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Fundamentals and Advanced Techniques in Derivatives Hedging pp 193–224Cite as

Hedging Under Loss Constraints

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Abstract

We present in this section a direct approach to obtain the hedging price of a contingent claim, in the almost sure sense of super-replication or in the sense of a risk criterion (quantile hedging, expected shortfall, utility indifference). This approach, based on the notion of stochastic target, was initiated by Soner and Touzi [55] for the super-replication criterion, and then extended by Bouchard, Elie and Touzi [10] for the hedging under risk control, see also [8, 13] and [14].

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Notes

  1. 1.

    To simplify the presentation.

  2. 2.

    See Exercise 5.2 for a rigorous formulation under specific assumptions.

References

  1. Alexander, C.O., Nogueira, L.: Stochastic Local Volatility, No. ICMA-dp2008-02. Henley Business School, Reading University (2008)

    Google Scholar 

  2. Alfonsi, A.: On the discretisation schemes for CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11 (4), 355–467 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avellaneda, M., Friedman,C., Holmes, R., Samperi, D.: Calibrating volatility surfaces via relative-entropy minimization. Appl. Math. Financ. 4 (1), 37–64 (1997)

    Article  MATH  Google Scholar 

  4. Andersen, L., Piterbarg, V.: Moment explosions in stochastic volatility models. Financ. Stoch. 11 (1), 29–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asym. Anal. 4 (3), 271–283 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Bates, D.: Jumps and stochastic volatility: Exchange rate process implicit in Deutschmark options. Rev. Financ. Stud. 9, 69–107 (1996)

    Article  Google Scholar 

  7. Bergomi, L.: Stochastic Volatility Modeling. Chapman and Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  8. Bouchard, B., Dang, N.M.: Generalized stochastic target problems for pricing and partial hedging under loss constraints – application in optimal book liquidation. Financ. Stoch. 17 (1), 31–72 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bouchard, B., Elie, R., Imbert, C.: Optimal control under stochastic target constraints. SIAM J. Control Optim. 48 (5), 3501–3531 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bouchard, B., Elie, R., Touzi N.: Stochastic target problems with controlled loss. SIAM J. Control Optim. 48 (5), 3123–3150 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bouchard, B., Moreau L., Nutz, M.: Stochastic target games with controlled loss. Ann. Appl. Probab. 24 (3), 899–934 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49 (3), 948–962 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bouchard, B., Vu, T.N.: The American version of the geometric dynamic programming principle: application to the pricing of American options under constraints. Appl. Math. Optim. 61 (2), 235–265 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bouchard, B., Vu, T.N.: A stochastic target approach for P&L matching problems. Math. Oper. Res. 37 (3), 526–558 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brézis, H.: Analyse Fonctionnelle. Masson, Paris (1983)

    MATH  Google Scholar 

  16. Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Financ. 2, 61–73 (1999)

    Google Scholar 

  17. Chassagneux, J.-F., Elie, R., Kharroubi, I.: When terminal facelift enforces Delta constraints. Financ. Stoch. 19, 329–362 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall, Boca Raton (2004)

    MATH  Google Scholar 

  19. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Am. Math. Soc. 27, 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Crépey, S.: Contribution à des méthodes numériques appliquées à la finance and aux jeux différentiels. Thèse de doctorat, École Polytechnique (2001)

    Google Scholar 

  22. Davis, M., Obloj, J.: Market completion using options. Prépublication (2008)

    Google Scholar 

  23. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of assand pricing. Math. Ann. 300, 463–520 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 (2), 215–250 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Delbaen, F., Shirakawa, H.: A note of option pricing for constant elasticity of variance model. Asia-Pac. Financ. Mark. 9 (2), 85–99 (2002)

    Article  MATH  Google Scholar 

  26. Dupire, B.: Pricing with a smile. Risk Mag. 7, 18–20 (1994)

    Google Scholar 

  27. Feynman, R.P., Hibbs, A.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  28. Föllmer, H., Kabanov, Y.: Optional decomposition and Lagrange multipliers. Financ. Stoch. 2, 69–81 (1998)

    MathSciNet  MATH  Google Scholar 

  29. Föllmer, H., Leukert, P.: Quantile Hedging. Financ. Stoch. 3 (3), 251–273 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Föllmer, H., Leukert, P.: Efficient hedging: cost versus shortfall risk. Financ. Stoch. 4, 117–146 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L.: Applications of Malliavin calculus to Monte Carlo methods in finance II. Financ. Stoch. 5, 201–236 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L., Touzi, N.: Applications of Malliavin calculus to Monte Carlo methods in finance. Financ. Stoch. 3, 391–412 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability, vol. 53. Springer, New York (2003)

    Google Scholar 

  34. Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. In: The Best of Wilmott, vol. 1, pp. 249–296. Jon Wiley & Sons, Chichester (2005)

    Google Scholar 

  35. Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6 (2), 327–343 (1993)

    Article  Google Scholar 

  36. Hodges, S.D., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Futures Mark. 8, 222–239 (1989)

    Google Scholar 

  37. Ishii, H.: On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcial. Ekvac 38 (1), 101–120 (1995)

    MathSciNet  MATH  Google Scholar 

  38. Jourdain, B.: Stochastic flows approach to Dupire’s formula. Financ. Stoch. 11 (4), 521–535 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kabanov, Y., Stricker, C.: A teachers’ note on no-arbitrage criteria. In: Séminaire de Probabilités, XXXV. Lecture Notes in Mathematics, vol. 1755, pp. 149–152. Springer, Berlin/London (2001)

    Google Scholar 

  40. Kac, M.: On some connections between probability theory and differential and integral equations. In: Proceedings 2nd Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkely/Los Angeles (1951)

    Google Scholar 

  41. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)

    MATH  Google Scholar 

  42. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, New York (1998)

    Book  MATH  Google Scholar 

  43. El Karoui, N.: Les aspects probabilistes du contrôle stochastique. École d’Été de Probabilités de Saint Flour IX. Lecture Notes in Mathematics, vol. 876. Springer (1979)

    Google Scholar 

  44. Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13 (4), 1504–1516 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lapeyre, B.J., Sulem, A., Talay, D.: Understanding Numerical Analysis for Financial Models. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  46. Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modeling. Stochastic Modelling and Applied Probability, vol. 36. Springer, Berlin/New York (2005)

    Google Scholar 

  47. Neveu, J.: Martingales à temps Discret. Masson, Paris (1974)

    Google Scholar 

  48. Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  49. Overhaus, M., et al.: Equity Hybrid Derivatives. Wiley Finance. Wiley, Hoboken (2007)

    Google Scholar 

  50. Pironneau, O.: Dupire-like identities for complex options. Comptes Rendus Mathematique 344 (2), 127–133 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  51. Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  52. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1990)

    MATH  Google Scholar 

  53. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  54. Schachermayer, W.: Optimal Investment in Incomplete Markets when Wealth may Become Negative. Ann. Appl. Probab. 11 (3), 694–734 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  55. Soner, H.M., Touzi, N.: Stochastic targetproblems, dynamic programming and viscosity solutions. SIAM J. Control Optim. 41, 404–424 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  56. Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Université Paris Dauphine, Paris, France

    Bruno Bouchard

  2. Université Paris Diderot, Paris, France

    Jean-François Chassagneux

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  1. Bruno Bouchard
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  2. Jean-François Chassagneux
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Bouchard, B., Chassagneux, JF. (2016). Hedging Under Loss Constraints. In: Fundamentals and Advanced Techniques in Derivatives Hedging. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-38990-5_6

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