On Honest Times in Financial Modeling

Ashkan Nikeghbali and Eckhard Platen ()
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Ashkan Nikeghbali: Institut fur Mathematik, Universitat Zurich

No 229, Research Paper Series from Quantitative Finance Research Centre, University of Technology, Sydney

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.

Keywords: jump diffusion market; honest times; growth optimal portfolio; benchmark approach; real world pricing; nonnegative local martingales (search for similar items in EconPapers)
JEL-codes: G10 G13 (search for similar items in EconPapers)
Pages: 27 pages
Date: 2008-08-01
New Economics Papers: this item is included in nep-fmk
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Citations: View citations in EconPapers (2)

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Working Paper: On honest times in financial modeling (2008)
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