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Contingent Claims Valued And Hedged By Pricing And Investing In A Basis

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  • Dilip B. Madan
  • Frank Milne

Abstract

Contingent claims with payoffs depending on finitely many asset prices are modeled as elements of a separable Hilbert space. Under fairly general conditions, including market completeness, it is shown that one may change measure to a reference measure under which asset prices are Gaussian and for which the family of Hermite polynomials serves as an orthonormal basis. Basis pricing synthesizes claim valuation and basis investment provides static hedging opportunities. For claims written as functions of a single asset price we infer from observed option prices the implicit prices of basis elements and use these to construct the implied equivalent martingale measure density with respect to the reference measure, which in this case is the Black‐Scholes geometric Brownian motion model. Data on S & P 500 options from the Wall Street Journal are used to illustrate the calculations involved. On this illustrative data set the equivalent martingale measure deviates from the Black‐Scholes model by relatively discounting the larger price movements with a compensating premia placed on the smaller movements.

Suggested Citation

  • Dilip B. Madan & Frank Milne, 1994. "Contingent Claims Valued And Hedged By Pricing And Investing In A Basis," Mathematical Finance, Wiley Blackwell, vol. 4(3), pages 223-245, July.
  • Handle: RePEc:bla:mathfi:v:4:y:1994:i:3:p:223-245
    DOI: 10.1111/j.1467-9965.1994.tb00093.x
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    File URL: https://doi.org/10.1111/j.1467-9965.1994.tb00093.x
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    References listed on IDEAS

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    1. Back, Kerry & Pliska, Stanley R., 1991. "On the fundamental theorem of asset pricing with an infinite state space," Journal of Mathematical Economics, Elsevier, vol. 20(1), pages 1-18.
    2. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    3. 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.
    4. Peter Lakner, 1993. "Martingale Measures For A Class of Right‐Continuous Processes," Mathematical Finance, Wiley Blackwell, vol. 3(1), pages 43-53, January.
    5. Freddy Delbaen, 1992. "Representing Martingale Measures When Asset Prices Are Continuous And Bounded," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 107-130, April.
    6. David C. Nachman, 1988. "Spanning and Completeness with Options," Review of Financial Studies, Society for Financial Studies, vol. 1(3), pages 311-328.
    7. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    8. Robert A. Jarrow & Dilip B. Madan, 1991. "A Characterization of Complete Security Markets On A Brownian Filtration1," Mathematical Finance, Wiley Blackwell, vol. 1(3), pages 31-43, July.
    9. John, Kose, 1981. "Efficient Funds in a Financial Market with Options: A New Irrelevance Proposition," Journal of Finance, American Finance Association, vol. 36(3), pages 685-695, June.
    10. Stephen A. Ross, 1976. "Options and Efficiency," The Quarterly Journal of Economics, Oxford University Press, vol. 90(1), pages 75-89.
    11. Amershi, Amin H, 1985. "A Complete Analysis of Full Pareto Efficiency in Financial Markets for Arbitrary Preferences," Journal of Finance, American Finance Association, vol. 40(4), pages 1235-1243, September.
    12. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    13. Banz, Rolf W & Miller, Merton H, 1978. "Prices for State-contingent Claims: Some Estimates and Applications," The Journal of Business, University of Chicago Press, vol. 51(4), pages 653-672, October.
    14. Michel Chatelain & Christophe Stricker, 1994. "ON COMPONENTWISE and VECTOR STOCHASTIC INTEGRATION," Mathematical Finance, Wiley Blackwell, vol. 4(1), pages 57-65, January.
    15. John, Kose, 1984. "Market Resolution and Valuation in Incomplete Markets," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 19(1), pages 29-44, March.
    16. Jarrow, Robert A., 1986. "A characterization theorem for unique risk neutral probability measures," Economics Letters, Elsevier, vol. 22(1), pages 61-65.
    17. Green, Richard C. & Jarrow, Robert A., 1987. "Spanning and completeness in markets with contingent claims," Journal of Economic Theory, Elsevier, vol. 41(1), pages 202-210, February.
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    More about this item

    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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