Abstract

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 multistage 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.

Highlights

  • The no-arbitrage paradigm is the cornerstone of mathematical finance

  • The determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics

  • 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 multistage stochastic optimization problem under model uncertainty

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Summary

Introduction

The no-arbitrage paradigm is the cornerstone of mathematical finance. The fundamental work of Harrison, Kreps and Pliska [13,14,15,22] and Delbaen and Schachermayer [6], to mention some of the most important contributions, paved the way for a sound theory for the pricing of contingent claims. Distributionally robust claim prices w.r.t. nested distance balls as ambiguity sets include a hedge under the true model with arbitrary high probability, depending on the available data. We provide a framework that allows for setting up bid and ask prices for a contingent claim which result from finding hedging strategies with truly calculated risks, since the important factor of model uncertainty is not neglected. We define the distributionally robust acceptability pricing problem and derive the dual problem formulation under rather general assumptions on the ambiguity set. 3 and the special stagewise structure of the nested distance by a sequential linear programming algorithm which yields approximate solutions to the originally semi-infinite non-convex problem In this way, we overcome the current state-of-the-art computational methods for multistage stochastic optimization problems under non-parametric model ambiguity.

Acceptable replications
Model ambiguity and distributional robustness
Acceptability pricing under model ambiguity
Nested distance balls as ambiguity sets: a large deviations result
Illustrative examples
Algorithmic solution
9: EndIteration
Conclusion
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