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Pricing options in incomplete equity markets via the instantaneous Sharpe ratio


  • Erhan Bayraktar


  • Virginia Young



We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. First, we apply our method to price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and Sa\'{a}-Requejo (2000) and of Bj\"{o}rk and Slinko (2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque, Papanicolaou, and Sircar (2000).
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Erhan Bayraktar & Virginia Young, 2008. "Pricing options in incomplete equity markets via the instantaneous Sharpe ratio," Annals of Finance, Springer, vol. 4(4), pages 399-429, October.
  • Handle: RePEc:kap:annfin:v:4:y:2008:i:4:p:399-429 DOI: 10.1007/s10436-007-0084-0

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    References listed on IDEAS

    1. Virginia R. Young, 2007. "Pricing Life Insurance under Stochastic Mortality via the Instantaneous Sharpe Ratio: Theorems and Proofs," Papers 0705.1297,
    2. Tim Leung & Ronnie Sircar, 2009. "Accounting For Risk Aversion, Vesting, Job Termination Risk And Multiple Exercises In Valuation Of Employee Stock Options," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 99-128.
    3. Marek Musiela & Thaleia Zariphopoulou, 2004. "An example of indifference prices under exponential preferences," Finance and Stochastics, Springer, vol. 8(2), pages 229-239, May.
    4. Schweizer, Martin, 2001. "From actuarial to financial valuation principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 31-47, February.
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    Cited by:

    1. Wang, Ting & Young, Virginia R., 2016. "Hedging pure endowments with mortality derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 238-255.
    2. repec:cup:jfinqa:v:52:y:2017:i:05:p:2251-2275_00 is not listed on IDEAS
    3. Bisetti, Emilio & Favero, Carlo A. & Nocera, Giacomo & Tebaldi, Claudio, 2017. "A Multivariate Model of Strategic Asset Allocation with Longevity Risk," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 52(05), pages 2251-2275, October.
    4. Bayraktar, Erhan & Milevsky, Moshe A. & David Promislow, S. & Young, Virginia R., 2009. "Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities," Journal of Economic Dynamics and Control, Elsevier, vol. 33(3), pages 676-691, March.
    5. Laurence Carassus & Emmanuel Temam, 2010. "Pricing and Hedging Basis Risk under No Good Deal Assumption," Working Papers hal-00498479, HAL.
    6. Young, Virginia R., 2008. "Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 691-703, April.
    7. repec:spr:mathme:v:86:y:2017:i:1:d:10.1007_s00186-017-0588-y is not listed on IDEAS
    8. Roman Kraeussl & Christian Wiehenkamp, 2012. "A call on art investments," Review of Derivatives Research, Springer, vol. 15(1), pages 1-23, April.
    9. L. Carassus & E. Temam, 2014. "Pricing and hedging basis risk under no good deal assumption," Annals of Finance, Springer, vol. 10(1), pages 127-170, February.
    10. Dirk Becherer & Klebert Kentia, 2016. "Hedging under generalized good-deal bounds and model uncertainty," Papers 1607.04488,, revised Apr 2017.

    More about this item


    Pricing derivative securities; Incomplete markets; Sharpe ratio; Correlated assets; Stochastic volatility; Non-linear partial differential equations; Good deal bounds; G13;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing


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