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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.

Suggested Citation

  • Ashkan Nikeghbali & Eckhard Platen, 2008. "On honest times in financial modeling," Papers 0808.2892, arXiv.org.
  • Handle: RePEc:arx:papers:0808.2892
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    References listed on IDEAS

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    1. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
    2. Norbert Hofmann & Eckhard Platen & Martin Schweizer, 1992. "Option Pricing Under Incompleteness and Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 2(3), pages 153-187.
    3. Shane Miller & Eckhard Platen, 2004. "A Two-Factor Model for Low Interest Rate Regimes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 107-133, March.
    4. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195.
    5. Madan, D. & Roynette, B. & Yor, Marc, 2008. "Option prices as probabilities," Finance Research Letters, Elsevier, vol. 5(2), pages 79-87, June.
    6. Platen, Eckhard, 2000. "A minimal financial market model," SFB 373 Discussion Papers 2000,91, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    7. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151.
    8. Eckhard Platen, 2002. "Benchmark Model with Intensity Based Jumps," Research Paper Series 81, Quantitative Finance Research Centre, University of Technology, Sydney.
    9. Amendinger, Jürgen & 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.
    10. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, June.
    11. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    12. Norbert Hofmann & Eckhard Platen & Martin Schweizer, 1992. "Option Pricing Under Incompleteness and Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 2(3), pages 153-187.
    13. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Robert Fernholz & Ioannis Karatzas, 2005. "Relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, vol. 1(2), pages 149-177, November.
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    Cited by:

    1. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.

    More about this item

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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