7,558 research outputs found
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
Duality for pathwise superhedging in continuous time
We provide a model-free pricing-hedging duality in continuous time. For a
frictionless market consisting of risky assets with continuous price
trajectories, we show that the purely analytic problem of finding the minimal
superhedging price of a path dependent European option has the same value as
the purely probabilistic problem of finding the supremum of the expectations of
the option over all martingale measures. The superhedging problem is formulated
with simple trading strategies, the claim is the limit inferior of continuous
functions, which allows for upper and lower semi-continuous claims, and
superhedging is required in the pathwise sense on a -compact sample
space of price trajectories. If the sample space is stable under stopping, the
probabilistic problem reduces to finding the supremum over all martingale
measures with compact support. As an application of the general results we
deduce dualities for Vovk's outer measure and semi-static superhedging with
finitely many securities
A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options
We consider the problem of superhedging under volatility uncertainty for an
investor allowed to dynamically trade the underlying asset, and statically
trade European call options for all possible strikes with some given maturity.
This problem is classically approached by means of the Skorohod Embedding
Problem (SEP). Instead, we provide a dual formulation which converts the
superhedging problem into a continuous martingale optimal transportation
problem. We then show that this formulation allows us to recover previously
known results about lookback options. In particular, our methodology induces a
new proof of the optimality of Az\'{e}ma-Yor solution of the SEP for a certain
class of lookback options. Unlike the SEP technique, our approach applies to a
large class of exotics and is suitable for numerical approximation techniques.Comment: Published in at http://dx.doi.org/10.1214/13-AAP925 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The maximum maximum of a martingale with given marginals
We obtain bounds on the distribution of the maximum of a martingale with
fixed marginals at finitely many intermediate times. The bounds are sharp and
attained by a solution to -marginal Skorokhod embedding problem in
Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many
marginals (2013) Preprint]. It follows that their embedding maximizes the
maximum among all other embeddings. Our motivating problem is superhedging
lookback options under volatility uncertainty for an investor allowed to
dynamically trade the underlying asset and statically trade European call
options for all possible strikes and finitely-many maturities. We derive a
pathwise inequality which induces the cheapest superhedging value, which
extends the two-marginals pathwise inequality of Brown, Hobson and Rogers
[Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by
elementary arguments, is derived by following the stochastic control approach
of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014)
312-336].Comment: Published at http://dx.doi.org/10.1214/14-AAP1084 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robust Optimal Control for a Consumption-investment Problem
We give an explicit PDE characterization for the solution of the problem of maximizing the utility of both terminal wealth and intertemporal consumption under model uncertainty. The underlying market model consists of a risky asset, whose volatility and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with risk aversion parameter 0Optimal Consumption, Robust Control, Model Uncertainty, Incomplete Markets, Stochastic Volatility, Coherent Risk Measures, Convex Duality
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