Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

Abstract

We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial market we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.JEL classification: C62, D52, D53

A financial market with interacting investors: does an equilibrium exist?

“…Cheridito et al [3] follow in the footsteps of Horst et al [11] to solve a problem of valuing a derivative in an incomplete market by equilibrium considerations. In Horst et al [11], the problem can be solved in a one-dimensional framework, since the derivative is assumed to complete the market.…”

“…In Horst et al [11], the problem can be solved in a one-dimensional framework, since the derivative is assumed to complete the market. Cheridito et al [3] do not impose this condition, which makes the analysis much more involved. The authors solve the problem in a discrete framework, but close their work with considerations on the continuous case.…”

“…Like above, we construct an auxiliary strategy π (3,1) for the third agent, replacing η 1 by η 3 , U 1 by U 3 , and F 1 by F 3 + λ 3 n−1 (X π (1,2) T + X π (2,2) T ). To account for the interdependence, we set λ n 1,2 :=…”

“…π (3,1) , π (2,3) := π (2,2) + λ n 1,2 π (3,3) , π (1,3) := π (1,2) + λ n 2,1 π (3,3) , achieving that (2,2) , (2,2) .…”

Equilibrium asset pricing with transaction costs

“…This can be interpreted as a fundamental liquidation value as in Kyle [42], a terminal dividend as in Kramkov [39], or the payoff of a derivative depending on an exogenous underlying as in Cheridito et al [15].…”