29 research outputs found
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
On the monotonicity principle of optimal Skorokhod embedding problem
In this paper, we provide an alternative proof of the monotonicity principle
for the optimal Skorokhod embedding problem established by Beiglb\"ock, Cox and
Huesmann. This principle presents a geometric characterization that reflects
the desired optimality properties of Skorokhod embeddings. Our proof is based
on the adaptation of the Monge-Kantorovich duality in our context together with
a delicate application of the optional cross-section theorem and a clever
conditioning argument
Optimal Skorokhod embedding under finitely-many marginal constraints
The Skorokhod embedding problem aims to represent a given probability measure
on the real line as the distribution of Brownian motion stopped at a chosen
stopping time. In this paper, we consider an extension of the optimal Skorokhod
embedding problem to the case of finitely-many marginal constraints. Using the
classical convex duality approach together with the optimal stopping theory, we
obtain the duality results which are formulated by means of probability
measures on an enlarged space. We also relate these results to the problem of
martingale optimal transport under multiple marginal constraints
Tightness and duality of martingale transport on the Skorokhod space
The martingale optimal transport aims to optimally transfer a probability
measure to another along the class of martingales. This problem is mainly
motivated by the robust superhedging of exotic derivatives in financial
mathematics, which turns out to be the corresponding Kantorovich dual. In this
paper we consider the continuous-time martingale transport on the Skorokhod
space of cadlag paths. Similar to the classical setting of optimal transport,
we introduce different dual problems and establish the corresponding dualities
by a crucial use of the S-topology and the dynamic programming principle
Some Results on Skorokhod Embedding and Robust Hedging with Local Time
In this paper, we provide some results on Skorokhod embedding with local time
and its applications to the robust hedging problem in finance. First we
investigate the robust hedging of options depending on the local time by using
the recently introduced stochastic control approach, in order to identify the
optimal hedging strategies, as well as the market models that realize the
extremal no-arbitrage prices. As a by-product, the optimality of Vallois'
Skorokhod embeddings is recovered. In addition, under appropriate conditions,
we derive a new solution to the two-marginal Skorokhod embedding as a
generalization of the Vallois solution. It turns out from our analysis that one
needs to relax the monotonicity assumption on the embedding functions in order
to embed a larger class of marginal distributions. Finally, in a full-marginal
setting where the stopping times given by Vallois are well-ordered, we
construct a remarkable Markov martingale which provides a new example of fake
Brownian motion
Generalised arbitrage-free SVI volatility surfaces
In this article we propose a generalisation of the recent work of Gatheral
and Jacquier on explicit arbitrage-free parameterisations of implied volatility
surfaces. We also discuss extensively the notion of arbitrage freeness and
Roger Lee's moment formula using the recent analysis by Roper. We further
exhibit an arbitrage-free volatility surface different from Gatheral's SVI
parameterisation.Comment: 20 pages, 4 figures. Corrected some typo
Strong equivalence between metrics of Wasserstein type
The sliced Wasserstein and more recently max-sliced Wasserstein metrics
\mW_p have attracted abundant attention in data sciences and machine learning
due to its advantages to tackle the curse of dimensionality. A question of
particular importance is the strong equivalence between these projected
Wasserstein metrics and the (classical) Wasserstein metric \Wc_p. Recently,
Paty and Cuturi have proved the strong equivalence of \mW_2 and \Wc_2. We
show that the strong equivalence also holds for , while we show that the
sliced Wasserstein metric does not share this nice property.Comment: To appear in Electronic Communications in Probabilit
Mean field game of mutual holding with defaultable agents, and systemic risk
We introduce the possibility of default in the mean field game of mutual
holding of Djete and Touzi [11]. This is modeled by introducing absorption at
the origin of the equity process. We provide an explicit solution of this mean
field game. Moreover, we provide a particle system approximation, and we derive
an autonomous equation for the time evolution of the default probability, or
equivalently the law of the hitting time of the origin by the equity process.
The systemic risk is thus described by the evolution of the default
probability
Robust pricing and hedging of options on multiple assets and its numerics
We consider robust pricing and hedging for options written on multiple assets
given market option prices for the individual assets. The resulting problem is
called the multi-marginal martingale optimal transport problem. We propose two
numerical methods to solve such problems: using discretisation and linear
programming applied to the primal side and using penalisation and deep neural
networks optimisation applied to the dual side. We prove convergence for our
methods and compare their numerical performance. We show how adding further
information about call option prices at additional maturities can be
incorporated and narrows down the no-arbitrage pricing bounds. Finally, we
obtain structural results for the case of the payoff given by a weighted sum of
covariances between the assets.Comment: Forthcoming in SIAM Journal on Financial Mathematic
Systemic robustness: a mean-field particle system approach
This paper is concerned with the problem of budget control in a large
particle system modeled by stochastic differential equations involving hitting
times, which arises from considerations of systemic risk in a regional
financial network. Motivated by Tang and Tsai (Ann. Probab., 46(2018), pp.
1597{1650), we focus on the number or proportion of surviving entities that
never default to measure the systemic robustness. First we show that both the
mean-field particle system and its limiting McKean-Vlasov equation are
well-posed by virtue of the notion of minimal solutions. We then establish a
connection between the proportion of surviving entities in the large particle
system and the probability of default in the limiting McKean-Vlasov equation as
the size of the interacting particle system N tends to infinity. Finally, we
study the asymptotic efficiency of budget control in different economy regimes:
the expected number of surviving entities is of constant order in a negative
economy; it is of order of the square root of N in a neutral economy; and it is
of order N in a positive economy where the budget's effect is negligible.Comment: 33 page