Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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COHERENT AND CONVEX HEDGING ON ORLICZ HEARTS IN INCOMPLETE MARKETS

  • Kim, Ju-Hong (Department of Mathematics, Sungshin Women's University)
  • Received : 2011.11.30
  • Accepted : 2012.03.06
  • Published : 2012.05.30
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Abstract

Every contingent claim is unable to be replicated in the incomplete markets. Shortfall risk is considered with some risk exposure. We show how the dynamic optimization problem with the capital constraint can be reduced to the problem to find an optimal modified claim $\tilde{\psi}H$ where$\tilde{\psi}H$ is a randomized test in the static problem. Convex and coherent risk measures defined in the Orlicz hearts spaces, $M^{\Phi}$, are used as risk measure. It can be shown that we have the same results as in [21, 22] even though convex and coherent risk measures defined in the Orlicz hearts spaces, $M^{\Phi}$, are used. In this paper, we use Fenchel duality Theorem in the literature to deduce necessary and sufficient optimality conditions for the static optimization problem using convex duality methods.

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Acknowledgement

Supported by : Sungshin Women's University

References

  1. T. Arai, Good deal bounds induced by shortfall risk, preprint (2009).
  2. P. Artzner, F. Delbaen, J.-M, Eber and D. Heath, Thinking coherently, RISK 10, November(1997), 68-71.
  3. P. Artzner, F. Delbaen, J.-M, Eber and D. Heath, Coherent measures of risk, Mathematical Finance 9 (1999), 203-223. https://doi.org/10.1111/1467-9965.00068
  4. F. Bellini, M. Fritteli, On the existence of minimax martingale measures, Mathematical Finance 12 (1999), 1-21.
  5. F. Biagini, M. Fritteli, Utility maximization in incomplete markets for unbounded processes, Finance and Stochastics 9 (1999), 493-517.
  6. F. Biagini, M. Fritteli, A unified framework for utility maximization problems: An Orlicz space approach, Annals of Applied Probability 18 (1999), 929-966.
  7. J. Borwein, Q. Zhu, Variational Methods in Convex Analysis, Journal of Global Optimization 35 (2006), 197-213. https://doi.org/10.1007/s10898-005-3835-3
  8. P. Cheridito and T. Li, Risk measures on Orlicz hearts, Mathematical Finance 19 (2009), 189-214. https://doi.org/10.1111/j.1467-9965.2009.00364.x
  9. F. Delbaen, Coherent risk measures on general probability spaces, Advances in finance and stochastics: Essays in honor of Dieter Sondermann (2002), Springer, 1-37.
  10. F. Delbaen and W. Schachermayer, A General version of the fundamental theorem of asset pricing, Mathematische Annalen 300 (1994), 463-520. https://doi.org/10.1007/BF01450498
  11. F. Delbaen and W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes, Mathematische Annalen 312 (1994), 215-250.
  12. I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland Publishing Company, Amsterdam, 1976.
  13. H. Follmer and P. Leukert, Quantile hedging, Finance and Stochastics 10 (1999), 163-181.
  14. H. Follmer and P. Leukert, Efficient hedging: Cost versus shortfall risk, Finance and Stochastics 4 (2000), 117-146. https://doi.org/10.1007/s007800050008
  15. H. Follmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, Springer-Verlag, New York, 2002.
  16. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991.
  17. N. El Karoui and 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
  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, Journal of Mathematical Analysis and Applications 293 (2004), 345-354. https://doi.org/10.1016/j.jmaa.2004.01.010
  20. W. Rudin, Real and Complex Analysis, McGraw-Hill Inc., New York, 1974.
  21. B. Rudloff, Convex hedging in incomplete markets, Applied Mathematical Finance 14 (2007), 437-452. https://doi.org/10.1080/13504860701352206
  22. B. Rudloff, Coherent hedging in incomplete markets, Quantitative Finance 9 (2009), 197-206 https://doi.org/10.1080/14697680802169787
  23. A. Schied, On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals, The Annals of Applied Probability 14 (2004), 1398-1423. https://doi.org/10.1214/105051604000000341
  24. K. Yosida, Functional Analysis, Springer, New York, 1980.