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A Discrete Time Equivalent Martingale Measure

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  • Robert J. Elliott
  • Dilip B. Madan

Abstract

An equivalent martingale measure selection strategy for discrete time, continuous state, asset price evolution models is proposed. The minimal martingale law is shown to generally fail to produce a probability law in this context. The proposed strategy, termed the extended Girsanov principle, performs a multiplicative decomposition of asset price movements into a predictable and martingale component with the measure change identifying the discounted asset price process to the martingale component. However, unlike the minimal martingale law, the resulting martingale law of the extended Girsanov principle leads to weak form efficient price processes. It is shown that the proposed measure change is relevant for economies in which investors adopt hedging strategies that minimize the variance of a risk adjusted discounted cost of hedging that uses risk adjusted asset prices in calculating hedging returns. Risk adjusted prices deflate asset prices by the asset's excess return. The explicit form of the change of measure density leads to tractable econometric strategies for testing the validity of the extended Girsanov principle. A number of interesting applications of the extended Girsanov principle are also developed. Copyright Blackwell Publishers 1998.

Suggested Citation

  • Robert J. Elliott & Dilip B. Madan, 1998. "A Discrete Time Equivalent Martingale Measure," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 127-152.
  • Handle: RePEc:bla:mathfi:v:8:y:1998:i:2:p:127-152
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    Cited by:

    1. Guégan, Dominique & Ielpo, Florian & Lalaharison, Hanjarivo, 2013. "Option pricing with discrete time jump processes," Journal of Economic Dynamics and Control, Elsevier, pages 2417-2445.
    2. Chorro, C. & Guégan, D. & Ielpo, F., 2010. "Martingalized historical approach for option pricing," Finance Research Letters, Elsevier, pages 24-28.
    3. Christophe Chorro & Dominique Guegan & Florian Ielpo, 2008. "Option pricing under GARCH models with generalized hyperbolic innovations (I) : methodology," Documents de travail du Centre d'Economie de la Sorbonne b08037, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Firoozi, Fathali, 2006. "On the martingale property of economic and financial instruments," International Review of Economics & Finance, Elsevier, vol. 15(1), pages 87-96.
    5. repec:hal:journl:halshs-00469529 is not listed on IDEAS
    6. Badescu Alex & Kulperger Reg & Lazar Emese, 2008. "Option Valuation with Normal Mixture GARCH Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, pages 1-42.
    7. Joan del Castillo & Juan-Pablo Ortega, 2011. "Hedging of time discrete auto-regressive stochastic volatility options," Papers 1110.6322, arXiv.org.
    8. Badescu, Alex & Elliott, Robert J. & Siu, Tak Kuen, 2009. "Esscher transforms and consumption-based models," Insurance: Mathematics and Economics, Elsevier, pages 337-347.
    9. repec:hal:journl:halshs-00611706 is not listed on IDEAS
    10. Liew, Chuin Ching & Siu, Tak Kuen, 2010. "A hidden Markov regime-switching model for option valuation," Insurance: Mathematics and Economics, Elsevier, pages 374-384.
    11. Huang, Shih-Feng & Tu, Ya-Ting, 2014. "Asymptotic distribution of the EPMS estimator for financial derivatives pricing," Computational Statistics & Data Analysis, Elsevier, pages 129-145.
    12. repec:hal:journl:halshs-00542475 is not listed on IDEAS
    13. Badescu, Alexandru & Elliott, Robert J. & Ortega, Juan-Pablo, 2014. "Quadratic hedging schemes for non-Gaussian GARCH models," Journal of Economic Dynamics and Control, Elsevier, vol. 42(C), pages 13-32.
    14. Badescu, Alexandru & Elliott, Robert J. & Ortega, Juan-Pablo, 2015. "Non-Gaussian GARCH option pricing models and their diffusion limits," European Journal of Operational Research, Elsevier, vol. 247(3), pages 820-830.
    15. Badescu, Alexandru M. & Kulperger, Reg J., 2008. "GARCH option pricing: A semiparametric approach," Insurance: Mathematics and Economics, Elsevier, pages 69-84.
    16. Badescu, Alexandru & Cui, Zhenyu & Ortega, Juan-Pablo, 2016. "A note on the Wang transform for stochastic volatility pricing models," Finance Research Letters, Elsevier, vol. 19(C), pages 189-196.
    17. : Andrea Gamba & Carmen Aranda Leon & Alessio Saretto, 2011. "Dynamic Capacity Choice, Dynamic Capital Structure and Credit Risk," Working Papers wpn11-03, Warwick Business School, Finance Group.
    18. Alexandru Badescu & Robert J. Elliott & Juan-Pablo Ortega, 2012. "Quadratic hedging schemes for non-Gaussian GARCH models," Papers 1209.5976, arXiv.org, revised Dec 2013.
    19. repec:hal:journl:halshs-00437927 is not listed on IDEAS
    20. Liu, Yanxin & Li, Johnny Siu-Hang & Ng, Andrew Cheuk-Yin, 2015. "Option pricing under GARCH models with Hansen's skewed-t distributed innovations," The North American Journal of Economics and Finance, Elsevier, vol. 31(C), pages 108-125.

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