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Analytic Pricing Of Contingent Claims Under The Real-World Measure

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Author Info

  • SHANE M. MILLER

    (Citigroup Global Markets Australia Pty. Ltd., 2 Park Street, Sydney NSW 2000, Australia)

  • ECKHARD PLATEN

    () (University of Technology Sydney, School of Finance & Economics and Department of Mathematical Sciences, PO Box 123, Broadway NSW 2007, Australia)

Abstract

This article derives a series of analytic formulae for various contingent claims under the real-world probability measure using the stylised minimal market model (SMMM). This model provides realistic dynamics for the growth optimal portfolio (GOP) as a well-diversified equity index. It captures both leptokurtic returns with correct tail properties and the leverage effect. Under the SMMM, the discounted GOP takes the form of a time-transformed squared Bessel process of dimension four. From this property, one finds that the SMMM possesses a special and interesting relationship to non-central chi-square random variables with zero degrees of freedom. The analytic formulae derived under the SMMM include options on the GOP, options on exchange prices and options on zero-coupon bonds. For options on zero-coupon bonds, analytic prices facilitate efficient calculation of interest rate caps and floors.

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Bibliographic Info

Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

Volume (Year): 11 (2008)
Issue (Month): 08 ()
Pages: 841-867

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Handle: RePEc:wsi:ijtafx:v:11:y:2008:i:08:p:841-867

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Related research

Keywords: Benchmark approach; real-world pricing; growth optimal portfolio; minimal market model; zero-coupon bonds; exchange prices; interest rate caps and floors;

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References

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  1. David Heath & Eckhard Platen, 2006. "Local volatility function models under a benchmark approach," Quantitative Finance, Taylor and Francis Journals, vol. 6(3), pages 197-206.
  2. Eckhard Platen, 2005. "An Alternative Interest Rate Term Structure Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(06), pages 717-735.
  3. Shane Miller & Eckhard Platen, 2004. "A Two-Factor Model for Low Interest Rate Regimes," Asia-Pacific Financial Markets, Springer, vol. 11(1), pages 107-133, March.
  4. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
  5. Hardy Hulley & Shane Miller & Eckhard Platen, 2005. "Benchmarking and Fair Pricing Applied to Two Market Models," Research Paper Series 155, Quantitative Finance Research Centre, University of Technology, Sydney.
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Cited by:
  1. Hardy Hulley & Shane Miller & Eckhard Platen, 2005. "Benchmarking and Fair Pricing Applied to Two Market Models," Research Paper Series 155, Quantitative Finance Research Centre, University of Technology, Sydney.

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