Capital asset market equilibrium with liquidity risk, portfolio constraints, and asset price bubbles

Abstract

This paper derives an equilibrium asset pricing model with endogenous liquidity risk, portfolio constraints, and asset price bubbles. Liquidity risk is modeled as a stochastic quantity impact on the price from trading, where the size of the impact depends on trade size. Asset price bubbles are generated by the existence of portfolio constraints, e.g. short sale prohibitions and margin requirements. Under a restrictive set of assumptions, we prove a unique equilibrium price process exists for our economy. We characterize the market’s state price density, which enables the derivation of the risk-return relation for the stock’s expected return including both liquidity risk and asset price bubbles. This yields a generalized intertemporal and consumption CAPM for our economy. In contrast to the traditional models without liquidity risk or asset price bubbles, there are additional systematic liquidity risk and asset price bubble factors which are related to the stock return’s covariation with liquidity risk and asset price bubbles.

Appendix

1.1 Proof of Theorem 9.

The proof needs the following lemma.

Lemma 14

(Existence of a Representative Trader Asset Market that Reflects the Asset Market’s Aggregate Optimal Demands) Given is an asset market $\big (\left( \varphi ,K,N\right) ,\big (\mathbb {P}_{i},U_{i},(x_{i},y_{i})\big ){}_{i=1}^{I}\big )$.

Consider the collection of representative trader asset markets

$\left( (\varphi ,K,N),(\mathbb {P},U(\lambda ),(0,N))\right)$ indexed by $\lambda \gg 0$.

Let S be a price process for all of these asset markets.

For each $\lambda \gg 0$, assume the existence of an equivalent martingale measure $\mathbb {M}(S-v^{\lambda })$ for the aggregate demands $\sum _{i=1}^{I}\Delta X_{t+1}^{i}$ where $\Delta X_{t+1}^{i}$ is the $i^{th}$ trader’s optimal trading strategy.

Assume there exists a unique $\lambda ^{*}\gg 0$ such that

$$\begin{aligned} \lambda _{i}^{*}=\frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}_{i}}\frac{1}{U_{i}'(Y_{T+1}^{i})}>0\qquad \mathrm {for\, all}\,i \end{aligned}$$

(28)

where $Y_{T+1}^{i}=-y_{i}-\varphi _{T}\left( -x_{i}-\sum _{t=0}^{T-1}\Delta X_{t+1}^{i}\right) S_{T}-\sum _{t=0}^{T-1}\varphi _{t}(\Delta X_{t+1}^{i})S_{t}$.

Then, the representative trader asset market indexed by this unique $\lambda ^{*}$ reflects the asset market’s aggregate optimal demands for the price process S.

Proof

The representative trader’s optimal trading strategy satisfies

$$\begin{aligned} E_{t}\left[ U'(Y_{T+1}^{\lambda },\lambda )\left( \varphi '_{t}(\Delta X_{t+1}^{\lambda })\left( S_{t}-v_{t}^{\lambda }\right) -\varphi '_{T}(\Delta X_{T+1}^{\lambda })\left( S_{t}-v_{t}^{\lambda }\right) \right) \right] =0 \end{aligned}$$

(29)

for $t\in \{0,\ldots ,T-1\}$

where $Y_{T+1}^{\lambda }=-\varphi _{T}\left( -N-\sum _{t=0}^{T-1}\Delta X_{t+1}^{\lambda }\right) S_{T}-\sum _{t=0}^{T-1}\varphi _{t}(\Delta X_{t+1}^{\lambda })S_{t}$.

From expression (16) we have

$$\begin{aligned} \lambda _{i}\frac{d\mathbb {P}_{i}}{d\mathbb {P}}U_{i}'({\tilde{Y}}_{T+1}^{i})=U'(Y_{T+1}^{\lambda },\lambda )\qquad \mathrm {for\,all}\,i \end{aligned}$$

where $Y_{T+1}^{\lambda }=\sum _{i=1}^{I}{\tilde{Y}}_{T+1}^{i}$.

Substitution into expression (29) gives

$$\begin{aligned} E_{t}^{i}\left[ \lambda _{i}U_{i}'({\tilde{Y}}_{T+1}^{i})\left( \varphi '_{t}(\Delta X_{t+1}^{\lambda })\left( S_{t}-v_{t}^{\lambda }\right) -\varphi '_{T}(\Delta X_{T+1}^{\lambda })(S_{t}-v_{t}^{\lambda })\right) \right] =0. \end{aligned}$$

(30)

Next, given ${\tilde{Y}}_{T+1}^{i}$, define $\Delta {\tilde{X}}_{t}^{i}$ as the solutions to

$$\begin{aligned} 0\in E_{t}^{i}\left[ U_{i}'({\tilde{Y}}_{T+1}^{i})\left( \varphi '_{t}(\Delta {\tilde{X}}_{t+1}^{i})S_{t}-\varphi '_{T}(\Delta {\tilde{X}}_{T+1}^{i})S_{T}\right) \right] +\mathcal {N}_{K_{t}}(\Delta {\tilde{X}}_{t+1}^{i}) \end{aligned}$$

for $t\in \{0,\ldots ,T-1\}$ where $\Delta {\tilde{X}}_{t}^{i}$ satisfies the self-financing condition (2).

By Theorem 2 we know a solution exists to these equations. By uniqueness of the optimal trading strategy, this implies $\Delta {\tilde{X}}_{t}^{i}$ are the solutions to trader i’s optimization problem.

To complete the proof, we need to show that $\Delta X_{t+1}^{\lambda }=\sum _{i=1}^{I}\Delta {\tilde{X}}_{t+1}^{i}$.

(Step 1) At time T because all shares must be liquidated, $-N=\Delta X_{T+1}^{\lambda }=\sum _{i=1}^{I}\Delta {\tilde{X}}_{T+1}^{i}$.

(Step 2) Since $\lambda \gg 0$ is arbitrary, choose $\lambda _{i}^{*}$ such that

$$\begin{aligned} \lambda _{i}^{*}=\frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}_{i}}\frac{1}{U_{i}'({\tilde{Y}}_{T+1}^{i})}>0\qquad \mathrm {for\,all}\,i \end{aligned}$$

where $\mathbb {M}(S-v^{\lambda ^{*}})$ is the equivalent probability measure assumed to exist in the hypothesis of the theorem.

Substitution of $\lambda ^{*}$ into expression (30) gives

$$\begin{aligned} E_{t}\left[ \frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}}\left( \varphi '_{t}(\Delta X_{t+1}^{\lambda ^{*}})(S_{t}-v_{t}^{\lambda ^{*}})-\varphi '_{T}(\Delta X_{T+1}^{\lambda ^{*}})(S_{T}-v_{T}^{\lambda ^{*}})\right) \right] =0 \end{aligned}$$

for $t\in \{0,\ldots ,T-1\}$. This implies

$$\begin{aligned} \varphi '_{t}(\Delta X_{t+1}^{\lambda ^{*}})(S_{t}-v_{t}^{\lambda ^{*}})=E_{t}\left[ \frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}}\varphi '_{T}(\Delta X_{T+1}^{\lambda ^{*}})(S_{T}-v_{T}^{\lambda ^{*}})\right] \end{aligned}$$

for $t\in \{0,\ldots ,T-1\}$.

Using the definition of $\mathbb {M}(S-v^{\lambda ^{*}})$ as an equivalent martingale measure for the aggregate demands we have that

$$\begin{aligned} \varphi '_{t}\left( \sum _{i=1}^{I}\Delta {\tilde{X}}_{t+1}^{i})(S_{t}-v_{t}^{\lambda ^{*}}\right) =E_{t}\left[ \frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}}\left( \varphi '_{T}\left( \sum _{i=1}^{I}\Delta {\tilde{X}}_{T+1}^{i}\right) (S_{T}-v_{T}^{\lambda ^{*}})\right) \right] \end{aligned}$$

for $t\in \{0,\ldots ,T\}$.

These last two equations combined with step 1 imply the result. This follows by backward induction.

Indeed, at time $T-1$ we have

$$\begin{aligned}&\varphi '_{T-1}(\Delta X_{T}^{\lambda ^{*}})(S_{T-1}-v_{T-1}^{\lambda ^{*}})=E_{T-1}\left[ \frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}}\left( \varphi '_{T}(\Delta X_{T+1}^{\lambda ^{*}})(S_{T}-v_{T}^{\lambda ^{*}})\right) \right] \\&\quad =E_{T-1}\left[ \frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}}\left( \varphi '_{T}\left( \sum _{i=1}^{I}\Delta {\tilde{X}}_{T+1}^{i}\right) (S_{T}-v_{T}^{\lambda ^{*}})\right) \right] \end{aligned}$$

since $\Delta X_{T+1}^{\lambda ^{*}}=\sum _{i=1}^{I}\Delta {\tilde{X}}_{T+1}^{i}$. But, this

$=\varphi '_{T-1}(\sum _{i=1}^{I}\Delta {\tilde{X}}_{T}^{i})(S_{T-1}-v_{T-1}^{\lambda ^{*}})$. This implies

$\varphi '_{T-1}(\Delta X_{T}^{\lambda ^{*}})=\varphi '_{T-1}(\sum _{i=1}^{I}\Delta {\tilde{X}}_{T}^{i})$.

Since $\varphi '_{T-1}$ is strictly increasing we get $\Delta X_{T}^{\lambda ^{*}}=\sum _{i=1}^{I}\Delta {\tilde{X}}_{T}^{i}$.

Next, we note that by the law of iterated expectations,

$$\begin{aligned}&\varphi '_{T-2}(\Delta X_{T-1}^{\lambda ^{*}})(S_{T-2}-v_{T-2}^{\lambda ^{*}})=E_{t}\left[ \frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}}\varphi '_{T-1}(\Delta X_{T}^{\lambda ^{*}})(S_{T-1}-v_{T-1}^{\lambda ^{*}})\right] \hbox { and }\\&\quad \varphi '_{T-2}\left( \sum _{i=1}^{I}\Delta {\tilde{X}}_{T-1}^{i}\right) (S_{T-2}-v_{T-2}^{\lambda ^{*}})\\&\qquad =E_{t}\left[ \frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}}\left( \varphi '_{T-1}\left( \sum _{i=1}^{I}\Delta {\tilde{X}}_{T}^{i}\right) (S_{T-1}-v_{T-1}^{\lambda ^{*}})\right) \right] . \end{aligned}$$

We can now repeat the previous argument at time $T-2$ to obtain $\Delta X_{T-1}^{\lambda ^{*}}=\sum _{i=1}^{I}\Delta {\tilde{X}}_{T-1}^{i}$. Continuing in this fashion to time 0 completes the proof. $\square$

This lemma is key in the proof for the existence of an economic equilibrium in the original asset market. We emphasize again that this lemma does not require the given price process S to be an equilibrium price process for the original asset market or the representative trader asset market.

Proof

By Lemma 14, given the equilibrium price process S, the representative agent asset market $\left( (\mathbb {P},K,N),(\mathbb {P},U(\lambda ^{*}),(0,N))\right)$ reflects the original market’s optimal wealths with $\lambda _{i}^{*}=\frac{d\mathbb {M}(S-v^{\lambda ^{*}})}{d\mathbb {P}_{i}}\frac{1}{U_{i}'({\tilde{Y}}_{T+1}^{i})}>0\qquad \mathrm {for\,all}\,i$, i.e.

$$\begin{aligned} \Delta X_{t+1}^{\lambda ^{*}}=\sum _{i=1}^{I}\Delta X_{t+1}^{i} \end{aligned}$$

for $t\in \{0,\ldots ,T-1\}$ a.s. $\mathbb {P}$ where $\Delta X_{t+1}^{\lambda ^{*}}$ are the optimal demands of the representative trader. But, in the asset market’s equilibrium, $0=\sum _{i=1}^{I}\Delta X_{t+1}^{i}$. Hence, $\Delta X_{t+1}^{\lambda ^{*}}=0$. This implies that $X_{t+1}^{\lambda ^{*}}=X_{t}^{\lambda ^{*}}\in int(K_{t})$, $v_{t}^{\lambda ^{*}}=0$ for all t, and $\lambda _{i}^{*}=\frac{d\mathbb {M}(S)}{d\mathbb {P}_{i}}\frac{1}{U_{i}'(Y_{T+1}^{i})}$. Hence, the representative trader asset market is in equilibrium with price process S. This completes the proof. $\square$

1.2 Proof of Theorem 10

Proof

(Existence) (Step 1) Consider the collection of representative trader asset markets

$\left( (\varphi ,K,N),(\mathbb {P},U(\lambda ),(0,N))\right)$ indexed by $\lambda \gg 0$. Fix an arbitrary $\lambda$. By Theorem 8, for the price process $S^{\lambda }$, the representative trader with the aggregate utility function weights $\lambda$ does not trade, i.e. $\Delta X_{t+1}^{\lambda }=0$. This implies that $X_{t+1}^{{\tilde{\lambda }}}=X_{t}^{{\tilde{\lambda }}}\in int(K_{t})$, $v_{t}^{{\tilde{\lambda }}}=0$ for all t.

Next, consider the asset market $\left( \left( \varphi ,K,N\right) ,\left( \mathbb {P}_{i},U_{i},(x_{i},y_{i})\right) {}_{i=1}^{I}\right)$ with the price process $S^{\lambda }$. Let $\Delta X_{t+1}^{i}(S^{\lambda })$ for $t\in \{0,\ldots ,T-1\}$ be the optimal trading strategy of trader i in this market. Note the dependence of trader’s demands on $S^{\lambda }$.

(Step 2) Fix the ${\tilde{\lambda }}\gg 0$ from expression (21). Jointly consider the asset market $\left( \left( \varphi ,K,N\right) ,\left( \mathbb {P}_{i},U_{i},(x_{i},y_{i})\right) {}_{i=1}^{I}\right)$ and the collection of representative trader asset markets $\left( (\varphi ,K,N),(\mathbb {P},U(\lambda ),(0,N))\right)$, all with the same price process $S^{{\tilde{\lambda }}}\equiv {\tilde{S}}$.

By the assumption regarding the existence of $\mathbb {M}({\tilde{S}}-v^{{\tilde{\lambda }}})$, the hypothesis of Theorem 14 is satisfied. Hence, there exists a $\lambda ^{*}$ such that a representative trader $U(\lambda ^{*})$ reflects these optimal demands, i.e.

$$\begin{aligned} \Delta X_{t+1}^{\lambda ^{*}}=\sum _{i=1}^{I}\Delta X_{t+1}^{i}({\tilde{S}}_{t}). \end{aligned}$$

The unique $\lambda ^{*}$ for the price process ${\tilde{S}}$ is

$$\begin{aligned} \lambda _{i}^{*}=\frac{d\mathbb {M}({\tilde{S}}-v^{\lambda ^{*}})}{d\mathbb {P}_{i}}\frac{1}{U_{i}'(Y_{T+1}^{i}({\tilde{S}}))}>0\qquad \mathrm {for\, all}\,i \end{aligned}$$

(31)

with $Y_{T+1}^{i}({\tilde{S}}_{t})=-y_{i}-\varphi _{T}\left( -x_{i}-\sum _{t=0}^{T-1}\Delta X_{t+1}^{i}\right) {\tilde{S}}_{T}-\sum _{t=0}^{T-1}\varphi _{t}(\Delta X_{t+1}^{i}){\tilde{S}}_{t}$.

(Step 3) We claim that $S^{{\tilde{\lambda }}}$ is an equilibrium price process for the asset market and the representative trader asset market $((\varphi ,K,N),(\mathbb {P},U({\tilde{\lambda }}),(0,N)))$.

(Part a) First, (Step 1) shows that $\Delta X_{t+1}^{{\tilde{\lambda }}}=0$ since the representative trader $U({\tilde{\lambda }})$’s optimal trading strategy is not to trade given ${\tilde{S}}=S^{{\tilde{\lambda }}}$. This implies that ${\tilde{S}}$ is an equilibrium price process for the representative trader asset market $\left( (\varphi ,N),(\mathbb {P},U({\tilde{\lambda }}),(0,N))\right)$.

(Part b) The uniqueness of the solution to expression (31) combined with expression (21) gives $\lambda ^{*}={\tilde{\lambda }}$.

This implies that $\Delta X_{t+1}^{{\tilde{\lambda }}}=\sum _{i=1}^{I}\Delta X_{t+1}^{i}(S^{{\tilde{\lambda }}})$. Hence, $0=\sum _{i=1}^{I}\Delta X_{t+1}^{i}(S^{{\tilde{\lambda }}})$, i.e. aggregate optimal demand equals supply. This proves that $S^{{\tilde{\lambda }}}$ is an equilibrium price process for the asset market.

(Step 4) By Theorem 7, since $\Delta X_{t+1}^{{\tilde{\lambda }}}=0$, $\varphi '_{t}(\Delta X_{t+1}^{{\tilde{\lambda }}}(S^{{\tilde{\lambda }}}))(S_{t}^{{\tilde{\lambda }}}+v^{{\tilde{\lambda }}})=S_{t}^{{\tilde{\lambda }}}$ for $t\in \{0,\ldots ,T-1\}$ is a martingale under $d\mathbb {Q}_{{\tilde{\lambda }}}=\frac{U'(-\varphi _{T}(-N)S_{T}^{{\tilde{\lambda }}},{\tilde{\lambda }})}{E\left[ U'(-\varphi _{T}(-N)S_{T}^{{\tilde{\lambda }}},{\tilde{\lambda }})\right] }d\mathbb {P}$ because $\Delta X_{t+1}^{{\tilde{\lambda }}}(S^{{\tilde{\lambda }}})=0$ for $t\in \{0,\ldots ,T-1\}$, $\varphi '_{t}(0)=1$, and $v^{{\tilde{\lambda }}}=0$.

(Characterization)

$S_{t}^{{\tilde{\lambda }}}$ for $t\in \{0,\ldots ,T-1\}$ a martingale under $\mathbb {Q}_{{\tilde{\lambda }}}$

implies that $S_{t}^{{\tilde{\lambda }}}\rho _{t}$ is a martingale under $\mathbb {P}$. Thus,

$$\begin{aligned}&E_{t}\left[ \frac{S_{t+1}^{{\tilde{\lambda }}}}{S_{t}^{{\tilde{\lambda }}}}\frac{\rho _{t+1}}{\rho _{t}}\right] =1\\&E_{t}\left[ \frac{S_{t+1}^{{\tilde{\lambda }}}}{S_{t}^{{\tilde{\lambda }}}}\right] E_{t}\left[ \frac{\rho _{t+1}}{\rho _{t}}\right] +cov_{t}\left( \frac{S_{t+1}^{{\tilde{\lambda }}}}{S_{t}^{{\tilde{\lambda }}}},\frac{\rho _{t+1}}{\rho _{t}}\right) =1\\ \end{aligned}$$

But, $E_{t}\left[ \frac{\rho _{t+1}}{\rho _{t}}\right] =1$. Yielding

$$\begin{aligned} E_{t}\left[ \frac{S_{t+1}^{{\tilde{\lambda }}}}{S_{t}^{{\tilde{\lambda }}}}-1\right] =-cov_{t}\left( \frac{S_{t+1}^{{\tilde{\lambda }}}}{S_{t}^{{\tilde{\lambda }}}},\frac{\rho _{t+1}}{\rho _{t}}\right) . \end{aligned}$$

This completes the proof. $\square$