Coherent Price Systems and Uncertainty-Neutral Valuation

We consider fundamental questions of arbitrage pricing arising when the uncertainty model is given by a set of possible mutually singular probability measures. With a single probability model, essential equivalence between the absence of arbitrage and the existence of an equivalent martingale measure is a folk theorem, see Harrison and Kreps (1979). We establish a microeconomic foundation of sublinear price systems and present an extension result. In this context we introduce a prior dependent notion of marketed spaces and viable price systems. We associate this extension with a canonically altered concept of equivalent symmetric martingale measure sets, in a dynamic trading framework under absence of prior depending arbitrage. We prove the existence of such sets when volatility uncertainty is modeled by a stochastic differential equation, driven by Peng's G-Brownian motions

Existence of Arrow-Debreu Equilibrium with Generalized Stochastic Differential Utility

Beißner P. Existence of Arrow-Debreu Equilibrium with Generalized Stochastic Differential Utility. Working Papers. Institute of Mathematical Economics. Vol 447. Bielefeld: Center for Mathematical Economics; 2011.This paper establishes, in the setting of Brownian information, a general equilibrium existence result under a stochastic differential for- mulation of intertemporal recursive utility. The present class of utility functionals is generated by a backward stochastic differential equation and incorporates preference for the local risk of the stochastic utility process. The setting contains models in which Knightian uncertainty is repre- sented in the subjective and objective sense

Radner Equilibria under Ambiguous Volatility

Beißner P. Radner Equilibria under Ambiguous Volatility. Center for Mathematical Economics Working Papers. Vol 493. Bielefeld: Center for Mathematical Economics; 2013.The present paper considers a class of general equilibrium economies when the primitive uncertainty model features uncertainty about continuous-time volatility. This requires a set of mutually singular priors, which do not share the same null sets. For this setting we introduce an appropriate commodity space and the dual of linear and continuous price systems. All agents in the economy are heterogeneous in their preference for uncer- tainty. Each utility functional is of variational type. The existence of equi- librium is approached by a generalized excess utility fixed point argument. Such Arrow-Debreu allocations can be implemented into a Radner economy with continuous-time trading. Effective completeness of the market spaces al- ters to an endogenous property. Only mean unambiguous claims equivalently satisfying the classical martingale representation property build the marketed space

Microeconomic theory of financial markets under volatility uncertainty

Beißner P. Microeconomic theory of financial markets under volatility uncertainty. Bielefeld: Bielefeld University; 2013

Knight-Walras equilibria

Beißner P, Riedel F. Knight-Walras equilibria. Center for Mathematical Economics Working Papers. Vol 558. Bielefeld: Center for Mathematical Economics; 2016.Knightian uncertainty leads naturally to nonlinear expectations. We introduce a corresponding equilibrium concept with sublinear prices and establish their existence. In general, such equilibria lead to Pareto inefficiency and coincide with Arrow-Debreu equilibria only if the values of net trades are ambiguity-free in the mean. Without aggregate uncertainty, inefficiencies arise generically. We introduce a constrained efficiency concept, uncertainty-neutral efficiency and show that Knight-Walras equilibrium allocations are efficient in this constrained sense. Arrow-Debreu equilibria turn out to be non-robust with respect to the introduction of Knightian uncertainty

Equilibria under Knightian Price Uncertainty

Beißner P, Riedel F. Equilibria under Knightian Price Uncertainty . Center for Mathematical Economics Working Papers. Vol 597. Bielefeld: Center for Mathematical Economics; 2018.We study economies in which agents face Knightian uncertainty about state prices. Knightian uncertainty leads naturally to nonlinear expectations. We introduce a corresponding equilibrium concept with sublinear prices and prove that equilibria exist under weak conditions. In general, such equilibria lead to Pareto inefficient allocations; the equilibria coincide with Arrow-Debreu equilibria only if the values of net trades are ambiguity-free in the mean. In economies without aggregate uncertainty, inefficiencies are generic. We introduce a constrained efficiency concept, uncertainty-neutral efficiency; equilibrium allocations under price uncertainty are efficient in this constrained sense. Arrow-Debreu equilibria turn out to be non-robust with respect to the introduction of Knightian uncertainty

Non-Implementability of Arrow-Debreu Equilibria by Continuous Trading under Knightian Uncertainty

Riedel F, Beißner P. Non-Implementability of Arrow-Debreu Equilibria by Continuous Trading under Knightian Uncertainty. Center for Mathematical Economics Working Papers. Vol 527. Bielefeld: Center for Mathematical Economics; 2014.Under risk, Arrow-Debreu equilibria can be implemented as Radner equilibria by continuous trading of few long-lived securities. We show that this result generically fails if there is Knightian uncertainty in the volatility. Implementation is only possible if all discounted net trades of the equilibrium allocation are mean ambiguity-free

Dynamically Consistent α-Maxmin Expected Utility

Beißner P, Lin Q, Riedel F. Dynamically Consistent α-Maxmin Expected Utility. Center for Mathematical Economics Working Papers. Vol 593. Bielefeld: Center for Mathematical Economics; 2017.The α-maxmin model is a prominent example of preferences under Knightian uncertainty as it allows to distinguish ambiguity and ambiguity attitude. These preferences are dynamically inconsistent for nontrivial versions of α. In this paper, we derive a recursive, dynamically consistent version of the α-maxmin model. In the continuous-time limit, the resulting dynamic utility function can be represented as a convex mixture between worst and best case, but now at the local, infinitesimal level. We study the properties of the utility function and provide an Arrow- Pratt approximation of the static and dynamic certainty equivalent. We derive a consumption-based capital asset pricing formula and study the implications for derivative valuation under indifference pricing

Brownian equilibria under Knightian uncertainty

Beißner P. Brownian equilibria under Knightian uncertainty. MATHEMATICS AND FINANCIAL ECONOMICS. 2015;9(1):39-56.This paper establishes, in the setting of Brownian information, a general equilibrium existence result in a heterogeneous agent economy. The existence is generic among income distributions. Agents differ moreover in their stochastic differential formulation of intertemporal recursive utility. The present class of utility functionals is generated by a recursive integral equation and incorporates preference for the local risk of the stochastic utility process. The setting contains models in which Knightian uncertainty is represented in terms of maxmin preferences of Chen and Epstein (Econometrica 70:1403-1443, 2002). Alternatively, Knightian decision making in terms of an inertia formulation from Bewley (Decis. Econ. Financ. 25:79-110, 2002) can be modeled as well

Brownian equilibria under Knightian uncertainty

This paper establishes, in the setting of Brownian information, a general equilibrium existence result in a heterogeneous agent economy. The existence is generic among income distributions. Agents differ moreover in their stochastic differential formulation of intertemporal recursive utility. The present class of utility functionals is generated by a recursive integral equation and incorporates preference for the local risk of the stochastic utility process. The setting contains models in which Knightian uncertainty is represented in terms of maxmin preferences of Chen and Epstein (2002). Alternatively, Knightian decision making in terms of an inertia formulation from Bewley (2002) can be modeled as well.generalized stochastic differential utility, super-gradients, properness, general equilibrium, Knightian uncertainty, generic existence, asset pricing