70 research outputs found
Duality for pathwise superhedging in continuous time
We provide a model-free pricing-hedging duality in continuous time. For a
frictionless market consisting of risky assets with continuous price
trajectories, we show that the purely analytic problem of finding the minimal
superhedging price of a path dependent European option has the same value as
the purely probabilistic problem of finding the supremum of the expectations of
the option over all martingale measures. The superhedging problem is formulated
with simple trading strategies, the claim is the limit inferior of continuous
functions, which allows for upper and lower semi-continuous claims, and
superhedging is required in the pathwise sense on a -compact sample
space of price trajectories. If the sample space is stable under stopping, the
probabilistic problem reduces to finding the supremum over all martingale
measures with compact support. As an application of the general results we
deduce dualities for Vovk's outer measure and semi-static superhedging with
finitely many securities
Non-asymptotic rates for the estimation of risk measures
Consider the problem of computing the riskiness of a financial
position written on the underlying with respect to a general law
invariant risk measure ; for instance, can be the average value at
risk. In practice the true distribution of is typically unknown and one
needs to resort to historical data for the computation. In this article we
investigate rates of convergence of to , where
is distributed as the empirical measure of with observations. We
provide (sharp) non-asymptotic rates for both the deviation probability and the
expectation of the estimation error. Our framework further allows for hedging,
and the convergence rates we obtain depend neither on the dimension of the
underlying stocks nor on the number of options available for trading
Optimal non-gaussian Dvoretzky-Milman embeddings
We construct the first non-gaussian ensemble that yields the optimal estimate
in the Dvoretzky-Milman Theorem: the ensemble exhibits almost Euclidean
sections in arbitrary normed spaces of the same dimension as the gaussian
embedding -- despite being very far from gaussian (in fact, it happens to be
heavy-tailed).Comment: This is part two of the paper "Structure preservation via the
Wasserstein distance" (arXiv:2209.07058v1) which was split into two part
Empirical approximation of the gaussian distribution in
Let be independent copies of the standard gaussian random
vector in . We show that there is an absolute constant such
that for any , with probability at least , for every , Here is the variance of and , where is determined
by an unexpected complexity parameter of that captures the set's geometry
(Talagrand's functional). The bound, the probability estimate, and
the value of are all (almost) optimal.
We use this fact to show that if is the random matrix that has as its rows, then the
structure of is far more rigid and
well-prescribed than was previously expected
Sensitivity of robust optimization problems under drift and volatility uncertainty
We examine optimization problems in which an investor has the opportunity to
trade in stocks with the goal of maximizing her worst-case cost of
cumulative gains and losses. Here, worst-case refers to taking into account all
possible drift and volatility processes for the stocks that fall within a
-neighborhood of predefined fixed baseline processes. Although
solving the worst-case problem for a fixed is known to be very
challenging in general, we show that it can be approximated as by the baseline problem (computed using the baseline processes) in the
following sense: Firstly, the value of the worst-case problem is equal to the
value of the baseline problem plus times a correction term. This
correction term can be computed explicitly and quantifies how sensitive a given
optimization problem is to model uncertainty. Moreover, approximately optimal
trading strategies for the worst-case problem can be obtained using optimal
strategies from the corresponding baseline problem
- …