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From Measure Changes to Time Changes in Asset Pricing

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  • Geman, Hélyette
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    Abstract

    The goal of the paper is to review the last 35 years of continuous-time finance by focusing on two major advances: (i) The powerful elegance of the martingale representation for primitive assets and attainable contingent claims in more and more general settings, thanks to the probabilistic tool of probability change and the economic flexibility in the choice of the numéraire relative to which prices are expressed. This numéraire evolved over time from the money market account to a zero-coupon bond or a stock price, lastly to strictly positive quantities involved in the Libor or swap market models and making the pricing of caps or swaptions quite efficient. (ii) The persistent central role of Brownian motion in finance across the 20th century: even when the underlying asset price is a very general semi-martingale, the no-arbitrage assumption and Monroe theorem [Monroe, I., 1978. Processes that can be embedded in Brownian motion. Annals of Probability 6, 42–56] allow us to write it as Brownian motion as long as we are willing to change the time. The appropriate stochastic clock can be shown empirically to be driven by the cumulative number of trades, hence by market activity. Consequently, starting with a general multidimensional stochastic process S defined on a probability space (Ω, , P) and representing the prices of primitive securities, the no-arbitrage assumption allows, for any chosen numéraire, to obtain a martingale representation for S under a probability measure QS equivalent to P. This route will be particularly beneficiary for the pricing of complex contingent claims. Alternatively, changing the clock, i.e., changing the filtration (), we can recover the Brownian motion and normality of returns. In all cases martingales appear as the central representation of asset prices, either through a measure change or through a time change.

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    Bibliographic Info

    Paper provided by Paris Dauphine University in its series Economics Papers from University Paris Dauphine with number 123456789/1388.

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    Date of creation: May 2005
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    Publication status: Published in Journal of Banking and Finance, 2005, Vol. 29, no. 11. pp. 2701-2722.Length: 21 pages
    Handle: RePEc:dau:papers:123456789/1388

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    Related research

    Keywords: Market activity; Numéraire change; Martingale representation; Stochastic clock; Stochastic volatility;

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    Cited by:
    1. Carol Alexander & Leonardo M. Nogueira, 2006. "Hedging Options with Scale-Invariant Models," ICMA Centre Discussion Papers in Finance icma-dp2006-03, Henley Business School, Reading University.
    2. Inderfurth, Karl & Kelle, Peter & Kleber, Rainer, 2013. "Dual sourcing using capacity reservation and spot market: Optimal procurement policy and heuristic parameter determination," European Journal of Operational Research, Elsevier, vol. 225(2), pages 298-309.
    3. Alexander, Carol & Nogueira, Leonardo M., 2007. "Model-free hedge ratios and scale-invariant models," Journal of Banking & Finance, Elsevier, vol. 31(6), pages 1839-1861, June.
    4. Chang, Carolyn W. & Chang, Jack S.K. & Lu, WeLi, 2010. "Pricing catastrophe options with stochastic claim arrival intensity in claim time," Journal of Banking & Finance, Elsevier, vol. 34(1), pages 24-32, January.
    5. Pippenger, John, 2008. "Freely Floating Exchange Rates Do Not Systematically Overshoot," University of California at Santa Barbara, Economics Working Paper Series qt97m8z6hw, Department of Economics, UC Santa Barbara.
    6. Chang, Carolyn W. & Chang, Jack S.K. & Lu, WeiLi, 2008. "Pricing catastrophe options in discrete operational time," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 422-430, December.
    7. Karl Inderfurth & Peter Kelle & Rainer Kleber, 2011. "Dual Sourcing Using Capacity Reservation and Spot Market: Optimal Procurement Policy and Heuristic Parameter Determination," FEMM Working Papers 110014, Otto-von-Guericke University Magdeburg, Faculty of Economics and Management.
    8. Klößner, Stefan & Becker, Martin & Friedmann, Ralph, 2012. "Modeling and measuring intraday overreaction of stock prices," Journal of Banking & Finance, Elsevier, vol. 36(4), pages 1152-1163.

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