39 research outputs found
On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale
We study the joint laws of a continuous, uniformly integrable martingale, its
maximum, and its minimum. In particular, we give explicit martingale
inequalities which provide upper and lower bounds on the joint exit
probabilities of a martingale, given its terminal law. Moreover, by
constructing explicit and novel solutions to the Skorokhod embedding problem,
we show that these bounds are tight. Together with previous results of Az\'ema
& Yor, Perkins, Jacka and Cox & Ob{\l}\'oj, this allows us to completely
characterise the upper and lower bounds on all possible exit/no-exit
probabilities, subject to a given terminal law of the martingale. In addition,
we determine some further properties of these bounds, considered as functions
of the maximum and minimum.Comment: 19 pages, 4 figures. This is the authors' accepted version of the
paper which will appear in Stochastic Processes and their Application
Pathwise Stochastic Calculus with Local Times
We study a notion of local time for a continuous path, defined as a limit of
suitable discrete quantities along a general sequence of partitions of the time
interval. Our approach subsumes other existing definitions and agrees with the
usual (stochastic) local times a.s. for paths of a continuous semimartingale.
We establish pathwise version of the It\^o-Tanaka, change of variables and
change of time formulae. We provide equivalent conditions for existence of
pathwise local time. Finally, we study in detail how the limiting objects, the
quadratic variation and the local time, depend on the choice of partitions. In
particular, we show that an arbitrary given non-decreasing process can be
achieved a.s. by the pathwise quadratic variation of a standard Brownian motion
for a suitable sequence of (random) partitions; however, such degenerate
behavior is excluded when the partitions are constructed from stopping times.Comment: minor change
Robust pricing--hedging duality for American options in discrete time financial markets
We investigate pricing-hedging duality for American options in discrete time
financial models where some assets are traded dynamically and others, e.g. a
family of European options, only statically. In the first part of the paper we
consider an abstract setting, which includes the classical case with a fixed
reference probability measure as well as the robust framework with a
non-dominated family of probability measures. Our first insight is that by
considering a (universal) enlargement of the space, we can see American options
as European options and recover the pricing-hedging duality, which may fail in
the original formulation. This may be seen as a weak formulation of the
original problem. Our second insight is that lack of duality is caused by the
lack of dynamic consistency and hence a different enlargement with dynamic
consistency is sufficient to recover duality: it is enough to consider
(fictitious) extensions of the market in which all the assets are traded
dynamically. In the second part of the paper we study two important examples of
robust framework: the setup of Bouchard and Nutz (2015) and the martingale
optimal transport setup of Beiglb\"ock et al. (2013), and show that our general
results apply in both cases and allow us to obtain pricing-hedging duality for
American options.Comment: 29 page
On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation
We study the class of Az\'ema-Yor processes defined from a general
semimartingale with a continuous running maximum process. We show that they
arise as unique strong solutions of the Bachelier stochastic differential
equation which we prove is equivalent to the drawdown equation. Solutions of
the latter have the drawdown property: they always stay above a given function
of their past maximum. We then show that any process which satisfies the
drawdown property is in fact an Az\'ema-Yor process. The proofs exploit group
structure of the set of Az\'ema-Yor processes, indexed by functions, which we
introduce. We investigate in detail Az\'ema-Yor martingales defined from a
nonnegative local martingale converging to zero at infinity. We establish
relations between average value at risk, drawdown function, Hardy-Littlewood
transform and its inverse. In particular, we construct Az\'ema-Yor martingales
with a given terminal law and this allows us to rediscover the Az\'ema-Yor
solution to the Skorokhod embedding problem. Finally, we characterize
Az\'ema-Yor martingales showing they are optimal relative to the concave
ordering of terminal variables among martingales whose maximum dominates
stochastically a given benchmark.Comment: Published in at http://dx.doi.org/10.1214/10-AOP614 the Annals of
Probability (http://www.imstat.org/aop/) 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