Continuous Trading Dynamically E ¤ ectively Complete Market with Heterogeneous Beliefs Zhenjiang

In a framework of heterogeneous beliefs, I investigate a two-date consumption model with continuous trading over the interval [0; T ], in which information on the aggregate consumption at time T is revealed by an Ornstein-Uhlenbeck Bridge. This information structure allows investors to speculate on the heterogeneous posterior variance of dividend throughout [0; T ). The market populated with many time-additive exponentialutility investors is dynamically e¤ectively complete, if investors are allowed to trade in only two long-lived securities continuously. The underlying mechanism is that these assumptions imply that the Pareto e¢ cient individual consumption plans are measurable with respect to the aggregate consumption. Hence, I may not need a dynamically complete market to facilitate a Pareto e¢ cient allocation of consumption, the securities only have to facilitate an allocation which is measurable with respect to the aggregate consumption. With normally distributed dividend, the equilibrium stock price is endogenized in a Radner equilibrium as a precision weighted average of the investors posterior mean minus a risk premium determined by the average posterior precision. The stock price is also a su¢ cient statistic for computation of the price of redundant dividend derivative and the equilibrium portfolios. The investors form their Pareto optimal trading strategies as if they intend to dynamically endogenously replicate the value of the dividend derivative. Financial support from the Center for Research in the Econometric Analysis of Time Series (CREATES) funded by the Danish National Research Foundation is gratefully acknowledged. yAddress correspondence: Department of Economics and Business, Building 1322, Bartholins Allé 10, Aarhus University, DK-8000 Aarhus C, Denmark; zqin@econ.au. dk, +45 8716 5287. Keywords: Heterogeneous Beliefs; Continuous Trading; Dynamically E¤ectively Complete Market; Asset Pricing JEL classi cation: G1, G11, G12 1 Introduction In a homogeneous belief setting, Du¢ e and Huang (1985) and Du¢ e and Zame (1989) demonstrate that continuous trading can play a role as a compensation of long lived securities to dynamically complete the nancial market, with nite number of securities. In other words, a continuous trading Radner equilibrium can implement the same Arrow-Debreu consumption allocations. These are two of the very few papers to address the issue of welfare consequences of continuous-trading opportunities in a few long-lived securities (Sundaresan 2000). However, Du¢ e and Zame (1989) model the endowment by Arithmetic Brownian Motion, and their information structure cannot be generalized to the heterogeneous beliefs case with heterogeneous volatility. Since it follows from Girsanovs Theorem that the instantaneous volatility of the endowment is identical across investors under both individual perceived dynamics and risk-neutral dynamics. Therefore, to investigate the e¤ects of speculation on the heterogeneity of perceived dividend variance, another information structure which allows di¤erence in perceived variance is in order. Furthermore, in Du¢ e and Huang (1985), information structure is modeled by a complete probability space which constitutes all possible states of the world that could exist at a terminal date. The investors can receive information which is not relevant to the Pareto optimal consumption. However, in the real world, investors usually receive only part of the information in the economy, for instance, information on the aggregate consumption. Naturally, questions arise: How to generalize Du¢ e and Huang (1985) to a case with heterogeneous beliefs and information only relevant to aggregate consumption? Can the Pareto consumption allocations in the Arrow-Debreu equilibrium with heterogeneous beliefs in Christensen and Qin (2012) be implemented by some counterpart continuous trading model? In this paper, I show that, fortunately, dynamically complete market can be substituted by additional assumptions about preferences, in order to yield e¤ectively identical results. I nd with more restrictive assumptions of utility, i.e., with time-additive negative-exponential utility, continuous trading can dynamically e¤ectively complete the nancial market with heterogeneous beliefs. This result can considered to be a consequence of continuous-time Pareto e¢ cient side-betting based on the heterogeneously perceived variance of dividend. This work is a heterogeneous-belief extension of Du¢ e and Huang (1985). I summarize the main results in the following three aspects. First, an information structure is constructed to allow heterogeneity in perceived variance of the terminal dividend. The investors can speculate on the variance of the dividend throughout the interval [0; T ). Second, with continuous-time Pareto e¢ cient side-betting on con dence among investors, the market is