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On the Use of Numeraires in Option pricing

Author

Listed:
  • Benninga, Simon

    (Faculty of Management, Tel Aviv University)

  • Björk, Tomas

    (Dept. of Finance, Stockholm School of Economics)

  • Wiener, Zvi

    (School of Business, Hebrew University of Jerusalem)

Abstract

In this paper we discuss the significant computational simplification that occurs when option pricing is approached through the change of numeraire technique. The original impetus was a recently published paper (Hoang, Powell, Shi 1999) on endowment options; in the present paper we extend these results to the case of stochastic interest rates. We also discuss four additional option pricing problems within the framework of a change of numeraire: 1. Pricing savings plans which incorporate a choice of linkage. 2. Pricing convertible bonds. 3. Pricing employee stock ownership plans 4. Pricing options where the strike price is in a currency different from the stock price.

Suggested Citation

  • Benninga, Simon & Björk, Tomas & Wiener, Zvi, 2002. "On the Use of Numeraires in Option pricing," SSE/EFI Working Paper Series in Economics and Finance 484, Stockholm School of Economics.
    • Handle: RePEc:hhs:hastef:0484
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    References listed on IDEAS

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    1. Brennan, M J & Schwartz, Eduardo S, 1977. "Convertible Bonds: Valuation and Optimal Strategies for Call and Conversion," Journal of Finance, American Finance Association, vol. 32(5), pages 1699-1715, December.
    2. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    3. Chip Heath & Steven Huddart & Mark Lang, 1999. "Psychological Factors and Stock Option Exercise," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(2), pages 601-627.
    4. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
    5. Fischer, Stanley, 1978. "Call Option Pricing when the Exercise Price Is Uncertain, and the Valuation of Index Bonds," Journal of Finance, American Finance Association, vol. 33(1), pages 169-176, March.
    6. Jamshidian, Farshid, 1989. " An Exact Bond Option Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 205-209, March.
    7. Huddart, Steven, 1994. "Employee stock options," Journal of Accounting and Economics, Elsevier, vol. 18(2), pages 207-231, September.
    8. Brenner, Menachem & Galai, Dan, 1978. "The determinants of the return on index bonds," Journal of Banking & Finance, Elsevier, vol. 2(1), pages 47-64, June.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Nikolai Dokuchaev, 2018. "On the implied market price of risk under the stochastic numéraire," Annals of Finance, Springer, vol. 14(2), pages 223-251, May.
    2. Yoram Landskroner & Alon Raviv, 2008. "The valuation of inflation‐indexed and FX convertible bonds," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 28(7), pages 634-655, July.
    3. Jui‐Jane Chang & Son‐Nan Chen & Ting‐Pin Wu, 2013. "Currency‐Protected Swaps and Swaptions with Nonzero Spreads in a Multicurrency LMM," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(9), pages 827-867, September.
    4. Hyong-Chol O & Dae-Sung Choe, 2019. "Pricing Formulae of Power Binary and Normal Distribution Standard Options and Applications," Papers 1903.04106, arXiv.org.
    5. Raquel M. Gaspar & Mariana Khapko, 2023. "In memoriam: Tomas Björk (1947–2021)," Finance and Stochastics, Springer, vol. 27(4), pages 867-885, October.
    6. Sergei Belkov & Igor V. Evstigneev & Thorsten Hens, 2020. "An evolutionary finance model with a risk-free asset," Annals of Finance, Springer, vol. 16(4), pages 593-607, December.
    7. Hyong-chol O & Yong-hwa Ro & Ning Wan, 2013. "The Use of Numeraires in Multi-dimensional Black-Scholes Partial Differential Equations," Papers 1310.8296, arXiv.org, revised Jul 2014.
    8. Hyong-chol O & Yong-hwa Ro & Ning Wan, 2014. "A Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes Equations," Papers 1406.2053, arXiv.org.
    9. Ravi Kashyap, 2022. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Annals of Operations Research, Springer, vol. 315(2), pages 1175-1215, August.
    10. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    11. Nikolai Dokuchaev, 2011. "On martingale measures and pricing for continuous bond-stock market with stochastic bond," Papers 1108.0719, arXiv.org, revised Sep 2014.
    12. Massoud Heidari & Liuren WU, 2002. "Are Interest Rate Derivatives Spanned by the Term Structure of Interest Rates?," Finance 0207013, University Library of Munich, Germany.
    13. Markus Leippold & Zvi Wiener, 2005. "Efficient Calibration of Trinomial Trees for One-Factor Short Rate Models," Review of Derivatives Research, Springer, vol. 7(3), pages 213-239, October.
    14. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742, Decembrie.
    15. Ravi Kashyap, 2016. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Papers 1609.01274, arXiv.org, revised Mar 2022.

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    More about this item

    Keywords

    Numeraire; option; convertible bond;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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    This paper has been announced in the following NEP Reports:

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