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Analytic Pricing of Contingent Claims Under the Real-World Measure

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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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Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 216.

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Length: 30
Date of creation: 01 Feb 2008
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Handle: RePEc:uts:rpaper:216

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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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  1. Shane Miller & Eckhard Platen, 2004. "Two-Factor Model for Low Interest Rate Regimes," Research Paper Series 130, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. Eckhard Platen, 2003. "An Alternative Interest Rate Term Structure Model," Research Paper Series 97, Quantitative Finance Research Centre, University of Technology, Sydney.
  3. 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.
  4. David Heath & Eckhard Platen, 2006. "Local volatility function models under a benchmark approach," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 197-206.
  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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