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Understanding the Implied Volatility Surface for Options on a Diversified Index

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  • David Heath
  • Eckhard Platen

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Abstract

This paper describes a two-factor model for a diversified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The proposed model, which includes a random initial condition for market activity, generates implied volatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for the proposed model and their Black-Scholes counterparts. Copyright Springer Science + Business Media, Inc. 2004

Suggested Citation

  • 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.
  • Handle: RePEc:kap:apfinm:v:11:y:2004:i:1:p:55-77
    DOI: 10.1007/s10690-005-4249-4
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    File URL: http://hdl.handle.net/10.1007/s10690-005-4249-4
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    References listed on IDEAS

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    1. Wolfgang Breymann & Leah Kelly & Eckhard Platen, 2005. "Intraday Empirical Analysis and Modeling of Diversified World Stock Indices," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 12(1), pages 1-28, March.
    2. David Heath & Simon Hurst & Eckhard Platen, 1999. "Modelling the Stochastic Dynamics of Volatility for Equity Indices," Research Paper Series 7, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    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. Das, Sanjiv Ranjan & Sundaram, Rangarajan K., 1999. "Of Smiles and Smirks: A Term Structure Perspective," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(02), pages 211-239, June.
    6. 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.
    7. 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.
    8. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, June.
    9. Joshua Rosenberg, 1999. "Implied Volatility Functions: A Reprise," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-027, New York University, Leonard N. Stern School of Business-.
    10. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July.
    11. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    12. Heynen, Ronald & Kemna, Angelien & Vorst, Ton, 1994. "Analysis of the Term Structure of Implied Volatilities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 29(01), pages 31-56, March.
    13. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
    14. Franks, Julian R & Schwartz, Eduardo S, 1991. "The Stochastic Behaviour of Market Variance Implied in the Prices of Index Options," Economic Journal, Royal Economic Society, vol. 101(409), pages 1460-1475, November.
    15. Eric Renault & Nizar Touzi, 1996. "Option Hedging And Implied Volatilities In A Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 279-302.
    16. repec:wsi:ijtafx:v:07:y:2004:i:04:n:s0219024904002499 is not listed on IDEAS
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    Citations

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    Cited by:

    1. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151.
    2. repec:wsi:ijtafx:v:08:y:2005:i:08:n:s0219024905003360 is not listed on IDEAS
    3. David Heath & Eckhard Platen, 2005. "Currency Derivatives under a Minimal Market Model with Random Scaling," Research Paper Series 154, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Shane Miller & Eckhard Platen, 2004. "A Two-Factor Model for Low Interest Rate Regimes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 107-133, March.

    More about this item

    Keywords

    index derivatives; minimal market model; random scaling; growth optimal portfolio; fair pricing; binary options;

    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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