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Local volatility function models under a benchmark approach

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  • David Heath
  • Eckhard Platen

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.

Suggested Citation

  • David Heath & Eckhard Platen, 2006. "Local volatility function models under a benchmark approach," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 197-206.
  • Handle: RePEc:taf:quantf:v:6:y:2006:i:3:p:197-206
    DOI: 10.1080/14697680600699787
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    References listed on IDEAS

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    1. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    2. H. Berestycki & J. Busca & I. Florent, 2002. "Asymptotics and calibration of local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 61-69.
    3. 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.
    4. 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 661-673, June.
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    6. Eckhard Platen, 2004. "Modeling The Volatility And Expected Value Of A Diversified World Index," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(04), pages 511-529.
    7. David Heath & Eckhard Platen, 2002. "Consistent pricing and hedging for a modified constant elasticity of variance model," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 459-467.
    8. Platen, Eckhard, 2000. "A minimal financial market model," SFB 373 Discussion Papers 2000,91, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    9. 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.
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    12. Lishang Jiang & Qihong Chen & Lijun Wang & Jin Zhang, 2003. "A new well-posed algorithm to recover implied local volatility," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 451-457.
    13. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
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    18. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    19. Eckhard Platen, 2002. "Benchmark Model with Intensity Based Jumps," Research Paper Series 81, Quantitative Finance Research Centre, University of Technology, Sydney.
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    Citations

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    Cited by:

    1. Ke Du & Eckhard Platen, 2011. "Three-Benchmarked Risk Minimization for Jump Diffusion Markets," Research Paper Series 296, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. David Heath & Eckhard Platen, 2004. "Understanding the Implied Volatility Surface for Options on a Diversified Index," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 55-77, March.
    3. repec:wsi:ijtafx:v:11:y:2008:i:08:n:s0219024908005056 is not listed on IDEAS
    4. Shane M. Miller & Eckhard Platen, 2008. "Analytic Pricing Of Contingent Claims Under The Real-World Measure," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(08), pages 841-867.
    5. Jing Zhao & Hoi Ying Wong, 2012. "A closed-form solution to American options under general diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 725-737, July.
    6. Mathias Barkhagen & Jörgen Blomvall & Eckhard Platen, 2016. "Recovering the real-world density and liquidity premia from option data," Quantitative Finance, Taylor & Francis Journals, vol. 16(7), pages 1147-1164, July.

    More about this item

    Keywords

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

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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