Incomplete financial markets and jumps in asset prices

Abstract

For incomplete financial markets, jumps in both prices and consumption can be unavoidable. We consider pure-exchange economies with infinite horizon, discrete time, uncertainty with a continuum of possible shocks at every date. The evolution of shocks follows a Markov process, and fundamentals depend continuously on shocks. It is shown that: (1) equilibria exist; (2) for effectively complete financial markets, asset prices depend continuously on shocks; and (3) for incomplete financial markets, there is an open set of economies ${\fancyscript{U}}$ such that for every equilibrium of every economy in ${\fancyscript{U}}$, asset prices at every date depend discontinuously on the shock at that date.

Appendix: Economies with finite time horizon

1.1 Setup

Consider a pure-exchange economy with finite horizon and uncertainty. There is a finite number of dates $t \in \{0,\ldots ,T\}$. The set of possible shocks at every date is $S = [0,1]$ with $s_t \in S$, and the shock at the first date is $s_0 \in S$. The evolution of shocks as time passes is described by a bounded and measurable time-independent transition density $\pi : S{\times }S \rightarrow \mathbb {R}_{++}$.

There are finite numbers of goods $\ell$ with $h \in \{1,\ldots ,\ell \}$, consumers $m$ with $i \in \{1,\ldots ,m\}$ and real assets $n$ with $j \in \{1,\ldots ,n\}$. Consumers have identical consumption sets $X = (\mathbb {R}_{++}^\ell )^{T+1}$. Consumers are described by their endowments $\omega _i = (\omega _i^t)$ with $\omega _i^t : S^t \rightarrow \mathbb {R}_{++}^\ell$, state utility functions $u_i : X \rightarrow \mathbb {R}$ and lower bounds on short sales of assets $\delta _i \in \mathbb {R}_{++}^n$. Assets are described by their dividends $a_j = (a_j^t)$ with $a_j^t : S^t \rightarrow \mathbb {R}_+^\ell$ for all $t$. An economy is a list of consumers and a list of assets ${\fancyscript{E}}^{\mathrm{Finite}} = ((\omega _i,u_i,\delta _i),(a_j))$.

1.2 Discontinuous financial market equilibria

Theorem 4

There exists an economy such that if $((\bar{p},\bar{q}),(\bar{x},\bar{\theta }))$ is a financial market equilibrium, then $\bar{q}$ is discontinuous in $s^T$.

Proof

Consider an economy with three dates $T = 2$, one good in each date $\ell = 1$, two consumers $m = 2$ and one asset $n = 1$. The dividend of the asset is supposed to be one unit of the good at the last date and otherwise before the last date. Endowments and asset dividends are assumed to depend on the shock $s$ at the second date $t = 1$ and to be independent of the shock at the third date $t=2$. The probability distribution on the set of shocks at date $t=1$ is the Lebesgue measure $\pi (s) = 1$ for all $s$.

Endowments at date $t=0$ are supposed to be identical $\omega _2^0 = \omega _1^0$ and endowments at the last two dates are supposed to be reverse in the sense that

$$\begin{aligned} \omega _2^1(s)= & {} \omega _1^2(1{-}s) \\ \omega _2^2(s)= & {} \omega _1^1(1{-}s). \end{aligned}$$

Similarly, utility functions are supposed to be identical for the first date and reverse for the last two dates such that

$$\begin{aligned} u_2(x^0,\alpha ,\beta ) = u_1(x^0,\beta ,\alpha ) \end{aligned}$$

for all positive real numbers $x^0,\alpha ,\beta > 0$. Denote by ${\fancyscript{E}}(s;(c_i^0)) = (e_i(s),v_i(\cdot ;c_i^0))$ the sub-economy with consumers $i\in \{1,2\}$, utility functions $v_i(\cdot ;c_i^0) : \mathbb {R}_{++}^2 \rightarrow \mathbb {R}$ defined by $v_i(x_i^1,x_i^2;c_i^0) \equiv u_i(c_i^0,x_i^1,x_i^2)$ and endowments $e_i(s)\in \mathbb {R}^2$.

For $c_i^0$, let $f_i(\cdot ;c_i^0)=(f^1_i(\cdot ;c_i^0),f^2_i(\cdot ;c_i^0)) : \mathbb {R}_{++}^2 \times \mathbb {R}_{++} \rightarrow \mathbb {R}_{++}^2$ denote the demand function for the consumer $i$ of the sub-economy ${\fancyscript{E}}(s;(c_i^0))$, so $f_i(p,p{\cdot }e_i(s);c^0_i)$ solves the problem

$$\begin{aligned}&\displaystyle \max _{(x^1,x^2)} \quad v_i(x^1,x^2;c_i) \\&\hbox {s.t.} \quad p{\cdot } (x{-}e_i(s)) \le 0. \end{aligned}$$

Then $(s,p)\in S{\times }\mathbb {R}_{++}^2$ is an equilibrium for the sub-economy ${\fancyscript{E}}(s;(c_i^0)_i)$ if and only if

$$\begin{aligned} f_1(p,p{\cdot } e_1(s);c_1^0) + f_2(p,p{\cdot } e_2(s);c_2^0) \ = \ e_1(s) + e_2(s). \end{aligned}$$

Clearly $(s,p_1,p_2)$ is an equilibrium for ${\fancyscript{E}}(s;(c_i^0))$ if and only if $(1{-}s,p_2,p_1)$ is an equilibrium for ${\fancyscript{E}}(1{-}s;(d_i^0))$, where $d_1^0 = c_2^0$ and $d_2^0 = c_1^0$.

For equilibrium, prices normalized such that the sum equals one let $W : S \rightarrow S{\times }\mathbb {R}_{++}^2$ be the Walras correspondence for the economies $({\fancyscript{E}}(s;(c_i^0)))$ with $c_i^0 = \omega _i^0$ for both $i$, so

$$\begin{aligned} W(s) = \{ (s,p_1,p_2) \vert (s,p_1,p_2) \hbox { is an equilibrium for } {\fancyscript{E}}(s;(\omega _i^0)) \}. \end{aligned}$$

Suppose that the graph of $W$ is $S$-shaped as shown in Fig. 2 and let $r : S \rightarrow \mathbb {R}_{++}^2$ be a selection from $W$ such that $r_1(s)$ is the lowest equilibrium price for $s < 1/2$, $r_1(s) = (1/2,1/2)$ for $s = 1/2$ and $r_1(s)$ is the highest equilibrium price for $s > 1/2$. In order to construct a financial market equilibrium: Let the allocation $\bar{x}$ be defined by $\bar{x}_i^0 = \omega _i^0$, and $\bar{x}_i^t(s) = f_i^t(r(s),r(s){\cdot }e_i(s);\omega _i^0)$ for both $i$ and $t \in \{1,2\}$; let the portfolio $\bar{\theta }$ be defined by $\bar{\theta }_i^0 = 0$ and

$$\begin{aligned} \bar{\theta }_i^1(s) = \frac{r_1(s)}{r_2(s)}\left( \omega _i^1(s){-}f_i^1(r(s),r(s){\cdot }e_i(s);\omega _i^0)\right) \! = \! f_i^2(r(s),r(s){\cdot }e_i(s);\omega _i^0)\!-\!\omega _i^2(s); \end{aligned}$$

let the price system $(\bar{p},\bar{q})$ be defined by $\bar{p}_0 = \bar{p}_1(s) = \bar{p}_2(s) = 1$ for all $s$, $\bar{q}_1(s) = r_2(s)/r_1(s)$ for all $s$ and $\bar{q}_0 > 0$ such that

$$\begin{aligned} \displaystyle \int \left( -\bar{q}_0\dfrac{\partial u_i(\bar{x}_i(s))}{\partial x_i^0} + \bar{q}_1(s) \dfrac{\partial u_i(\bar{x}_i(s))}{\partial x_i^1} \right) \ \mathrm{{d}} s \ = \ 0. \end{aligned}$$

Then $((\bar{p},\bar{q}),(\bar{x},\bar{\theta }))$ is a financial market equilibrium and the asset price at date 1 is discontinuous at $s = 1/2$.

Fig. 2

The Walras correspondence $W$ and the selection $r$

Since $q_1(s)\theta _i^0 = (x_i^1(s){-}\omega _i^1(s))+q_1(s)(x_i^2(s){-}\omega _i^2(s))$ for both $i$ and almost all $s$, the portfolio $\bar{\theta }_1^0$ of consumer $1$ at date 0 satisfies $\theta ^1_0\in [\delta ^L,\delta ^U]$ for $\delta ^L = -\inf _s (\omega _1^1(s)/q_1(s) {+} \omega _1^2(s))$ and $\delta ^U = \inf _s (\omega _1^1(s)/q_1(s) {+} \omega _1^2(s))$. Assume $\omega _1^1(s),\omega _1^2(s) <\varepsilon$ for $s \in \{0,1\}$ and the marginal rates of substitution at the Pareto optimal allocations in the sub-economies for $s \in \{0,1\}$ are bounded away from zero and infinity. Then $\lim _{\varepsilon \rightarrow 0} \delta ^L =\lim _{\varepsilon \rightarrow 0} \delta ^U = 0$. Moreover there is $\bar{\varepsilon }> 0$ such that if $\varepsilon \le \bar{\varepsilon }$, then the set of equilibria for the collection of sub-economies is $S$-shaped for all feasible date $0$ portfolios. Therefore $\bar{q}$ is discontinuous in $s$. $\square$

Remark

The proof of Theorem 4 reveals that any measurable selection $r : S \rightarrow \mathbb {R}_{++}^2$ such that $r_1(s) = 1{-}r_1(1{-}s)$ and $r_2(s) = 1{-}r_2(1{-}s)$ is part of a financial market equilibrium. Therefore as shown in Mas-Colell (1991), there is a continuum of financial market equilibria.

Remark

An crucial difference between the examples in Mas-Colell (1991) and in the proof of Theorem 4 is the number of dates. In Mas-Colell (1991), there are two dates and there is a single asset traded at the first date. Hence the asset price at the first date is unaffected by the shock at the second date, so jumps in asset prices are impossible. In the proof of Theorem 4, there are three dates and a single asset traded at the two first dates. Thus the asset price on the second date is affected by the shock at that date making jumps in asset prices possible.