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
Journal:
Theor. Probability and Math. Statist. 78 (2009), 75-82
MSC (2000):
Primary 60H30, 62P05, 91B28; Secondary 60J75, 60G44, 91B30
DOI:
https://doi.org/10.1090/S0094-9000-09-00763-7
Published electronically:
August 4, 2009
MathSciNet review:
2446850
Full-text PDF Free Access
Abstract |
References |
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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.
References
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References
- 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)
- J. Bardhan and X. Chao, Pricing options on securities with discontinuous returns, Stochastic Process. Appl. 48 (1993), 123–137. MR 1237171 (94g:90011)
- 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
- Y. Bergman, Option pricing with different interest rates for borrowing and for lending, Working Paper, University of California, Berkeley 109 (1981).
- D. Colwell and R. Elliott, Discontinuous asset prices and non-attainable contingent claims and corporate policy, Math. Finance 3 (1993), 295–318.
- J. Cvitanić, Optimal trading under constraints, Lectures Notes in Mathematics, vol. 1656, Springer-Verlag, Berlin, 1997, pp. 123–190. MR 1478201
- 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
- J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolio, Annals Appl. Probab. 3(3) (1993), 652–681. MR 1233619 (95c:90022)
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- R. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer-Verlag, Berlin, 1998. MR 2098795 (2005g:91001)
- H. Föllmer and D. O. Kramkov, Optional decompositions under constraints, Probab. Theory Related Fields 109 (1997), 1–25. MR 1469917 (98j:60065)
- H. Föllmer and P. Leukert, Efficient hedging: Cost versus shortfall risk, Finance Stoch. 4 (2000), 117–146. MR 1780323 (2001f:91054)
- 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)
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- R. C. Merton, Continuous-time finance, Basil–Blackwell, Oxford, 1990.
- Y. Nakano, Minimization of shortfall risk in a jump-diffusion model, Statist. Probab. Lett. 67 (2004), 87–95. MR 2039936 (2005b:62137)
- 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
Email:
skane@ualberta.ca
Alexander Melnikov
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G2G1 Canada
Email:
melnikov@ualberta.ca
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