34 research outputs found
Continuity of the martingale optimal transport problem on the real line
We show continuity of the martingale optimal transport optimisation problem
as a functional of its marginals. This is achieved via an estimate on the
projection in the nested/causal Wasserstein distance of an arbitrary coupling
on to the set of martingale couplings with the same marginals. As a corollary
we obtain an independent proof of sufficiency of the monotonicity principle
established in [Beiglboeck, M., & Juillet, N. (2016). On a problem of optimal
transport under marginal martingale constraints. Ann. Probab., 44 (2016), no.
1, 42106]. On a problem of optimal transport under marginal martingale
constraints. Ann. Probab., 44 (2016), no. 1, 42-106] for cost functions of
polynomial growth
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
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
The uniform diversification strategy is optimal for expected utility maximization under high model ambiguity
We investigate an expected utility maximization problem under model
uncertainty in a one-period financial market. We capture model uncertainty by
replacing the baseline model with an adverse choice from a
Wasserstein ball of radius around in the space of probability
measures and consider the corresponding Wasserstein distributionally robust
optimization problem. We show that optimal solutions converge to the uniform
diversification strategy when uncertainty is increasingly large, i.e. when the
radius tends to infinity
Distributionally robust portfolio maximization and marginal utility pricing in one period financial markets
We consider the optimal investment and marginal utility pricing problem of a risk averse agent and quantify their exposure to model uncertainty. Specifically, we compute explicitly the first-order sensitivity of their value function, optimal investment policy and Davis' option prices to model uncertainty. To achieve this, we capture model uncertainty by replacing the baseline model P with an adverse choice from a small Wasserstein ball around P in the space of probability measures. Our sensitivities are thus fully non-parametric. We show that the results entangle the baseline model specification and the agent's risk attitudes. The sensitivities can behave in a non-monotone way as a function of the baseline model's Sharpe's ratio, the relative weighting of assets in the agent's portfolio can change and marginal prices can both increase or decrease when the agent faces model uncertainty
On concentration of the empirical measure for general transport costs
Let be a probability measure on and its
empirical measure with sample size . We prove a concentration inequality for
the optimal transport cost between and for cost functions with
polynomial local growth, that can have superpolynomial global growth. This
result generalizes and improves upon estimates of Fournier and Guillin. The
proof combines ideas from empirical process theory with known concentration
rates for compactly supported . By partitioning into
annuli, we infer a global estimate from local estimates on the annuli and
conclude that the global estimate can be expressed as a sum of the local
estimate and a mean-deviation probability for which efficient bounds are known
Robust uncertainty sensitivity analysis
We consider sensitivity of a generic stochastic optimization problem to model
uncertainty. We take a non-parametric approach and capture model uncertainty
using Wasserstein balls around the postulated model. We provide explicit
formulae for the first order correction to both the value function and the
optimizer and further extend our results to optimization under linear
constraints. We present applications to statistics, machine learning,
mathematical finance and uncertainty quantification. In particular, we provide
explicit first-order approximation for square-root LASSO regression
coefficients and deduce coefficient shrinkage compared to the ordinary least
squares regression. We consider robustness of call option pricing and deduce a
new Black-Scholes sensitivity, a non-parametric version of the so-called Vega.
We also compute sensitivities of optimized certainty equivalents in finance and
propose measures to quantify robustness of neural networks to adversarial
examples