Endogenous Completeness of Diffusion Driven Equilibrium Markets

Abstract

We study the existence of equilibria with endogenously complete markets in a continuous-time, heterogenous agents economy driven by a multidimensional diffusion process. Our main results show that if prices are real analytic as functions of time and the state variables of the model then a sufficient condition for market completeness is that the volatility of dividends be nondegenerate. In contrast to previous research, our formulation does not require that securities pay terminal dividends and thus allows for both finite or infinite horizon economies. We illustrate our results by providing easily applicable conditions for market completeness in two benchmark cases: that where the state variables are given by a vector autoregressive process and that where they are given by a vector of autonomous diffusion processes. We also provide counterexamples which show that real analyticity cannot be dispensed with if one is to deduce dynamic market completeness from the structure of dividends.

A Simple Characterization of Dynamic Completeness in Continuous Time

“…Which is to say of course that, although very important in general, the extension undertaken by Hugonnier et al [16] to introduce a money market account as an explicit security is rather meaningless in the particular economy the present paper analyzes. For if, in the presence of the zero-coupon bond, one of the securities indexed by 1, .…”

“…For it obtains by restricting the functions and to be, respectively, zero and deterministic (hence, either trivially analytic) while all terminal dividends in (4) At first glance, this necessity might seem at odds with the study of Hugonnier et al [16] who characterize dynamic completeness in a setting which is, in some dimensions more general than that in Anderson and Raimondo [1], in others more restrictive. On the one hand, the evolution of the state variable y ∈ ℝ need not be dictated by a Brownian motion but a general diffusion process.…”

“…Raimondo [1] for the corresponding setting (for an example, see Section B in the Supplement on Hugonnier et al [16]). The authors argue, however, that is not also necessary for dynamic completeness.…”

“…Yet, a fundamental assumption for the analysis in Hugonnier et al [16] is that the instantaneous variance-covariance matrix of the underlying stochastic state-process is non-degenerate -see their Assumption A(a). And within the confines of the setting studied in the present paper (namely, that the state-variable follows a geometric Brownian motion), this amounts to requiring nothing other than condition (ii) of Theorem 1 above.…”

Radner Equilibria Under Ambiguous Volatility

“…This approach is based on a representative agent, see Huang (1987). We also refer to Hugonnier, Malamud, and Trubowitz (2012) and Herzberg and Riedel (2013) for a recent discussion of endogenous completeness in continuoustime finance models. Existence of equilibria in incomplete markets for a finite state space is well developed, starting with the seminal paper by Duffie and Shafer (1985).…”