44 research outputs found
An explicit Skorokhod embedding for functionals of Markovian excursions
We develop an explicit non-randomized solution to the Skorokhod embedding
problem in an abstract setup of signed functionals of Markovian excursions. Our
setting allows to solve the Skorokhod embedding problem, in particular, for
diffusions and their (signed, scaled) age processes, for Azema's martingale,
for spectrally one-sided Levy processes and their reflected versions, for
Bessel processes of dimension smaller than 2, and for their age processes, as
well as for the age process of excursions of Cox-Ingersoll-Ross processes. This
work is a continuation and an important generalization of Obloj and Yor (SPA
110) [35]. Our methodology, following [35], is based on excursion theory and
the solution to the Skorokhod embedding problem is described in terms of the
Ito measure of the functional. We also derive an embedding for positive
functionals and we correct a mistake in the formula in [35] for measures with
atoms.Comment: 50 page
Fine-tune your smile: Correction to Hagan et al
In this small note we use results derived in Berestycki et al. to correct the
celebrated formulae of Hagan et al. We derive explicitly the correct zero order
term in the expansion of the implied volatility in time to maturity. The new
term is consistent as . Furthermore, numerical simulations show
that it reduces or eliminates known pathologies of the earlier formula.Comment: Typos and reference corrected. Eq (3) valid for all x no
A complete characterization of local martingales which are functions of Brownian motion and its maximum
We prove the max-martingale conjecture given in recent article with Marc Yor.
We show that for a continuous local martingale and a function
, is a local martingale if and
only if there exists a locally integrable function such that
. This implies readily, via Levy's
equivalence theorem, an analogous result with the maximum process replaced by
the local time at 0
On local martingale and its supremum: harmonic functions and beyond
We discuss certain facts involving a continuous local martingale and its
supremum . A complete characterization of -harmonic
functions is proposed. This yields an important family of martingales, the
usefulness of which is demonstrated, by means of examples involving the
Skorokhod embedding problem, bounds on the law of the supremum, or the local
time at 0, of a martingale with a fixed terminal distribution, or yet in some
Brownian penalization problems. In particular we obtain new bounds on the law
of the local time at 0, which involve the excess wealth order
Robust estimation of superhedging prices
We consider statistical estimation of superhedging prices using historical
stock returns in a frictionless market with d traded assets. We introduce a
plugin estimator based on empirical measures and show it is consistent but
lacks suitable robustness. To address this we propose novel estimators which
use a larger set of martingale measures defined through a tradeoff between the
radius of Wasserstein balls around the empirical measure and the allowed norm
of martingale densities. We establish consistency and robustness of these
estimators and argue that they offer a superior performance relative to the
plugin estimator. We generalise the results by replacing the superhedging
criterion with acceptance relative to a risk measure. We further extend our
study, in part, to the case of markets with traded options, to a multiperiod
setting and to settings with model uncertainty. We also study convergence rates
of estimators and convergence of superhedging strategies.Comment: This work will appear in the Annals of Statistics. The above version
merges the main paper to appear in print and its online supplemen
An explicit Skorokhod embedding for spectrally negative Levy processes
We present an explicit solution to the Skorokhod embedding problem for
spectrally negative L\'evy processes. Given a process and a target measure
satisfying an explicit admissibility condition we define functions
\f_\pm such that the stopping time T = \inf\{t>0: X_t \in \{-\f_-(L_t),
\f_+(L_t)\}\} induces . We also treat versions of which take
into account the sign of the excursion straddling time . We prove that our
stopping times are minimal and we describe criteria under which they are
integrable. We compare our solution with the one proposed by Bertoin and Le Jan
(1992) and we compute explicitly their general quantities in our setup.
Our method relies on some new explicit calculations relating scale functions
and the It\^o excursion measure of . More precisely, we compute the joint
law of the maximum and minimum of an excursion away from 0 in terms of the
scale function.Comment: This is the final version of the paper that has been accepted for
publication in J. Theor. Probab. In this new version several typos were
corrected and Lemma 6(iii) [now Lemma 5(iii)] was modified. The original
publication is available at http://www.springerlink.co
Efficient discretisation of stochastic differential equations
The aim of this study is to find a generic method for generating a path of
the solution of a given stochastic differential equation which is more
efficient than the standard Euler-Maruyama scheme with Gaussian increments.
First we characterize the asymptotic distribution of pathwise error in the
Euler-Maruyama scheme with a general partition of time interval and then, show
that the error is reduced by a factor (d+2)/d when using a partition associated
with the hitting times of sphere for the driving d-dimensional Brownian motion.
This reduction ratio is the best possible in a symmetric class of partitions.
Next we show that a reduction which is close to the best possible is achieved
by using the hitting time of a moving sphere which is easier to implement
Computational Methods for Martingale Optimal Transport problems
We establish numerical methods for solving the martingale optimal transport
problem (MOT) - a version of the classical optimal transport with an additional
martingale constraint on transport's dynamics. We prove that the MOT value can
be approximated using linear programming (LP) problems which result from a
discretisation of the marginal distributions combined with a suitable
relaxation of the martingale constraint. Specialising to dimension one, we
provide bounds on the convergence rate of the above scheme. We also show a
stability result under only partial specification of the marginal
distributions. Finally, we specialise to a particular discretisation scheme
which preserves the convex ordering and does not require the martingale
relaxation. We introduce an entropic regularisation for the corresponding LP
problem and detail the corresponding iterative Bregman projection. We also
rewrite its dual problem as a minimisation problem without constraint and solve
it by computing the concave envelope of scattered data
The robust superreplication problem: a dynamic approach
In the frictionless discrete time financial market of Bouchard et al.(2015)
we consider a trader who, due to regulatory requirements or internal risk
management reasons, is required to hedge a claim in a risk-conservative
way relative to a family of probability measures . We first
describe the evolution of - the superhedging price at time of
the liability at maturity - via a dynamic programming principle and
show that can be seen as a concave envelope of
evaluated at today's prices. Then we consider an optimal investment problem for
a trader who is rolling over her robust superhedge and phrase this as a robust
maximisation problem, where the expected utility of inter-temporal consumption
is optimised subject to a robust superhedging constraint. This utility
maximisation is carrried out under a new family of measures ,
which no longer have to capture regulatory or institutional risk views but
rather represent trader's subjective views on market dynamics. Under suitable
assumptions on the trader's utility functions, we show that optimal investment
and consumption strategies exist and further specify when, and in what sense,
these may be unique