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A Benchmark Approach to Risk-Minimization under Partial Information

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  • Claudia Ceci
  • Katia Colaneri
  • Alessandra Cretarola

Abstract

In this paper we study a risk-minimizing hedging problem for a semimartingale incomplete financial market where d+1 assets are traded continuously and whose price is expressed in units of the num\'{e}raire portfolio. According to the so-called benchmark approach, we investigate the (benchmarked) risk-minimizing strategy in the case where there are restrictions on the available information. More precisely, we characterize the optimal strategy as the integrand appearing in the Galtchouk-Kunita-Watanabe decomposition of the benchmarked claim under partial information and provide its description in terms of the integrands in the classical Galtchouk-Kunita-Watanabe decomposition under full information via dual predictable projections. Finally, we apply the results in the case of a Markovian jump-diffusion driven market model where the assets prices dynamics depend on a stochastic factor which is not observable by investors.

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Paper provided by arXiv.org in its series Papers with number 1307.6036.

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Date of creation: Jul 2013
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Handle: RePEc:arx:papers:1307.6036

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  1. Eckhard Platen, 2008. "A Unifying Approach to Asset Pricing," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 227, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. Hardy Hulley & Martin Schweizer, 2010. "M6 - On Minimal Market Models and Minimal Martingale Measures," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 280, Quantitative Finance Research Centre, University of Technology, Sydney.
  3. Thomas Møller, 2001. "Risk-minimizing hedging strategies for insurance payment processes," Finance and Stochastics, Springer, vol. 5(4), pages 419-446.
  4. Eckhard Platen, 2004. "Diversified Portfolios with Jumps in a Benchmark Framework," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 129, Quantitative Finance Research Centre, University of Technology, Sydney.
  5. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
  6. Martin Schweizer, 1994. "Risk-Minimizing Hedging Strategies Under Restricted Information," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 4(4), pages 327-342.
  7. Constantinos Kardaras, 2012. "Market viability via absence of arbitrage of the first kind," Finance and Stochastics, Springer, vol. 16(4), pages 651-667, October.
  8. Francesca Biagini & Jan Widenmann, 2012. "Pricing Of Unemployment Insurance Products With Doubly Stochastic Markov Chains," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., World Scientific Publishing Co. Pte. Ltd., vol. 15(04), pages 1250025-1-1.
  9. Ke Du & Eckhard Platen, 2011. "Three-Benchmarked Risk Minimization for Jump Diffusion Markets," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 296, Quantitative Finance Research Centre, University of Technology, Sydney.
  10. Claudia Ceci & Anna Gerardi, 2009. "Pricing For Geometric Marked Point Processes Under Partial Information: Entropy Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., World Scientific Publishing Co. Pte. Ltd., vol. 12(02), pages 179-207.
  11. Riesner, Martin, 2006. "Hedging life insurance contracts in a Lévy process financial market," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 599-608, June.
  12. Vandaele, Nele & Vanmaele, Michèle, 2008. "A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1128-1137, June.
  13. Morten Christensen & Eckhard Platen, 2004. "A General Benchmark Model for Stochastic Jump Sizes," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 139, Quantitative Finance Research Centre, University of Technology, Sydney.
  14. Claudia Ceci & Anna Gerardi, 2006. "A Model For High Frequency Data Under Partial Information: A Filtering Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., World Scientific Publishing Co. Pte. Ltd., vol. 9(04), pages 555-576.
  15. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 16(1), pages 131-151.
  16. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, Econometric Society, vol. 41(5), pages 867-87, September.
  17. Francesca Biagini & Alessandra Cretarola & Eckhard Platen, 2012. "Local Risk-Minimization under the Benchmark Approach," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 319, Quantitative Finance Research Centre, University of Technology, Sydney.
  18. Tomas Björk & Yuri Kabanov & Wolfgang Runggaldier, 1997. "Bond Market Structure in the Presence of Marked Point Processes," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 7(2), pages 211-239.
  19. Rüdiger Frey, 2000. "Risk Minimization with Incomplete Information in a Model for High-Frequency Data," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 10(2), pages 215-225.
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Cited by:
  1. Jan Baldeaux & Man Chung Fung & Katja Ignatieva & Eckhard Platen, 2014. "A Hybrid Model for Pricing and Hedging of Long Dated Bonds," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 343, Quantitative Finance Research Centre, University of Technology, Sydney.

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