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On honest times in financial modeling

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  • Ashkan Nikeghbali
  • Eckhard Platen

Abstract

This paper demonstrates the usefulness and importance of the concept of honest times to financial modeling. It studies a financial market with asset prices that follow jump-diffusions with negative jumps. The central building block of the market model is its growth optimal portfolio (GOP), which maximizes the growth rate of strictly positive portfolios. Primary security account prices, when expressed in units of the GOP, turn out to be nonnegative local martingales. In the proposed framework an equivalent risk neutral probability measure need not exist. Derivative prices are obtained as conditional expectations of corresponding future payoffs, with the GOP as numeraire and the real world probability as pricing measure. The time when the global maximum of a portfolio with no positive jumps, when expressed in units of the GOP, is reached, is shown to be a generic representation of an honest time. We provide a general formula for the law of such honest times and compute the conditional distributions of the global maximum of a portfolio in this framework. Moreover, we provide a stochastic integral representation for uniformly integrable martingales whose terminal values are functions of the global maximum of a portfolio. These formulae are model independent and universal. We also specialize our results to some examples where we hedge a payoff that arrives at an honest time.

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File URL: http://arxiv.org/pdf/0808.2892
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 0808.2892.

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Date of creation: Aug 2008
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Handle: RePEc:arx:papers:0808.2892

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  1. Amendinger, J├╝rgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," SFB 373 Discussion Papers 1998,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  2. Madan, D. & Roynette, B. & Yor, Marc, 2008. "Option prices as probabilities," Finance Research Letters, Elsevier, vol. 5(2), pages 79-87, June.
  3. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  4. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151.
  5. Eckhard Platen, 2002. "Benchmark Model with Intensity Based Jumps," Research Paper Series 81, Quantitative Finance Research Centre, University of Technology, Sydney.
  6. Eckhard Platen, 2001. "A Minimal Financial Market Model," Research Paper Series 48, Quantitative Finance Research Centre, University of Technology, Sydney.
  7. Shane Miller & Eckhard Platen, 2004. "Two-Factor Model for Low Interest Rate Regimes," Research Paper Series 130, Quantitative Finance Research Centre, University of Technology, Sydney.
  8. Robert Fernholz & Ioannis Karatzas, 2005. "Relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, vol. 1(2), pages 149-177, November.
  9. N. Hofmann & E. Platen & M. Schweizer, 1992. "Option Pricing under Incompleteness and Stochastic Volatility," Discussion Paper Serie B 209, University of Bonn, Germany.
  10. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
  11. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195.
  12. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
  13. repec:wop:humbsf:1998-25 is not listed on IDEAS
  14. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, July.
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