Advertisement

Hedging Under Loss Constraints

  • Bruno Bouchard
  • Jean-François Chassagneux
Chapter
Part of the Universitext book series (UTX)

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].

Keywords

Risky Asset Polynomial Growth Martingale Measure Price Equation Hedging Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alexander, C.O., Nogueira, L.: Stochastic Local Volatility, No. ICMA-dp2008-02. Henley Business School, Reading University (2008)Google Scholar
  2. 2.
    Alfonsi, A.: On the discretisation schemes for CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11 (4), 355–467 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avellaneda, M., Friedman,C., Holmes, R., Samperi, D.: Calibrating volatility surfaces via relative-entropy minimization. Appl. Math. Financ. 4 (1), 37–64 (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Andersen, L., Piterbarg, V.: Moment explosions in stochastic volatility models. Financ. Stoch. 11 (1), 29–50 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asym. Anal. 4 (3), 271–283 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bates, D.: Jumps and stochastic volatility: Exchange rate process implicit in Deutschmark options. Rev. Financ. Stud. 9, 69–107 (1996)CrossRefGoogle Scholar
  7. 7.
    Bergomi, L.: Stochastic Volatility Modeling. Chapman and Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton (2016)zbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bouchard, B., Elie, R., Imbert, C.: Optimal control under stochastic target constraints. SIAM J. Control Optim. 48 (5), 3501–3531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bouchard, B., Elie, R., Touzi N.: Stochastic target problems with controlled loss. SIAM J. Control Optim. 48 (5), 3123–3150 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bouchard, B., Moreau L., Nutz, M.: Stochastic target games with controlled loss. Ann. Appl. Probab. 24 (3), 899–934 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49 (3), 948–962 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bouchard, B., Vu, T.N.: A stochastic target approach for P&L matching problems. Math. Oper. Res. 37 (3), 526–558 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brézis, H.: Analyse Fonctionnelle. Masson, Paris (1983)zbMATHGoogle Scholar
  16. 16.
    Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Financ. 2, 61–73 (1999)Google Scholar
  17. 17.
    Chassagneux, J.-F., Elie, R., Kharroubi, I.: When terminal facelift enforces Delta constraints. Financ. Stoch. 19, 329–362 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall, Boca Raton (2004)zbMATHGoogle Scholar
  19. 19.
    Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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. 22.
    Davis, M., Obloj, J.: Market completion using options. Prépublication (2008)Google Scholar
  23. 23.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of assand pricing. Math. Ann. 300, 463–520 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 (2), 215–250 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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)CrossRefzbMATHGoogle Scholar
  26. 26.
    Dupire, B.: Pricing with a smile. Risk Mag. 7, 18–20 (1994)Google Scholar
  27. 27.
    Feynman, R.P., Hibbs, A.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)zbMATHGoogle Scholar
  28. 28.
    Föllmer, H., Kabanov, Y.: Optional decomposition and Lagrange multipliers. Financ. Stoch. 2, 69–81 (1998)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Föllmer, H., Leukert, P.: Quantile Hedging. Financ. Stoch. 3 (3), 251–273 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Föllmer, H., Leukert, P.: Efficient hedging: cost versus shortfall risk. Financ. Stoch. 4, 117–146 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Glasserman, P.: Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability, vol. 53. Springer, New York (2003)Google Scholar
  34. 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. 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)CrossRefGoogle Scholar
  36. 36.
    Hodges, S.D., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Futures Mark. 8, 222–239 (1989)Google Scholar
  37. 37.
    Ishii, H.: On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcial. Ekvac 38 (1), 101–120 (1995)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Jourdain, B.: Stochastic flows approach to Dupire’s formula. Financ. Stoch. 11 (4), 521–535 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 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. 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. 41.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)zbMATHGoogle Scholar
  42. 42.
    Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  43. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Lapeyre, B.J., Sulem, A., Talay, D.: Understanding Numerical Analysis for Financial Models. Cambridge University Press, Cambridge (2003)Google Scholar
  46. 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. 47.
    Neveu, J.: Martingales à temps Discret. Masson, Paris (1974)Google Scholar
  48. 48.
    Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Berlin (1995)CrossRefzbMATHGoogle Scholar
  49. 49.
    Overhaus, M., et al.: Equity Hybrid Derivatives. Wiley Finance. Wiley, Hoboken (2007)Google Scholar
  50. 50.
    Pironneau, O.: Dupire-like identities for complex options. Comptes Rendus Mathematique 344 (2), 127–133 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  52. 52.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1990)zbMATHGoogle Scholar
  53. 53.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  54. 54.
    Schachermayer, W.: Optimal Investment in Incomplete Markets when Wealth may Become Negative. Ann. Appl. Probab. 11 (3), 694–734 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Soner, H.M., Touzi, N.: Stochastic targetproblems, dynamic programming and viscosity solutions. SIAM J. Control Optim. 41, 404–424 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Bruno Bouchard
    • 1
  • Jean-François Chassagneux
    • 2
  1. 1.Université Paris DauphineParisFrance
  2. 2.Université Paris DiderotParisFrance

Personalised recommendations