r[(1-\alpha)\mu+\alpha p(\alpha)],\,(1-\alpha)^2r^2\sigma^2\right)$.

Note

Solution ((5.1)→(5.2)→(5.5): CARA simplification, pricing scheme) Simplify (5.1) with the normal MGF (He 2019a Prop 27.2): $$\max_\alpha -e^{-r[(1-\alpha)\mu+\alpha p(\alpha)]+\frac12(1-\alpha)^2r^2\sigma^2}\ \Longleftrightarrow\ \max_\alpha (1-\alpha)\mu+\alpha p(\alpha)-\tfrac12(1-\alpha)^2 r\sigma^2,$$ with first-order condition (5.2): $$-\mu+\alpha p'(\alpha)+p(\alpha)+(1-\alpha)r\sigma^2=0.$$ The four terms: $-\mu$ the mean loss from owning less; $\alpha p'(\alpha)$ the price-pressure loss; $p(\alpha)$ the revenue benefit; $(1-\alpha)r\sigma^2$ the benefit of reduced risk loading. In competitive equilibrium the rational investor offers (5.3) $p(\alpha^\star(\mu))=\mu$. Plugging into (5.2): $$\alpha p'(\alpha)+(1-\alpha)r\sigma^2=0\ \Rightarrow\ p'(\alpha)=\Big(1-\tfrac1\alpha\Big)r\sigma^2\ \Rightarrow\ p(\alpha)=C+r\sigma^2(\alpha-\ln\alpha)\tag{5.4}$$ Pin down $C$ with the initial condition $\alpha^\star(\underline\mu)=1$ (the worst type gets the worst price anyway, so sells the entire asset): $p(1)=C+r\sigma^2\Rightarrow C=\underline\mu-r\sigma^2$. So the separating-equilibrium price scheme (5.5): $$p(\alpha)=\underline\mu+r\sigma^2(\alpha-\ln\alpha-1).\tag{5.5}$$ $\blacksquare$

(5.5) is the Pareto-optimal separating equilibrium (the lowest type $\underline\mu$ sells the entire asset, Pareto-dominating all separating equilibria with $\alpha^\star(\underline\mu)<1$). And $p'(\alpha)=r\sigma^2(1-\tfrac1\alpha)<0$: the more you sell, the worse type you are, so the market gives a lower price — consistent with adverse selection.

Tip

Remark 5.1 The separating-equilibrium price scheme (5.5) does not involve any information about the distribution of type, intuitively because market investors can accurately identify the private type $\mu$ by observing $\alpha^\star(\mu)$ for every $\mu$, so they don't care about the distribution of $\mu$.

5.1.4 Comparative Statics

From (5.5) (with $p(\alpha)=\mu$ in equilibrium), define the implicit function $F=\alpha-\ln\alpha-1-\dfrac{\mu-\underline\mu}{r\sigma^2}$. Differentiating w.r.t. both $\sigma^2$ and $\alpha$:

$$\frac{\partial\alpha}{\partial\sigma^2}=-\frac{\partial F/\partial\sigma^2}{\partial F/\partial\alpha}=\frac{\dfrac{\mu-\underline\mu}{r(\sigma^2)^2}}{1-\dfrac1\alpha}>0$$

即更高的风险会令风险厌恶代理人卖出更大比例的资产，非常直观。

5.1.5 Implication

Leland-Pyle (1977) 常用于 IPO 问题。
IPO 的另一重要议题是「IPO 折价 (under-pricing)」。Rock (1986) 设有一组对公司与不知情投资者都有优越信息的成熟投资者：公司好时成熟者大量买入挤出不知情者，公司坏时撤单。故不知情者面临赢者诅咒；为使其有动机参与（至少回本），新股必须折价。
数值例：公司各半概率好/坏，好则 IPO 后 $+20\%$、坏则 $-5\%$。不知情者只知分布、成熟者知确切类型；不知情者下单买 4 股。公司好时成熟者大量下单挤出、不知情者只得 1 股；公司坏时成熟者撤单、不知情者得全部 4 股。不知情者期望回报 $1\times20\%+4\times(-5\%)=0$，回本愿参与。注意为使其愿参与，IPO 后期望回报 $0.5\times20\%+0.5\times(-5\%)=7.5\%>0$，意味对风险中性投资者新股折价了 $7.5\%$。
Leland-Pyle (1977) 假设未卖部分须由代理人持有至故事结束。美国有 180 天锁定期，但其后（甚至期内）内部人可能找到出售办法。故需在信息不对称下允许动态出售，Daley and Green (2012) 为此建立动态模型。

5.2 Equity Selling in Fraction: DeMarzo and Duffie (1999)

DeMarzo-Duffie (1999) 讨论股权流动性模型，信号为待售股权比例，类似 Leland-Pyle (1977)。他们讨论证券设计，主张在单调契约约束下债务最优、能帮代理人更有效地出售资产。

5.2.1 Setup

风险资产随机收益 $\tilde X$。风险中性代理人拥有它、私下观察信号 $Z$，$Z$ 越高 $\tilde X$ 分布越好，即 $\mathbb E[\tilde X\mid Z]$ 随 $Z$ 递增。
代理人最优选择待售比例 $q$，售予风险中性投资者、价格 $P(q)$。
代理人有持有资产的成本，体现在贴现因子 $\delta\in(0,1)$（其现金有更好用途、持有资产有机会成本）。

5.2.2 Directly Sell the Asset

代理人直接卖 $q$，求解 (5.6) $\max_{q\in[0,1]} q\,P(q)+\delta(1-q)\mathbb E[\tilde X\mid Z]$，一阶条件 (5.7)：

i.e. higher risk leads the risk-averse agent to sell a greater fraction of the asset, which is very intuitive.

5.1.5 Implication

Leland-Pyle (1977) is always used for IPO problems.
Another important issue about IPO is "IPO under-pricing." Rock (1986) sets up a group of sophisticated investors with superior information to both the firm and uninformed investors: when the firm is good, the sophisticated crowd out the uninformed; when bad, they withdraw orders. So uninformed investors face a winner's curse; to give them an incentive to buy (at least break even), new shares must be discounted.
Numerical example: the firm is good/bad with equal probability, good gives $+20\%$ after IPO, bad gives $-5\%$. The uninformed only know the distribution, the sophisticated know the exact type; the uninformed order 4 shares. If good, the sophisticated put a huge order to crowd out, so the uninformed only get 1 share; if bad, the sophisticated withdraw, so the uninformed get all 4 shares. The uninformed's expected return is $1\times20\%+4\times(-5\%)=0$, so they break even and participate. Note that to make them participate, the expected return after IPO is $0.5\times20\%+0.5\times(-5\%)=7.5\%>0$, meaning the new shares are under-priced by $7.5\%$ to a risk-neutral investor.
Leland-Pyle (1977) assumes the fraction not sold must be held by the agent until the story ends. In the US there is a 180-day lock-up period, but after that period (even within it) insiders may find a way to sell. So we need to allow dynamic selling under asymmetric information; Daley and Green (2012) develop a dynamic model for this.

5.2 Equity Selling in Fraction: DeMarzo and Duffie (1999)

DeMarzo-Duffie (1999) discuss an equity-liquidity model where the signal is the fraction of equity for selling, similar to Leland-Pyle (1977). They discuss security design to claim that debt is optimal in helping the agent sell the asset more efficiently under the monotone contract constraint.

5.2.1 Setup

A risky asset with random payoff $\tilde X$. A risk-neutral agent owns it and privately observes a signal $Z$; higher $Z$ implies a better distribution of $\tilde X$, i.e. $\mathbb E[\tilde X\mid Z]$ increases in $Z$.
The agent optimally chooses the fraction $q$ to sell to risk-neutral market investors, at price $P(q)$.
The agent has a cost of holding the asset, reflected in his discount factor $\delta\in(0,1)$ (he has better use for his cash, so an opportunity cost of holding).

5.2.2 Directly Sell the Asset

The agent directly sells $q$, solving (5.6) $\max_{q\in[0,1]} q\,P(q)+\delta(1-q)\mathbb E[\tilde X\mid Z]$, with first-order condition (5.7):

$$q\,P'(q)+P(q)-\delta\,\mathbb E[\tilde X\mid Z]=0\tag{5.7}$$

三项：$qP'(q)$ 价格压力成本、$P(q)$ 出售收入、$-\delta\mathbb E[\tilde X\mid Z]$ 持有成本下降的收益。竞争分离均衡中投资者回本 (5.8) $P(q^\star(Z))=\mathbb E[\tilde X\mid Z]$。设 $Z$ 下界 $Z_0$。

Note

求解 / Solution（(5.9)/(5.10)：分离均衡价格与最优比例）把 (5.8) 代入 (5.7)： $$q P'(q)+(1-\delta)P(q)=0\ \Rightarrow\ \frac{d\ln P(q)}{dq}=\frac{\delta-1}{q}\ \Rightarrow\ P(q)=e^C q^{\delta-1}.$$ 用最低类型 $q^\star(Z_0)=1$ 定 $C$：$P(1)=\mathbb E[\tilde X\mid Z_0]\Rightarrow e^C=\mathbb E[\tilde X\mid Z_0]$。故 (5.9)/(5.10)： $$P(q)=\mathbb E[\tilde X\mid Z_0]\,q^{\delta-1},\qquad q^\star(Z)=\left(\frac{\mathbb E[\tilde X\mid Z]}{\mathbb E[\tilde X\mid Z_0]}\right)^{\frac{1}{\delta-1}}.$$ $\blacksquare$

5.2.3 Design and Sell a Security

代理人可改为设计一个由 $\tilde X$ 支撑的证券 $\tilde F\le\tilde X$ 并卖出其比例 $q$。时序：先设计 $\tilde F$ 并承诺其收益；再观察私有信号 $Z$，面对整资产价格表 $P_{\tilde F}(q)$。代理人求解 (5.11) $\max_q q\,P_F(q)+\delta(1-q)\mathbb E[\tilde F\mid Z]$，与 (5.9)/(5.10) 同形：

$$P_F(q)=\mathbb E[\tilde F\mid Z_0]\,q^{\delta-1},\qquad q^\star(Z)=\left(\frac{\mathbb E[\tilde F\mid Z]}{\mathbb E[\tilde F\mid Z_0]}\right)^{\frac{1}{\delta-1}}.$$

在单调证券与一致最坏情形假设下（存在 $Z_0$ 使对任意 $Z$，条件分布 $X\mid Z$ 一阶随机占优 $X\mid Z_0$），债务最优：

The three terms: $qP'(q)$ the price-pressure cost, $P(q)$ the selling revenue, $-\delta\mathbb E[\tilde X\mid Z]$ the benefit of reduced holding cost. In the competitive separating equilibrium investors break even (5.8) $P(q^\star(Z))=\mathbb E[\tilde X\mid Z]$. Suppose the lower bound of $Z$ is $Z_0$.

Note

Solution ((5.9)/(5.10): separating-equilibrium price and optimal fraction) Plugging (5.8) into (5.7): $$q P'(q)+(1-\delta)P(q)=0\ \Rightarrow\ \frac{d\ln P(q)}{dq}=\frac{\delta-1}{q}\ \Rightarrow\ P(q)=e^C q^{\delta-1}.$$ Pin down $C$ with the lowest type $q^\star(Z_0)=1$: $P(1)=\mathbb E[\tilde X\mid Z_0]\Rightarrow e^C=\mathbb E[\tilde X\mid Z_0]$. So (5.9)/(5.10): $$P(q)=\mathbb E[\tilde X\mid Z_0]\,q^{\delta-1},\qquad q^\star(Z)=\left(\frac{\mathbb E[\tilde X\mid Z]}{\mathbb E[\tilde X\mid Z_0]}\right)^{\frac{1}{\delta-1}}.$$ $\blacksquare$

5.2.3 Design and Sell a Security

The agent can instead design a security $\tilde F\le\tilde X$ backed by $\tilde X$ and sell a fraction $q$ of it. Timing: first design $\tilde F$ and commit its payoff; then observe the private signal $Z$, facing the whole-asset price schedule $P_{\tilde F}(q)$. The agent solves (5.11) $\max_q q\,P_F(q)+\delta(1-q)\mathbb E[\tilde F\mid Z]$, of the same form as (5.9)/(5.10):

$$P_F(q)=\mathbb E[\tilde F\mid Z_0]\,q^{\delta-1},\qquad q^\star(Z)=\left(\frac{\mathbb E[\tilde F\mid Z]}{\mathbb E[\tilde F\mid Z_0]}\right)^{\frac{1}{\delta-1}}.$$

Under the monotone security and uniform worst-case assumptions (there exists $Z_0$ such that for any $Z$, the conditional distribution $X\mid Z$ first-order stochastically dominates $X\mid Z_0$), debt is optimal:

Note

证明 / Proof（债务 $\tilde F=\min\{\tilde X,d\}$ 最优）考虑任意单调证券 $\tilde G=\varphi(\tilde X)$（$\varphi'\ge0$、$\varphi(\tilde X)\le\tilde X$）与标准债务 $\tilde F=\min\{\tilde X,d\}$（单调且 $\tilde F\le\tilde X$）。可选 $d$ 使 $\mathbb E[\tilde F\mid Z_0]=\mathbb E[\tilde G\mid Z_0]$（因 $\mathbb E[\min\{\tilde X,d\}\mid Z_0]$ 随 $d$ 连续，$d$ 大时超过 $\varphi$、$d$ 小时低于 $\varphi$）。记 $\tilde H=\tilde G-\tilde F$、$h(Z)\equiv\mathbb E[\tilde H\mid Z]$，则 $h(Z_0)=0$。由 $\varphi\le\tilde X$、$\varphi'\ge0$，存在唯一 $X^\star$ 使 $\varphi(\tilde X)\le\min\{\tilde X,d\}$（$XX^\star$）。对 $Z>Z_0$ 定义似然比 $\pi_Z(\tilde X)\equiv f(\tilde X\mid Z)/f(\tilde X\mid Z_0)$，由 FOSD 它在 $X$ 中递增。则 $$h(Z)=\mathbb E[\tilde H\,\pi_Z(\tilde X)\mid Z_0]=\mathbb E\!\left[(\varphi-\min)\pi_Z\mid Z_0\right]\ge\mathbb E\!\left[(\varphi-\min)\pi_Z(X^\star)\mid Z_0\right]=\pi_Z(X^\star)\,\underbrace{\mathbb E[\tilde H\mid Z_0]}_{=h(Z_0)=0}=0,$$ 不等式因：$X不高于 \(\tilde F$。对任意证券 $\tilde G$ 及其最优比例 $q^\star_G$，代理人总可改用 $\tilde F$、卖同样的 $q^\star_G$ 获更高利润，再进一步优化 $q^\star_F$ 获更高利润。$\tilde G$ 任意，故债务在最大化代理人利润意义下最优。$\blacksquare$

Tip

Remark 5.2 与其把项目像股权那样线性整售，经理只以债务形式卖出项目的一部分，这信号化了项目是好的。该结果简单地意味着分级 (tranching) 对发行人有利。

5.3 Insider's Trading and Price Effect with Market Maker: Kyle (1985)

Leland-Pyle (1977) 与 DeMarzo-Duffie (1999) 中股权非流动性由信息不对称引起。Kyle (1985) 同样用信息不对称解释，但在两方之间引入第三方——做市商 (market maker)。此外，前两者是典型公司金融论文，Kyle (1985) 带有资产定价特征。

5.3.1 Setup

三方：
知情代理人：风险中性、非原子 (non-atomistic)（非价格接受者，可通过交易影响价格）、无限期且前瞻（内部化今日交易对未来价格的影响）。
噪声交易者 (noise traders)：与知情者同时向做市商提交订单。
风险中性做市商：据收到的订单决定价格。
两期 $t=0,1$。$t=0$ 知情者与噪声者向做市商下单，噪声订单 $\tilde u\sim\mathcal N(0,\sigma_u^2)$、$\tilde u\perp\tilde v$，知情订单 $\tilde x$；$t=1$ 做市商释放交易价 $\tilde p$。
一风险资产价值 $\tilde v\sim\mathcal N(p_0,\sigma_0^2)$：知情者在 $t=0,1$ 都知 $\tilde v$ 的确切实现值；做市商与噪声者只知分布。

5.3.2 The Two Stage Game

知情者须有市场势力，否则若 $\tilde p<\tilde v$ 知情者会买 $\tilde x=\infty$，无法成均衡。
知情者看不到 $\tilde u$，故基于私有信息有需求函数 $\tilde x=X(\tilde v)$，最大化利润 $\tilde\pi=\tilde x(\tilde v-\tilde p)=X(\tilde v)(\tilde v-P(\tilde x+\tilde u))$，即 $\tilde\pi=\Pi(X,P)$。
做市商观察到 $\tilde x+\tilde u$（无法分辨二者），设定定价函数 $\tilde p=P(\tilde x+\tilde u)$ 使其回本（做市业竞争）。
均衡条件：知情者选最优需求 $X$ (5.12) $\mathbb E[\Pi(X,P)\mid\tilde v=v]\ge\mathbb E[\Pi(X',P)\mid\tilde v=v]$；做市商回本 (5.13) $P(\tilde x+\tilde u)=\mathbb E[\tilde v\mid\tilde x+\tilde u]$。

5.3.3 The Linear Equilibrium

聚焦线性形式 $P(y)=\mu+\lambda y$、$X(v)=\alpha+\beta v$，求 $\mu,\lambda,\alpha,\beta$。

Note

Proof (debt $\tilde F=\min\{\tilde X,d\}$ is optimal) Consider any monotone security $\tilde G=\varphi(\tilde X)$ ($\varphi'\ge0$, $\varphi(\tilde X)\le\tilde X$) and standard debt $\tilde F=\min\{\tilde X,d\}$ (monotone and $\tilde F\le\tilde X$). We can choose $d$ such that $\mathbb E[\tilde F\mid Z_0]=\mathbb E[\tilde G\mid Z_0]$ (since $\mathbb E[\min\{\tilde X,d\}\mid Z_0]$ is continuous in $d$, exceeding $\varphi$ for large $d$ and below it for small $d$). Denote $\tilde H=\tilde G-\tilde F$, $h(Z)\equiv\mathbb E[\tilde H\mid Z]$, so $h(Z_0)=0$. By $\varphi\le\tilde X$, $\varphi'\ge0$, there is a unique $X^\star$ with $\varphi(\tilde X)\le\min\{\tilde X,d\}$ for $XX^\star$. For $Z>Z_0$ define the likelihood ratio $\pi_Z(\tilde X)\equiv f(\tilde X\mid Z)/f(\tilde X\mid Z_0)$, increasing in $X$ by FOSD. Then $$h(Z)=\mathbb E[\tilde H\,\pi_Z(\tilde X)\mid Z_0]=\mathbb E\!\left[(\varphi-\min)\pi_Z\mid Z_0\right]\ge\mathbb E\!\left[(\varphi-\min)\pi_Z(X^\star)\mid Z_0\right]=\pi_Z(X^\star)\,\underbrace{\mathbb E[\tilde H\mid Z_0]}_{=h(Z_0)=0}=0,$$ the inequality because for $Xno higher than for \(\tilde F$ at any fraction $q$. For any security $\tilde G$ with its optimal fraction $q^\star_G$, the agent can always switch to $\tilde F$ and sell the same $q^\star_G$ for higher profit, then further optimize $q^\star_F$ for even higher profit. Since $\tilde G$ is arbitrary, debt is optimal in maximizing the agent's profit. $\blacksquare$

Tip

Remark 5.2 Instead of selling the project linearly as equity, the manager sells only part of the project in a debt form, which signals that the project is good. This result simply means that tranching is good for the issuer.

5.3 Insider's Trading and Price Effect with Market Maker: Kyle (1985)

In Leland-Pyle (1977) and DeMarzo-Duffie (1999), equity illiquidity is caused by information asymmetry. Kyle (1985) also explains it with information asymmetry, but introduces a third party between the two — the market maker. In addition, the first two are typical corporate finance papers, while Kyle (1985) has asset-pricing characteristics.

5.3.1 Setup

Three parties:
Informed agent: risk-neutral, non-atomistic (not a price taker, can change the price by trading), infinite-horizon and forward-looking (internalizes the effect of today's trading on future prices).
Noise traders: submit orders simultaneously with the informed agent to the market maker.
Risk-neutral market maker: based on the orders received, determines the price.
Two periods $t=0,1$. At $t=0$ the informed agent and noise traders submit orders, the noise order $\tilde u\sim\mathcal N(0,\sigma_u^2)$, $\tilde u\perp\tilde v$, the informed order $\tilde x$; at $t=1$ the market maker releases the trading price $\tilde p$.
One risky asset with value $\tilde v\sim\mathcal N(p_0,\sigma_0^2)$: the informed agent knows the exact realized $\tilde v$ at both $t=0,1$; the market maker and noise traders only know the distribution.

5.3.2 The Two Stage Game

The informed agent must have some market power, otherwise if $\tilde p<\tilde v$ he would buy $\tilde x=\infty$, which cannot be an equilibrium.
The informed agent cannot see $\tilde u$, so has a demand function based on private information $\tilde x=X(\tilde v)$, maximizing profit $\tilde\pi=\tilde x(\tilde v-\tilde p)=X(\tilde v)(\tilde v-P(\tilde x+\tilde u))$, i.e. $\tilde\pi=\Pi(X,P)$.
The market maker observes $\tilde x+\tilde u$ (cannot tell them apart) and sets a pricing function $\tilde p=P(\tilde x+\tilde u)$ such that he breaks even (competitive market-making).
Equilibrium conditions: the informed agent chooses the optimal demand $X$ (5.12) $\mathbb E[\Pi(X,P)\mid\tilde v=v]\ge\mathbb E[\Pi(X',P)\mid\tilde v=v]$; the market maker breaks even (5.13) $P(\tilde x+\tilde u)=\mathbb E[\tilde v\mid\tilde x+\tilde u]$.

5.3.3 The Linear Equilibrium

Focus on the linear form $P(y)=\mu+\lambda y$, $X(v)=\alpha+\beta v$, and solve for $\mu,\lambda,\alpha,\beta$.

Note

求解 / Solution（线性均衡 (5.14)–(5.18)） 知情者（如 5.12）： $$\max_X \mathbb E[X(\tilde v-P(X+\tilde u))\mid\tilde v=v]=\max_X X(v-\mu-\lambda X),$$ 一阶条件 $v-\mu-2\lambda X=0$ (5.14)：$X=-\dfrac\mu{2\lambda}+\dfrac1{2\lambda}v$，故 $\alpha=-\dfrac\mu{2\lambda}$、$\beta=\dfrac1{2\lambda}$，得 (5.15) $\lambda\beta=\dfrac12$。 做市商回本 (5.13)：$\mu+\lambda y=\mathbb E[\tilde v\mid\alpha+\beta\tilde v+\tilde u=y]$。由 Kalman 滤波（He 2019a §8），$\tilde y\equiv\alpha+\beta\tilde v+\tilde u\sim\mathcal N(\alpha+\beta p_0,\beta^2\sigma_0^2+\sigma_u^2)$，$m_1=\dfrac{\beta\sigma_0^2}{\beta^2\sigma_0^2+\sigma_u^2}$，更新后 $\tilde v\sim\mathcal N(p_1,\sigma_1^2)$，$p_1=\underbrace{\dfrac{\beta\sigma_0^2}{\beta^2\sigma_0^2+\sigma_u^2}}_{\lambda}y+\underbrace{\dfrac{p_0\sigma_u^2-\beta\sigma_0^2\alpha}{\beta^2\sigma_0^2+\sigma_u^2}}_{\mu}$，即定价函数 (5.16)。后验方差 (5.17)：$\sigma_1^2=\sigma_0^2-m_1^2(\beta^2\sigma_0^2+\sigma_u^2)=\dfrac{\sigma_0^2\sigma_u^2}{\beta^2\sigma_0^2+\sigma_u^2}$。由 (5.15) $\lambda\beta=\dfrac{\beta^2\sigma_0^2}{\beta^2\sigma_0^2+\sigma_u^2}=\dfrac12\Rightarrow\beta^2\sigma_0^2=\sigma_u^2$，代入 (5.17) 得 (5.18)： $$\sigma_1^2=\tfrac12\sigma_0^2.$$ $\blacksquare$

Tip

Remark 5.3 / 5.4 / 5.5 / 5.6 5.3：线性需求 $X$ 与线性定价 $P$ 的初始猜测被 (5.14)、(5.16) 验证，故这是「猜测-验证」型题目。 5.4：只聚焦线性结果不代表只有线性解，可能存在非线性解；但线性格式下解唯一。 5.5：由 (5.18) 交易把市场方差减半，即知情者向市场泄露了一半私有信息。信息不免费——私人主体释放信息纯粹为最大化自身利润。 5.6：(5.18) 与噪声方差 $\sigma_u^2$ 无关，即信息披露量与噪声交易无关。但噪声方差越高，定价函数对总订单越不敏感（做市商对含太多噪声的信息更不在意）；且噪声方差越高、需求函数越激进（$\beta$ 越高），因知情者能更好地藏在噪声交易者之后。非常直观。

Note

Solution (linear equilibrium (5.14)–(5.18)) Informed agent (as in 5.12): $$\max_X \mathbb E[X(\tilde v-P(X+\tilde u))\mid\tilde v=v]=\max_X X(v-\mu-\lambda X),$$ with f.o.c. $v-\mu-2\lambda X=0$ (5.14): $X=-\dfrac\mu{2\lambda}+\dfrac1{2\lambda}v$, so $\alpha=-\dfrac\mu{2\lambda}$, $\beta=\dfrac1{2\lambda}$, giving (5.15) $\lambda\beta=\dfrac12$. Market maker breaks even (5.13): $\mu+\lambda y=\mathbb E[\tilde v\mid\alpha+\beta\tilde v+\tilde u=y]$. By Kalman filtering (He 2019a §8), $\tilde y\equiv\alpha+\beta\tilde v+\tilde u\sim\mathcal N(\alpha+\beta p_0,\beta^2\sigma_0^2+\sigma_u^2)$, $m_1=\dfrac{\beta\sigma_0^2}{\beta^2\sigma_0^2+\sigma_u^2}$, updated $\tilde v\sim\mathcal N(p_1,\sigma_1^2)$, $p_1=\underbrace{\dfrac{\beta\sigma_0^2}{\beta^2\sigma_0^2+\sigma_u^2}}_{\lambda}y+\underbrace{\dfrac{p_0\sigma_u^2-\beta\sigma_0^2\alpha}{\beta^2\sigma_0^2+\sigma_u^2}}_{\mu}$, i.e. the pricing function (5.16). Posterior variance (5.17): $\sigma_1^2=\sigma_0^2-m_1^2(\beta^2\sigma_0^2+\sigma_u^2)=\dfrac{\sigma_0^2\sigma_u^2}{\beta^2\sigma_0^2+\sigma_u^2}$. By (5.15) $\lambda\beta=\dfrac{\beta^2\sigma_0^2}{\beta^2\sigma_0^2+\sigma_u^2}=\dfrac12\Rightarrow\beta^2\sigma_0^2=\sigma_u^2$, plug into (5.17) to get (5.18): $$\sigma_1^2=\tfrac12\sigma_0^2.$$ $\blacksquare$

Tip

Remark 5.3 / 5.4 / 5.5 / 5.6 5.3: the initial guesses of linear demand $X$ and linear pricing $P$ are supported by the verified results (5.14), (5.16), so this is a "guess and verify" question. 5.4: focusing on linear results doesn't mean there are only linear results; there could be non-linear ones, but the solution in linear format is unique. 5.5: from (5.18), trading reduces the market variance by one half, i.e. the informed agent discloses half of his private information to the market. Information does not come for free — private agents release it purely to maximize their own profit. 5.6: (5.18) is not related to the noise trader's variance $\sigma_u^2$, i.e. the amount of information disclosure is unrelated to noise trading. But the higher the noise variance, the less sensitive the pricing function to total orders (the market maker cares less about information containing too much noise); and the higher the noise variance, the more aggressive the demand function (higher $\beta$), since insiders can better hide behind noise traders. Very intuitive.

References

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DeMarzo, P. and D. Duffie (1999). A liquidity-based model of security design. Econometrica 67(1), 65–99.
He, X. (2019a). Econometrics Notes by Xindi He.
Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica, 1315–1335.
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Rock, K. (1986). Why new issues are underpriced. Journal of Financial Economics 15(1-2), 187–212.