445 research outputs found
On the Hedging of American Options in Discrete Time Markets with Proportional Transaction Costs
In this note, we consider a general discrete time financial market with
proportional transaction costs as in Kabanov and Stricker (2001), Kabanov et
al. (2002), Kabanov et al. (2003) and Schachermayer (2004). We provide a dual
formulation for the set of initial endowments which allow to super-hedge some
American claim. We show that this extends the result of Chalasani and Jha
(2001) which was obtained in a model with constant transaction costs and risky
assets which evolve on a finite dimensional tree. We also provide fairly
general conditions under which the expected formulation in terms of stopping
times does not work
Strong Approximations of BSDEs in a domain
We study the strong approximation of a Backward SDE with finite stopping time
horizon, namely the first exit time of a forward SDE from a cylindrical domain.
We use the Euler scheme approach of Bouchard and Touzi, Zhang 04}. When the
domain is piecewise smooth and under a non-characteristic boundary condition,
we show that the associated strong error is at most of order h^{\frac14-\eps}
where denotes the time step and \eps is any positive parameter. This rate
corresponds to the strong exit time approximation. It is improved to
h^{\frac12-\eps} when the exit time can be exactly simulated or for a weaker
form of the approximation error. Importantly, these results are obtained
without uniform ellipticity condition.Comment: 35 page
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
Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
We study a class of stochastic target games where one player tries to find a
strategy such that the state process almost-surely reaches a given target, no
matter which action is chosen by the opponent. Our main result is a geometric
dynamic programming principle which allows us to characterize the value
function as the viscosity solution of a non-linear partial differential
equation. Because abstract mea-surable selection arguments cannot be used in
this context, the main obstacle is the construction of measurable
almost-optimal strategies. We propose a novel approach where smooth
supersolutions are used to define almost-optimal strategies of Markovian type,
similarly as in ver-ification arguments for classical solutions of
Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed
by an exten-sion of Krylov's method of shaken coefficients. We apply our
results to a problem of option pricing under model uncertainty with different
interest rates for borrowing and lending.Comment: To appear in MO
Optimal consumption in discrete-time financial models with industrial investment opportunities and nonlinear returns
We consider a general discrete-time financial market with proportional
transaction costs as in [Kabanov, Stricker and R\'{a}sonyi Finance and
Stochastics 7 (2003) 403--411] and [Schachermayer Math. Finance 14 (2004)
19--48]. In addition to the usual investment in financial assets, we assume
that the agents can invest part of their wealth in industrial projects that
yield a nonlinear random return. We study the problem of maximizing the utility
of consumption on a finite time period. The main difficulty comes from the
nonlinearity of the nonfinancial assets' return. Our main result is to show
that existence holds in the utility maximization problem. As an intermediary
step, we prove the closedness of the set of attainable claims under a
robust no-arbitrage property similar to the one introduced in [Schachermayer
Math. Finance 14 (2004) 19--48] and further discussed in [Kabanov, Stricker and
R\'{a}sonyi Finance and Stochastics 7 (2003) 403--411]. This allows us to
provide a dual formulation for .Comment: Published at http://dx.doi.org/10.1214/105051605000000467 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Barrier Option Hedging under Constraints: A Viscosity Approach
We study the problem of finding the minimal initial capital needed in order to hedge without risk a barrier option when the vector of proportions of wealth invested in each risky asset is constraint to lie in a closed convex domain. In the context of a Brownian diffusion model, we provide a PDE characterization of the super-hedging price. This extends the result of Broadie, Cvitanic and Soner (1998) and Cvitanic, Pham and Touzi (1999) which was obtained for plain vanilla options, and provides a natural numerical procedure for computing the corresponding super-hedging price. As a by-product, we obtain a comparison theorem for a class of parabolic PDE with relaxed Dirichet conditions involving a constraint on the gradient.Super-replication, barrier options, portfolio constraints, viscosity solutions
Explicit characterization of the super-replication strategy in financial markets with partial transaction costs
We consider a continuous time multivariate financial market with proportional transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi (2000) except that some of the assets can be exchanged freely, i.e. without paying transaction costs. This is the so-called non-effcient friction case. To our knowledge, this is the first time that such a model is considered in a continuous time setting. In this context, we generalize the result of the above paper and prove that the super-replication price is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time.Transaction costs, hedging options, viscosity solutions
Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs
We propose a continuous time model for financial markets with proportional
transactions costs and a continuum of risky assets. This is motivated by bond
markets in which the continuum of assets corresponds to the continuum of
possible maturities. Our framework is well adapted to the study of no-arbitrage
properties and related hedging problems. In particular, we extend the
Fundamental Theorem of Asset Pricing of Guasoni, R\'asonyi and L\'epinette
(2012) which concentrates on the one dimensional case. Namely, we prove that
the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the
existence of a Strictly Consistent Price System. Interestingly, the presence of
transaction costs allows a natural definition of trading strategies and avoids
all the technical and un-natural restrictions due to stochastic integration
that appear in bond models without friction. We restrict to the case where
exchange rates are continuous in time and leave the general c\`adl\`ag case for
further studies.Comment: 41 page
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