Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets

Abstract

We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and of a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information. We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy.

Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

“…Results on market completeness in more general equilibrium models can be found in Magill and Shafer (1990) and Anderson and Raimondo (2009). …”

On securitization, market completion and equilibrium risk transfer

“…Hence it is potentially complete, but there is no a-priori reason to assume completeness in equilibrium 3 . The problem of dynamic completion of financial markets has recently been addressed by Anderson & Raimondo (2008). They provide a non-degeneracy condition on the terminal security dividends to insure completeness in equilibrium.…”

“…This allows us to deduce market completeness in equilibrium. 2 Market completeness in equilibrium is key. Our model comprises of two sources of uncertainties and two assets.…”

“…Allowing for such a feedback is one of the main contributions of this paper. 2 The question of when trading in derivative securities completes the market has been addressed by many authors including Davis (2004), Davis & Obloj (2008) and Romano and Touzi (1997). In their models derivative prices are given as conditional expected payoff under an exogenous measure (usually the real-world measure) while in our model the pricing measure is derived endogenously by an equilibrium condition.…”

Hyperfinite Construction of G-Expectation

“…This approach is very promising because it also allows, for instance, to study 1 continuous-time stochastic processes as formally finite objects. Many authors have applied nonstandard analysis to problems in measure theory, probability theory and mathematical economics (see for example, Anderson and Raimondo [3] and the references therein or the contribution in Berg and Neves [4]), especially after Loeb [12] converted nonstandard measures (i.e. the images of standard measures under the nonstandard embedding * ) into real-valued, countably additive measures, by means of the standard part operator and Caratheodory's extension theorem.…”