Optimal arbitrage under model uncertainty

In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.Comment: Published in at http://dx.doi.org/10.1214/10-AAP755 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian Verification under Model Uncertainty

Machine learning enables systems to build and update domain models based on runtime observations. In this paper, we study statistical model checking and runtime verification for systems with this ability. Two challenges arise: (1) Models built from limited runtime data yield uncertainty to be dealt with. (2) There is no definition of satisfaction w.r.t. uncertain hypotheses. We propose such a definition of subjective satisfaction based on recently introduced satisfaction functions. We also propose the BV algorithm as a Bayesian solution to runtime verification of subjective satisfaction under model uncertainty. BV provides user-definable stochastic bounds for type I and II errors. We discuss empirical results from an example application to illustrate our ideas.Comment: Accepted at SEsCPS @ ICSE 201

Nested models and model uncertainty

Uncertainty about the appropriate choice among nested models is a central concern for optimal policy when policy prescriptions from those models differ. The standard procedure is to specify a prior over the parameter space ignoring the special status of some sub-models, e.g. those resulting from zero restrictions. This is especially problematic if a model's generalization could be either true progress or the latest fad found to fit the data. We propose a procedure that ensures that the specified set of sub-models is not discarded too easily and thus receives no weight in determining optimal policy. We find that optimal policy based on our procedure leads to substantial welfare gains compared to the standard practice.Optimal monetary policy, model uncertainty, Bayesian model estimation

Consistent Price Systems under Model Uncertainty

We develop a version of the fundamental theorem of asset pricing for discrete-time markets with proportional transaction costs and model uncertainty. A robust notion of no-arbitrage of the second kind is defined and shown to be equivalent to the existence of a collection of strictly consistent price systems.Comment: 19 page

Model Uncertainty and Liquidity

Extreme market outcomes are often followed by a lack of liquidity and a lack of trade. This market collapse seems particularly acute for markets where traders rely heavily on a specific empirical model such as in derivative markets. Asset pricing and trading, in these cases, are intrinsically model dependent. Moreover, the observed behavior of traders and institutions that places a large emphasis on 'worst-case scenarios'' through the use of 'stress testing'' and 'value-at-risk'' seems different than Savage rationality (expected utility) would suggest. In this paper we capture model-uncertainty explicitly using an Epstein-Wang (1994) uncertainty-averse utility function with an ambiguous underlying asset-returns distribution. To explore the connection of uncertainty with liquidity, we specify a simple market where a monopolist financial intermediary makes a market for a propriety derivative security. The market-maker chooses bid and ask prices for the derivative, then, conditional on trade in this market, chooses an optimal portfolio and consumption. We explore how uncertainty can increase the bid-ask spread and, hence, reduces liquidity. In addition, 'hedge portfolios'' for the market-maker, an important component to understanding spreads, can look very different from those implied by a model without Knightian uncertainty. Our infinite-horizon example produces short, dramatic decreases in liquidity even though the underlying environment is stationary.