Costis Skiadas

Continuous-time security pricing: A utility gradient approach

Abstract We consider a (not necessarily complete) continuous-time security market with semimartingale prices and general information filtration. In such a setting, we show that the first-order conditions for optimality of an agent maximizing a ‘smooth’ (but not necessarily additive) utility can be formulated as the martingale property of prices, after normalization by a ‘state-price’ process. The latter is given explicitly in terms of the agents utility gradient, which is in turn computed in closed form for a wide class of dynamic utilities, including stochastic differential utility, habit-forming utilities, and extensions.

Robust control and recursive utility

Abstract. This paper shows that a finite-horizon version of the robust control criterion appearing in recent papers by Hansen, Sargent, and their coauthors can be described as recursive utility, which in continuous time takes the form of the Stochastic Differential Utility (SDU) of Duffie and Epstein (1992). While it has previously been noted that Bellman equations arising in robust control settings are of the same form as Bellman equations arising from SDU maximization, here this connection is shown directly without reference to any underlying dynamics, or Markov structure.

Conditioning and Aggregation of Preferences

This paper develops a general framework for modeling choice under uncertainty that extends subjective expected utility to include nonseparabilities, state-dependence, and the effect of subjective or ill defined consequences. This is accomplished by not including consequences among the formal primitives. Instead, the effect of consequences is modeled indirectly, through conditional preferences over acts. The main results concern the aggregation of conditional utilities to form an unconditional utility, including the case of additive aggregation. Applications, obtained by further specifying the structure of acts and conditional preferences, include disappointment, regret, and the subjective value of information.

Efficient and equilibrium allocations with stochastic differential utility

Abstract This paper presents results on the existence and characterization of Pareto efficient and of equilibrium allocations in a continuous-time setting under uncertainty in which agents have stochastic differential utility, a version of recursive utility. In order to characterize equilibrium and efficient allocations in terms of pointwise first-order conditions, uniform properness conditions on preferences are avoided.

Smooth Ambiguity Aversion toward Small Risks and Continuous-Time Recursive Utility

Assuming Brownian/Poisson uncertainty, a certainty equivalent (CE) based on the smooth second-order expected utility of Klibanoff, Marinacci, and Mukerji is shown to be approximately equal to an expected-utility CE. As a consequence, the corresponding continuous-time recursive utility form is the same as for Kreps-Porteus utility. The analogous conclusions are drawn for a smooth divergence CE, based on a formulation of Maccheroni, Marinacci, and Rustichini, but only under Brownian uncertainty. Under Poisson uncertainty, a smooth divergence CE can be approximated with an expected-utility CE if and only if it is of the entropic type. A nonentropic divergence CE results in a new class of continuous-time recursive utilities that price Brownian and Poissonian risks differently.

A term structure model with preferences for the timing of resolution of uncertainty

SummaryIn this paper we present a model of the term structure of interest rates with imperfect information and stochastic differential utility, a form of non-additive recursive utility. A principal feature of recursive utility, that distinguishes it from time-separable expected utility, is its dependence on the timing of resolution of uncertainty. In our model, we parametrize the nonlinearity of recursive utility in a way that corresponds to preferences for the timing of resolution. This way we show explicitly the dependence of prices on the rate of information, as a consequence of the nature of utilities. State prices and the term structure of interest rates are obtained in closed form, and are shown to have a form in which derivative asset pricing is tractable. Comparative statics relating to the dependence of the term structure on the rate of information are also discussed.

Optimality and State Pricing in Constrained Financial Markets with Recursive Utility Under Continuous and Discontinuous Information

We study marginal pricing and optimality conditions for an agent maximizing generalized recursive utility in a financial market with information generated by Brownian motion and marked point processes. The setting allows for convex trading constraints, non-tradable income, and non-linear wealth dynamics. We show that the FBSDE system of the general optimality conditions reduces to a single BSDE under translation or scale invariance assumptions, and we identify tractable applications based on quadratic BSDEs. An appendix relates the main optimality conditions to duality.

Scale‐invariant uncertainty‐averse preferences and source‐dependent constant relative risk aversion

Preferences are defined over payoffs that are contingent on a finite number of states representing a horse race (Knightian uncertainty) and a roulette wheel (objective risk). The class of scale-invariant (SI) ambiguity-averse preferences, in a broad sense, is uniquely characterized by a multiple-prior utility representation. Adding a weak certainty-independence axiom is shown to imply either unit coefficient of relative risk aversion (CRRA) toward roulette risk or SI maxmin expected utility. Removing the weak independence axiom but adding a separability assumption on preferences over pure horse-race bets leads to source-dependent constant-relative-risk-aversion expected utility with a higher CRRA assigned to horse-race uncertainty than to roulette risk. The multiple-prior representation in this case is shown to generalize entropic variational preferences. An appendix characterizes the functional forms associated with SI ambiguity-averse preferences in terms of suitable weak independence axioms in place of scale invariance.

Chapter 19 Dynamic Portfolio Choice and Risk Aversion

Abstract This chapter presents a theory of optimal lifetime consumption-portfolio choice in a continuous information setting, with emphasis on the modeling of risk aversion through generalized recursive utility. A novel contribution is a decision theoretic development of the notions of source-dependent first- or second-order risk aversion. Backward stochastic differential equations (BSDEs) are explained heuristically as continuous-information versions of backward recursions on an information tree, and are used to formulate utility functions as well as optimality conditions. The role of scale invariance and quadratic BSDEs in obtaining tractable solutions is explained. A final section outlines extensions, including optimality conditions under trading constraints, and tractable formulations with nontradeable income.