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

  • Frank Milne


    (Queen's University)

  • Dilip Madan

    (University of Maryland)

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.

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Paper provided by Queen's University, Department of Economics in its series Working Papers with number 1158.

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Length: 23 pages
Date of creation: Jul 1994
Date of revision:
Publication status: Published in Mathematical Finance, Vol. 4, No. 3 (July 1994), 223-245
Handle: RePEc:qed:wpaper:1158
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  1. 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 685-95, June.
  2. Ross, Stephen A, 1976. "Options and Efficiency," The Quarterly Journal of Economics, MIT Press, vol. 90(1), pages 75-89, February.
  3. 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.
  4. David C. Nachman, 1988. "Spanning and Completeness with Options," Review of Financial Studies, Society for Financial Studies, vol. 1(3), pages 311-328.
  5. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
  6. Peter Lakner, 1993. "Martingale Measures For A Class of Right-Continuous Processes," Mathematical Finance, Wiley Blackwell, vol. 3(1), pages 43-53.
  7. 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 653-72, October.
  8. 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.
  9. Robert A. Jarrow & Dilip B. Madan, 1991. "A Characterization of Complete Security Markets On A Brownian Filtration," Mathematical Finance, Wiley Blackwell, vol. 1(3), pages 31-43.
  10. 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 621-51, October.
  11. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-92.
  12. John, Kose, 1984. "Market Resolution and Valuation in Incomplete Markets," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 19(01), pages 29-44, March.
  13. Freddy Delbaen, 1992. "Representing Martingale Measures When Asset Prices Are Continuous And Bounded," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 107-130.
  14. 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 1235-43, September.
  15. 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.
  16. Jarrow, Robert A., 1986. "A characterization theorem for unique risk neutral probability measures," Economics Letters, Elsevier, vol. 22(1), pages 61-65.
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