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Local volatility function models under a benchmark approach

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Author Info
David Heath
Eckhard Platen

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Abstract

Without requiring the existence of an equivalent risk-neutral probability measure this paper studies a class of one-factor local volatility function models for stock indices under a benchmark approach. It is assumed that the dynamics for a large diversified index approximates that of the growth optimal portfolio. Fair prices for derivatives when expressed in units of the index are martingales under the real-world probability measure. Different to the classical approach that derives risk-neutral probabilities the paper obtains the transition density for the index with respect to the real-world probability measure. Furthermore, the Dupire formula for the underlying local volatility function is recovered without assuming the existence of an equivalent risk-neutral probability measure. A modification of the constant elasticity of variance model and a version of the minimal market model are discussed as specific examples together with a smoothed local volatility function model that fits a snapshot of S&P500 index options data.

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Publisher Info
Article provided by Taylor and Francis Journals in its journal Quantitative Finance.

Volume (Year): 6 (2006)
Issue (Month): 3 (June)
Pages: 197-206
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Handle: RePEc:taf:quantf:v:6:y:2006:i:3:p:197-206

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Related research
Keywords: Local volatility function; Index derivatives; Growth optimal portfolio; Benchmark approach; Fair pricing; Dupire formula; Modified CEV model; Minimal market model;

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July. [Downloadable!] (restricted)
  2. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, September.
  3. David Heath & Eckhard Platen, 2002. "Consistent Pricing and Hedging for a Modified Constant Elasticity of Variance Model," Research Paper Series 78, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
    Other versions:
  4. 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. [Downloadable!]
  5. Leif Andersen, Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor and Francis Journals, vol. 7(1), pages 1-32, March. [Downloadable!] (restricted)
  6. David Heath & Eckhard Platen, 2002. "A Variance Reduction Technique Based on Integral Representations," Research Paper Series 75, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  7. Beckers, Stan, 1980. " The Constant Elasticity of Variance Model and Its Implications for Option Pricing," Journal of Finance, American Finance Association, vol. 35(3), pages 661-73, June. [Downloadable!] (restricted)
  8. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July. [Downloadable!] (restricted)
  9. Schroder, Mark Douglas, 1989. " Computing the Constant Elasticity of Variance Option Pricing Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 211-19, March. [Downloadable!] (restricted)
  10. Eckhard Platen, 2003. "Modeling the Volatility and Expected Value of a Diversified World Index," Research Paper Series 103, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  11. Eckhard Platen, 2001. "A Minimal Financial Market Model," Research Paper Series 48, Quantitative Finance Research Centre, University of Technology, Sydney.
    Other versions:
  12. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October. [Downloadable!] (restricted)
  13. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  14. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166. [Downloadable!] (restricted)
  15. Eckhard Platen, 2002. "Benchmark Model with Intensity Based Jumps," Research Paper Series 81, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  16. Kevin Fergusson & Eckhard Platen, 2005. "On the Distributional Characterization of Log-returns of a World Stock Index," Research Paper Series 153, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  17. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June. [Downloadable!] (restricted)
  18. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley. [Downloadable!]
  19. David Heath & Eckhard Platen, 2003. "Pricing of Index Options Under a Minimal Market Model with Lognormal Scaling," Research Paper Series 101, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
Full references

Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. David Heath & Eckhard Platen, 2004. "Understanding the Implied Volatility Surface for Options on a Diversified Index," Asia-Pacific Financial Markets, Springer, vol. 11(1), pages 55-77, March. [Downloadable!] (restricted)
    Other versions:
  2. Shane Miller & Eckhard Platen, 2008. "Analytic Pricing of Contingent Claims Under the Real-World Measure," Research Paper Series 216, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
    Other versions:
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