Semimartingale properties of the lower Snell envelope in optimal stopping under model uncertainty
Erick Treviño*, Universidad de Guanajuato
Optimal stopping under model uncertainty is a recent topic under research. The classical approach to characterize the solution of optimal stopping is based on the Snell envelope which can be seen as the value process as time runs. The analogous concept under model uncertainty is the so-called lower Snell envelope and in this paper, we investigate its structural properties. We give conditions under which it is a semimartingale with respect to one of the underlying probability
measures and show how to identify the finite variation process by a limiting procedure. An example illustrates that without our conditions, the semimartingale property does not hold in general.
On de Finetti’s problem under a time of ruin constraint
Mauricio Junca*, Universidad de Los Andes
We consider the classic de Finetti’s problem when the reserves are assumed to follow a spectrally negative Levy process subject to a constraint on the time of ruin. We introduce the dual problem and show that the complementary slackness condition is satisfied, thus there is no duality gap. Therefore the optimal value function can be obtained as the point-wise
infimum of auxiliary value functions indexed by Lagrange multipliers. We also present a series of numerical examples.
Joint work with Camilo Hernández
Utility maximization in a multi-dimensional semi-martingale setting with nonlinear wealth dynamics
Rafael Serrano*, Universidad del Rosario.
We explore martingale and convex duality techniques to study optimal investment strategies that maximize expected risk-averse utility from consumption and terminal wealth in a multi-dimensional semimartingale market model with absolutely continuous characteristics and non-linear wealth dynamics. This allows to take account of market frictions such as different borrowing and lending interest rates or short positions with cash collateral and negative rebate rates. Our main result is a sufficient condition for existence of optimal policies and their explicit chracterization in the case of CRRA utility functions. We present numerical examples and some preliminary results for the case in which the investor’s final wealth is liable to deferred capital gains taxes or subject to further downside or expected loss
Optimizing the exercise boundary for the holder of an American option over a parametric family.
José Vidal Alcalá* , Centro de Investigación en Matemáticas, CIMAT
In the setting of American option pricing, we introduce an efficient stochastic optimization algorithm to find the optimal exercise boundary among a parametric family. We use the Calculus of Variations to write down a probabilistic representation of the payout sensitivity with respect to the exercise boundary parameter.
This representation is used in a Monte Carlo estimator after the development of an accurate SDE discretization scheme for stopped diffusions. As an intermediate result, we are able to approximate deltas at the boundary for barrier options. Numerical simulations/analysis of the algorithms used are presented.