26 research outputs found
Stochastic programs without duality gaps
This paper studies dynamic stochastic optimization problems parametrized by a
random variable. Such problems arise in many applications in operations
research and mathematical finance. We give sufficient conditions for the
existence of solutions and the absence of a duality gap. Our proof uses
extended dynamic programming equations, whose validity is established under new
relaxed conditions that generalize certain no-arbitrage conditions from
mathematical finance
Superhedging in illiquid markets
We study contingent claims in a discrete-time market model where trading
costs are given by convex functions and portfolios are constrained by convex
sets. In addition to classical frictionless markets and markets with
transaction costs or bid-ask spreads, our framework covers markets with
nonlinear illiquidity effects for large instantaneous trades. We derive dual
characterizations of superhedging conditions for contingent claim processes in
a market without a cash account. The characterizations are given in terms of
stochastic discount factors that correspond to martingale densities in a market
with a cash account. The dual representations are valid under a topological
condition and a weak consistency condition reminiscent of the ``law of one
price'', both of which are implied by the no arbitrage condition in the case of
classical perfectly liquid market models. We give alternative sufficient
conditions that apply to market models with nonlinear cost functions and
portfolio constraints
A note on super-hedging for investor-producers
We study the situation of an agent who can trade on a financial market and
can also transform some assets into others by means of a production system, in
order to price and hedge derivatives on produced goods. This framework is
motivated by the case of an electricity producer who wants to hedge a position
on the electricity spot price and can trade commodities which are inputs for
his system. This extends the essential results of Bouchard & Nguyen Huu (2011)
to continuous time markets. We introduce the generic concept of conditional
sure profit along the idea of the no sure profit condition of R\`asonyi (2009).
The condition allows one to provide a closedness property for the set of
super-hedgeable claims in a very general financial setting. Using standard
separation arguments, we then deduce a dual characterization of the latter and
provide an application to power futures pricing
Arbitrage and deflators in illiquid markets
This paper presents a stochastic model for discrete-time trading in financial
markets where trading costs are given by convex cost functions and portfolios
are constrained by convex sets. The model does not assume the existence of a
cash account/numeraire. In addition to classical frictionless markets and
markets with transaction costs or bid-ask spreads, our framework covers markets
with nonlinear illiquidity effects for large instantaneous trades. In the
presence of nonlinearities, the classical notion of arbitrage turns out to have
two equally meaningful generalizations, a marginal and a scalable one. We study
their relations to state price deflators by analyzing two auxiliary market
models describing the local and global behavior of the cost functions and
constraints
Consistent Price Systems and Arbitrage Opportunities of the Second Kind in Models with Transaction Costs.
In contrast with the classical models of frictionless financial markets, market models with proportional transaction costs, even satisfying usual no-arbitrage properties, may admit arbitrage opportunities of the second kind. This means that there are self-financing portfolios with initial endowments laying outside the solvency region but ending inside. Such a phenomenon was discovered by M. R´asonyi in the discrete-time framework. In this note we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for the property which we call No Free Lunch of the 2nd Kind, NFL2. We provide a number of equivalent conditions elucidating, in particular, the financial meaning of the property B which appeared as an indispensable âtechnicalâ hypothesis in previous papers on hedging (super-replication) of contingent claims under transaction costs. We show that it is equivalent to another condition on the ârichnessâ of the set of consistent price systems, close to the condition PCE introduced by R´asonyi. In the last section we deduce the R´asonyi theorem from our general result using specific features of discrete-time models.Consistent price systems; No Free Lunch; Arbitrage; Transaction costs; Martingales; Set-valued processes;