3,915 research outputs found
Martingale Optimal Transport and Robust Hedging in Continuous Time
The duality between the robust (or equivalently, model independent) hedging
of path dependent European options and a martingale optimal transport problem
is proved. The financial market is modeled through a risky asset whose price is
only assumed to be a continuous function of time. The hedging problem is to
construct a minimal super-hedging portfolio that consists of dynamically
trading the underlying risky asset and a static position of vanilla options
which can be exercised at the given, fixed maturity. The dual is a
Monge-Kantorovich type martingale transport problem of maximizing the expected
value of the option over all martingale measures that has the given marginal at
maturity. In addition to duality, a family of simple, piecewise constant
super-replication portfolios that asymptotically achieve the minimal
super-replication cost is constructed
Robust Hedging with Proportional Transaction Costs
Duality for robust hedging with proportional transaction costs of path
dependent European options is obtained in a discrete time financial market with
one risky asset. Investor's portfolio consists of a dynamically traded stock
and a static position in vanilla options which can be exercised at maturity.
Both the stock and the option trading is subject to proportional transaction
costs. The main theorem is duality between hedging and a Monge-Kantorovich type
optimization problem. In this dual transport problem the optimization is over
all the probability measures which satisfy an approximate martingale condition
related to consistent price systems in addition to the usual marginal
constraints
Dual formulation of second order target problems
This paper provides a new formulation of second order stochastic target
problems introduced in [SIAM J. Control Optim. 48 (2009) 2344-2365] by
modifying the reference probability so as to allow for different scales. This
new ingredient enables us to prove a dual formulation of the target problem as
the supremum of the solutions of standard backward stochastic differential
equations. In particular, in the Markov case, the dual problem is known to be
connected to a fully nonlinear, parabolic partial differential equation and
this connection can be viewed as a stochastic representation for all nonlinear,
scalar, second order, parabolic equations with a convex Hessian dependence.Comment: Published in at http://dx.doi.org/10.1214/12-AAP844 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Weak Approximation of G-Expectations
We introduce a notion of volatility uncertainty in discrete time and define
the corresponding analogue of Peng's G-expectation. In the continuous-time
limit, the resulting sublinear expectation converges weakly to the
G-expectation. This can be seen as a Donsker-type result for the G-Brownian
motion.Comment: 14 page
Approximating stochastic volatility by recombinant trees
A general method to construct recombinant tree approximations for stochastic
volatility models is developed and applied to the Heston model for stock price
dynamics. In this application, the resulting approximation is a four tuple
Markov process. The first two components are related to the stock and
volatility processes and take values in a two-dimensional binomial tree. The
other two components of the Markov process are the increments of random walks
with simple values in . The resulting efficient option pricing
equations are numerically implemented for general American and European options
including the standard put and calls, barrier, lookback and Asian-type
pay-offs. The weak and extended weak convergences are also proved.Comment: Published in at http://dx.doi.org/10.1214/13-AAP977 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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