Incorporating statistical model error into the calculation of acceptability prices of contingent claims

The determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics. Given this model, optimal bid and ask prices can be found by stochastic optimization. However, the model for the underlying asset price process is typically based on data and found by a statistical estimation procedure. We define a confidence set of possible estimated models by a nonparametric neighborhood of a baseline model. This neighborhood serves as ambiguity set for a multi-stage stochastic optimization problem under model uncertainty. We obtain distributionally robust solutions of the acceptability pricing problem and derive the dual problem formulation. Moreover, we prove a general large deviations result for the nested distance, which allows to relate the bid and ask prices under model ambiguity to the quality of the observed data.Comment: 27 pages, 2 figure

05031 Abstracts Collection -- Algorithms for Optimization with Incomplete Information

From 16.01.05 to 21.01.05, the Dagstuhl Seminar 05031 ``Algorithms for Optimization with Incomplete Information\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

The distortion principle for insurance pricing: properties, identification and robustness

Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia

Multistage Optimization

We provide a new identity for the multistage Average Value-at-Risk. The identity is based on the conditional Average Value-at-Risk at random level, which is introduced. It is of interest in situations, where the information available increases over time, so it is – among other applications – customized to multistage optimization. The identity relates to dynamic programming and is adapted to problemswhich involve the Average Value-at-Risk in its objective. We elaborate further dynamic programming equations for specific multistage optimization problems and derive a characterizing martingale property for the value function. The concept solves a particular aspect of time consistency and is adapted for situations, wheredecisions are planned and executed consecutively in subsequent instants of time. We discuss theapproach for other risk measures, which are in frequent use for decision making under uncertainty,particularly for financial decisions

Risk measures for income streams

A new measure of risk is introduced for a sequence of random incomes adapted to some filtration. This measure is formulated as the optimal net present value of a stream of adaptively planned commitments for consumption. The calculation of the new measure is done by solving a stochastic dynamic linear optimization problem, which, in case of a finite filtration, reduces to a simple deterministic linear program. We show properties of the new measure by exploiting the convexity and duality structure of the stochastic dynamic linear problem. The measure depends on the full distribution of the income process (not only on its marginal distribution) as well as on the filtration, which is interpreted as the available information about the future

Convergence of the Smoothed Empirical Process in Nested Distance

The nested distance, also process distance, provides a quantitative measure of distance for stochastic processes. It is the crucial and determining distance for stochastic optimization problems.In this paper we demonstrate first that the empirical measure, which is built from observed sample paths, does not converge in nested distance to its underlying distribution. We show that smoothing convolutions, which are appropriately adapted from classical density estimation using kernels, can be employed to modify the empirical measure in order to obtain stochastic processes, which converge in nested distance to the underlying process. We employ the results to estimate transition probabilities at each time moment. Finally we construct processes with discrete sample space from observed empirical paths, which approximate well the original stochastic process as they converge in nested distance

Electricity Swing Option Pricing by Stochastic Bilevel Optimization: a Survey and New Approaches

We demonstrate how the problem of determining the ask price for electricityswing options can be considered as a stochastic bilevel program with asymmetricinformation. Unlike as for financial options, there is no way for basingthe pricing method on no-arbitrage arguments. Two main situations are analyzed:if the seller has strong market power he/she might be able to maximizehis/her utility, while in fully competitive situations he/she will just look for aprice which makes profit and has acceptable risk. In both cases the seller hasto consider the decision problem of a potential buyer - the valuation problemof determining a fair value for a specific option contract - and anticipate thebuyer's optimal reaction to any proposed strike price. We also discuss somemethods for finding numerical solutions of stochastic bilevel problems witha special emphasis on using duality gap penalizations