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On investment and minimization of shortfall risk for a diffusion model with jumps and two interest rates via market completion

Authors: Selly Kane and Alexander Melnikov
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 75-82
MSC (2000): Primary 60H30, 62P05, 91B28; Secondary 60J75, 60G44, 91B30
Published electronically: August 4, 2009
MathSciNet review: 2446850
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the problems of investment and shortfall risk minimization in the framework of a two-factor diffusion model with jumps and with different credit and deposit rates. The optimal strategies are derived by means of auxiliary completions of the initial market.

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  • 1. K. K. Aase, Contingent claim valuation when the security price is a combination of an Ito process and a random point process, Stochastic Process. Appl. 28 (1988), 185-220. MR 952829 (89k:90015)
  • 2. J. Bardhan and X. Chao, Pricing options on securities with discontinuous returns, Stochastic Process. Appl. 48 (1993), 123-137. MR 1237171 (94g:90011)
  • 3. Y. Bart, Option hedging in the binomial model with differing interest rates, Uspekhi Math. Nauk 53 (1998), no. 5, 227-228; English transl. in Russian Math. Surveys 53 (1998), 1084-1085. MR 1691190
  • 4. Y. Bergman, Option pricing with different interest rates for borrowing and for lending, Working Paper, University of California, Berkeley 109 (1981).
  • 5. D. Colwell and R. Elliott, Discontinuous asset prices and non-attainable contingent claims and corporate policy, Math. Finance 3 (1993), 295-318.
  • 6. J. Cvitanić, Optimal trading under constraints, Lectures Notes in Mathematics, vol. 1656, Springer-Verlag, Berlin, 1997, pp. 123-190. MR 1478201
  • 7. J. Cvitanić, Theory of Portfolio Optimization in Markets with Frictions, Handbooks in Math. Finance: Option Pricing, Interest Rates and Risk Management (E. Jouini and M. Musiela, eds.), Cambridge University Press, 2001. MR 1848563
  • 8. J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolio, Annals Appl. Probab. 3(3) (1993), 652-681. MR 1233619 (95c:90022)
  • 9. J. Cvitanić, H. Pham, and N. Touzi, Super-replication in stochastic volatility models under portfolio constraints, J. Appl. Probab. 36 (1999), 523-545. MR 1724796 (2001a:91048)
  • 10. R. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer-Verlag, Berlin, 1998. MR 2098795 (2005g:91001)
  • 11. H. Föllmer and D. O. Kramkov, Optional decompositions under constraints, Probab. Theory Related Fields 109 (1997), 1-25. MR 1469917 (98j:60065)
  • 12. H. Föllmer and P. Leukert, Efficient hedging: Cost versus shortfall risk, Finance Stoch. 4 (2000), 117-146. MR 1780323 (2001f:91054)
  • 13. S. Kane and A. Melnikov, On pricing contingent claims in a two interest rates jump diffusion model via market completions, Theory Probab. Math. Statist. 77 (2007), 57-69. MR 2432772 (2009f:91062)
  • 14. I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. MR 1640352 (2000e:91076)
  • 15. R. Korn, Contingent claim valuation in a market with different interest rates, Math. Methods Oper. Res. 42 (1995), 255-274. MR 1358829
  • 16. R. Krutchenko and A. V. Melnikov, Quantile hedging for a jump-diffusion financial market, Trends in Mathematics (M. Kohlmann, ed.), Birkhäuser Verlag, Basel/Switzerland, 2001, pp. 215-229. MR 1882833
  • 17. A. V. Melnikov, M. Nechaev, and S. Volkov, Mathematics of Financial Obligations, American Mathematical Society, Providence, R.I., 2002. MR 1918716 (2003f:91055)
  • 18. F. Mercurio and W. Runggaldier, Option pricing for jump-diffusions: Approximations and their interpretation, Math. Finance 3 (1993), 191-200.
  • 19. R. C. Merton, Continuous-time finance, Basil-Blackwell, Oxford, 1990.
  • 20. Y. Nakano, Minimization of shortfall risk in a jump-diffusion model, Statist. Probab. Lett. 67 (2004), 87-95. MR 2039936 (2005b:62137)
  • 21. H. Soner and N. Touzi, Superreplication under gamma constraints, J. Control Optim. 39 (2000), 73-96. MR 1780909 (2002h:91068)

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Additional Information

Selly Kane
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G2G1 Canada

Alexander Melnikov
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G2G1 Canada

Keywords: Constrained market, completion, hedging and pricing, diffusion with jumps, different interest rates
Received by editor(s): January 9, 2007
Published electronically: August 4, 2009
Additional Notes: The paper was supported by the discovery grant NSERC #261855
Article copyright: © Copyright 2009 American Mathematical Society

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