Optimal arbitrage under model uncertainty.

By Daniel Fernholz, Ioannis Karatzas. 2011

53 citations recorded by the source corpus.

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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/ae9b22fe48b6b76c78775f842e052ea53081e07a

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Citations in the corpus

UTILITY MAXIMIZATION UNDER MODEL UNCERTAINTY IN DISCRETE TIME (2013, background): “Turning to different but related nondominated problems in the mathematical finance literature, [3] studies robust maximization of asymptotic growth under covariance uncertainty, [15] analyzes optimal arbitrage under model uncertainty and [7] and [34] consider risk management under model uncertainty.”
ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES (2014, background): “[5, 16, 20, 30, 36, 38, 39, 41, 44, 45], among others, or when all paths are considered possible for the stock and options are also traded; see, e.”
Arbitrage and duality in nondominated discrete-time models (2013, background): “In the setting where no options are traded and P is supposed to consist of martingale measures in the rst place, there is a substantial literature on superhedging under volatility uncertainty, where S is a (continuous-time) process with continuous trajectories; see, among others, [8, 27, 40, 41, 42,”
Quasi-sure Stochastic Analysis through Aggregation (2010, background): “We also mention a related recent paper by Fernholz and Karatzas [8] whose existence results are obtained in the Markov case with a continuity assumption on the corresponding value function.”
Financial markets with volatility uncertainty (2010, background): “Fernholz and Karatzas (2010) consider the question of outperforming the market when the assumption of no–arbitrage is not imposed and incorporate volatility uncertainty.”
Distribution of the time to explosion for one-dimensional diffusions (2013, result): “The following characterization of this function is analogous to the results in Problem 3.5.1 of McKean (1969) and in Exercise 4.4.7 of Karatzas and Shreve (1991); see also Fernholz and Karatzas (2010, 2011) and Ruf (2013a).”
The geometry of relative arbitrage (2014, background): “rage have been done in [Str12]. A recent trend in mathematical ﬁnance is to study models with model uncertainty and robust pricing and hedging of contingent claims. For example, Fernholz and Karatzas [FK11] characterizes optimal relative arbitrage when the covariance matrix is uncertain. The theory ”