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Pricing options in incomplete equity markets via the instantaneous Sharpe ratio

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  • Erhan Bayraktar

    ()

  • Virginia Young

    ()

Abstract

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).

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File URL: http://hdl.handle.net/10.1007/s10436-007-0084-0
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Bibliographic Info

Article provided by Springer in its journal Annals of Finance.

Volume (Year): 4 (2008)
Issue (Month): 4 (October)
Pages: 399-429

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Handle: RePEc:kap:annfin:v:4:y:2008:i:4:p:399-429

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Web page: http://www.springerlink.com/link.asp?id=112370

Related research

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

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References

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  1. Schweizer, Martin, 2001. "From actuarial to financial valuation principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 31-47, February.
  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, 05.
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Cited by:
  1. Bauer, Daniel & Börger, Matthias & Ruß, Jochen, 2010. "On the pricing of longevity-linked securities," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 139-149, February.
  2. 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.
  3. Kraeussl, Roman & Wiehenkamp, Christian, 2010. "A call on Art investments," CFS Working Paper Series 2010/03, Center for Financial Studies (CFS).
  4. Akuzawa, Toshinao & Nishiyama, Yoshihiko, 2013. "Implied Sharpe ratios of portfolios with options: Application to Nikkei futures and listed options," The North American Journal of Economics and Finance, Elsevier, vol. 25(C), pages 335-357.
  5. Emilio Bisetti & Carlo A. Favero & Giacomo Nocera & Claudio Tebaldi, 2013. "A Multivariate Model of Strategic Asset Allocation with Longevity Risk," Working Papers 503, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
  6. 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.
  7. 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.

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