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

Applied Mathematics & Optimization volume 72, pages 353–389 (2015)Cite this article

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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References

  1. Ansel, J.P., Stricker, C. (1993). Décomposition de Kunita-Watanabe. In Azema, J., Meyer, P. A., Yor, M. (eds.). Séminaire de probabilité \(XXVII\), vol. 1557. Lecture Notes in Mathematics, pp. 30–32. Springer (1993)

  2. Ansel, J.P., Stricker, C.: Lois de martingale, densités et décomposition de Föllmer-Schweizer. Annales de l’Institut Henri-Poincaré Probabilités et Statistiques 28(3), 375–392 (1992)

    MATH  MathSciNet  Google Scholar 

  3. Asmussen, S., Rosinski, J.: Approximations of small jump Lévy processes with a view towards simulation. J. Appl. Probab. 38, 482–493 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  4. Benth, F.E., Di Nunno, G., Khedher, A.: A note on convergence of option prices and their Greeks for Lévy models. Stoch. Int. J. Probab. Stoch. Proces. 85(6), 1015–1039 (2013)

  5. Benth, F.E., Di Nunno, G., Khedher, A.: Robustness of option prices and their deltas in markets modelled by jump-diffusions. Commun. Stoch. Anal. 5(2), 285–307 (2011)

    MathSciNet  Google Scholar 

  6. Bismut, J.M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973)

    MathSciNet  Article  Google Scholar 

  7. Bouchard, B., Touzi, N.: Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl. 111, 175–206 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  8. Bouchard, B., Elie, R.: Discrete time approximation of decoupled forward backward SDE with jumps. Stoch. Process. Appl. 118, 53–75 (2008)

    MATH  MathSciNet  Article  Google Scholar 

  9. Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Math. 134, 111–183 (1975)

    MATH  MathSciNet  Article  Google Scholar 

  10. Carbone, R., Ferrario, B., Santacroce, M.: Backward stochastic differential equations driven by càdlàg martingales. Theory Probab. Appl. 52(2), 304–314 (2008)

    MATH  MathSciNet  Article  Google Scholar 

  11. Choulli, T., Krawczyk, L., Stricker, C.: \(\cal E\)-martingales and their applications in mathematical finance. Ann. Probab. 26(2), 853–876 (1998)

    MATH  MathSciNet  Article  Google Scholar 

  12. Choulli, T., Vandaele, N., Vanmaele, M.: The Föllmer-Schweizer decomposition: comparison and description. Stoch. Process. Appl. 120(6), 853–872 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  13. Cont, R., Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman Hall

  14. Daveloose, C., Khedher, A., Vanmaele, M. (2013). Robustness of quadratic hedging strategies in finance via Fourier transform. Submitted paper. E-print: http://mediatum.ub.tum.de/doc/1198117/1198117

  15. Föllmer, H., Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliot, R.J. (eds.). Applied Stochastic Analysis, pp. 389–414.

  16. Föllmer, H., Sondermann, D. (1986). Hedging of non redundant contingent claims. In: Hildenbrand, W., Mas-Collel, A. (eds.) Contributions to Mathematical Economics, pp. 205–223. North-Holland, Elsevier (1986)

  17. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)

    MATH  Book  Google Scholar 

  18. Jeanblanc, M., Mania, M., Santacroce, M., Schweizer, M.: Mean-variance hedging via stochastic control and BSDEs for general semimartingales. Ann. Appl. Probab. 22(6), 2388–2428 (2012)

    MATH  MathSciNet  Article  Google Scholar 

  19. El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Fin. 7(1), 1–71 (1997)

    MATH  MathSciNet  Article  Google Scholar 

  20. Khedher, A., Schulz, T., Vanmaele, M. (2014). Model risk and discretisation of hedging strategies in incomplete markets. Working paper.

  21. Kohatsu-Higa, A., Tankov, P.: Jump-adapted discretisation schemes for Lévy-driven SDEs. Stoch. Process Appl. 120(11), 2258–2285 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  22. Kunita, H., Watanabe, S.: On square integrable martingales. Nagoya Math. 30, 209–245 (1967)

    MATH  MathSciNet  Google Scholar 

  23. Di Nunno, G.: Stochastic integral representations, stochastic derivatives and minimal variance hedging. Stoch. Stoch. Rep. 73, 181–198 (2001)

    Article  Google Scholar 

  24. Di Nunno, G.: Random Fields: non-anticipating derivative and differentiation formulas. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(3), 465–481 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  25. Di Nunno, G., Eide, I.B.: Minimal variance hedging in large financial markets: random fields approach. Stoch. Anal. Appl. 28, 54–85 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  26. Øksendal, B., Zhang, T. (2009). Backward stochastic differential equations with respect to general filtrations and applications to insider finance. Preprint No. 19, September, Department of Mathematics, University of Oslo, Norway

  27. El Otmani, M.: Reflected BSDE driven by a Lévy process. J. Theor. Probab. 22, 601–619 (2009)

    MATH  Article  Google Scholar 

  28. Protter, P. (2005) Stochastic Integration and Differential Equations, 2nd edn., Version 2.1. Springer, Berlin (2005)

  29. Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanic, J., Musiela, M. (eds.) Option Pricing, Interest Rates and Risk Management, pp. 538–574. Cambridge University Press.

  30. Schweizer, M.: Approximating random variables by stochastic integrals. Ann. Probab. 22(3), 1536–1575 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  31. Tang, S., Li., X. , : Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1994)

  32. Vandaele, N., Vanmaele, M.: A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market. Insur. Math. Econ. 42(3), 1128–1137 (2008)

    MATH  MathSciNet  Article  Google Scholar 

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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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Authors and Affiliations

  1. Center of Mathematics for Applications, University of Oslo, PO Box 1053 , Blindern, 0316, Oslo, Norway

    Giulia Di Nunno

  2. Norwegian School of Economics and Business Administration, Helleveien 30, 5045, Bergen, Norway

    Giulia Di Nunno

  3. Chair of Mathematical Finance, Technische Universität München, Parkring 11, 85748, Garching-Hochbruck, Germany

    Asma Khedher

  4. Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan, 281 S9, 9000, Ghent, Belgium

    Michèle Vanmaele

Authors
  1. Giulia Di Nunno
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  2. Asma Khedher
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  3. Michèle Vanmaele
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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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  • DOI: https://doi.org/10.1007/s00245-014-9283-z

Keywords

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