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SHORTFALL RISK MINIMIZATION: THE DUAL APPROACH

  • Kim, Ju-Hong (Department of Mathematics, Sungshin Women's University)
  • Received : 2012.04.05
  • Accepted : 2012.05.15
  • Published : 2012.05.31

Abstract

We find the solution minimizing the shortfall risk by using the Lagrange-multiplier method. The conventional duality method in the expected utility maximization problem is used and we get the same results as in the paper [21].

Keywords

References

  1. T. Arai: Good deal bounds induced by shortfall risk. preprint (2009).
  2. M. Avriel: Nonlinear Programming: Analysis and Methods. Dover Publications, Inc., New York, 2003.
  3. P. Artzner, F. Delbaen, J.-M, Eber & D. Heath: Coherent measures of risk. Mathematical Finance 9(1999), 203-223. https://doi.org/10.1111/1467-9965.00068
  4. F. Bellini & M. Fritelli: Utility maximization in incomplete markets for unbounded processes. Mathematical Finance 21 (2002), 1-21.
  5. F. Bellini & M. Fritelli: On the existence of minimax martingale measures. Finance Stoch. 9 (2005), 493-517. https://doi.org/10.1007/s00780-005-0163-x
  6. J. Cvitanic: Minimizing expected loss of hedging in incomplete and constrained markets. SIAM J. Control Optim. 38 (2000), 1050-1066. https://doi.org/10.1137/S036301299834185X
  7. J. Cvitanic & I. Karatzas: On dynamic measures of risk. Finance Stoch. 3 (1999), 451-482. https://doi.org/10.1007/s007800050071
  8. F. Delbaen: Coherent risk measures on general probability spaces. Advances in finance and stochastics: Essays in honor of Dieter Sondermann (2002), Springer, 1-37.
  9. F. Delbaen & W. Schachermayer: A General version of the fundamental theorem of asset pricing. Mathematische Annalen 300 (1994), 463-520. https://doi.org/10.1007/BF01450498
  10. F. Delbaen & W. Schachermayer: The fundamental theorem of asset pricing for unbounded stochastic processes. Mathematische Annalen 312 (1994), 463-520.
  11. H. Follmer & P. Leukert: Quantile hedging. Finance and Stochastics 10 (1999), 215-250.
  12. H. Follmer & P. Leukert: Efficient hedging: Cost versus shortfall risk. Finance and Stochastics 4 (2000), 117-146. https://doi.org/10.1007/s007800050008
  13. H. Follmer & A. Schied: Stochastic Finance: An Introduction in Discrete Time. Springer-Verlag, New York, 2002.
  14. I. Karatzas & S.E. Shreve: Brownian Motion and Stochastic Calculus. Springer-Verlag, New York, 1991.
  15. N. El Karoui & M.C. Quenez: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control and Optimization 33 (1995), 29-66. https://doi.org/10.1137/S0363012992232579
  16. I. Karatzas, S.E. Shreve, J. Lehoczky & G. Xu: Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control and Optimization 29 (1991), 702-730. https://doi.org/10.1137/0329039
  17. Ju Hong Kim: Optimal partial hedging coherent measure of risk. J. Appl. Math. & Informatics 29 (2011), 987-1000.
  18. D.O. Kramkov: Optional decomposition of supermartingales and hedging contingent claims in an incomplete security market. Probability Theory and Related Fields 105 (1996), 459-479. https://doi.org/10.1007/BF01191909
  19. Y. Nakano: Efficient hedging with coherent risk measures. J. Math. Anal. Appl. 293 (2004), 345-354. https://doi.org/10.1016/j.jmaa.2004.01.010
  20. H. Pham: Dynamic $L^P$-hedging in descrete-time under cone constraints. SIAM J. Control and Optimization 38 (2000), 665-682. https://doi.org/10.1137/S0363012998341095
  21. H. Pham: Minimizing shortfall risk and applications to finance and insurance problems. Annals of Applied Probability 12 (2002), 143-172. https://doi.org/10.1214/aoap/1015961159
  22. R.T. Rockfellar: Conjugate Duality and Optimization. Conference Board of Math. Sciences Series, SIAM Publications, No. 16, 1974.
  23. G. Spivak & J. Cvitanic: Maximizing the probability of a perfect hedge. Annals of Applied Probability 9 (2002), 1303-1328.

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