Robustness of Quadratic Hedging Strategies in Finance via Backward Stochastic Differential Equations with Jumps

Abstract

We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate-approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poisson random measure with infinite activity and models in which these small variations are modeled with a Brownian motion or are cut off. Using the convergence results on BSDEJs, we show that quadratic hedging strategies are robust towards the approximation of the market prices and we derive an estimation of the model risk.

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Acknowledgments

The authors acknowledge the Centre of Advanced Study (CAS) at the Norwegian Royal Academy of Science and Letters (Program SEFE) for providing occasions of research discussions at the finalising stage of this paper. Also they acknowledge the valuable suggestions of the referee and Associate Editor. Asma Khedher thanks the KPMG Center of Excellence in Risk Management for the financial support. Michèle Vanmaele acknowledges the Research Foundation Flanders (FWO) and the Special Research Fund (BOF) of the Ghent University for providing the possibility to go on sabbatical leave to CAS.

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Correspondence to Asma Khedher.

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Di Nunno, G., Khedher, A. & Vanmaele, M. Robustness of Quadratic Hedging Strategies in Finance via Backward Stochastic Differential Equations with Jumps. Appl Math Optim 72, 353–389 (2015). https://doi.org/10.1007/s00245-014-9283-z

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Keywords

  • Quadratic hedging strategies
  • Backward stochastic differential equations
  • Jump-diffusions
  • Robustness