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Understanding the Implied Volatility Surface for Options on a Diversified Index

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Author Info
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

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File URL: http://hdl.handle.net/10.1007/s10690-005-4249-4
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Publisher Info
Article provided by Springer in its journal Asia-Pacific Financial Markets.

Volume (Year): 11 (2004)
Issue (Month): 1 (March)
Pages: 55-77
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Handle: RePEc:kap:apfinm:v:11:y:2004:i:1:p:55-77

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Web page: http://springerlink.metapress.com/link.asp?id=102851

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Related research
Keywords: index derivatives; minimal market model; random scaling; growth optimal portfolio; fair pricing; binary options;

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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. Wolfgang Breymann & Leah Kelly & Eckhard Platen, 2004. "Intraday Empirical Analysis and Modeling of Diversified World Stock Indices," Research Paper Series 125, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
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  2. 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. [Downloadable!]
  3. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, September.
  4. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford. [Downloadable!]
  5. 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-. [Downloadable!]
  6. David Heath & S. 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.
  7. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July. [Downloadable!] (restricted)
  8. 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!]
  9. Eckhard Platen, 2001. "A Minimal Financial Market Model," Research Paper Series 48, Quantitative Finance Research Centre, University of Technology, Sydney.
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  10. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  11. 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 1460-75, November. [Downloadable!] (restricted)
  12. 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. [Downloadable!]
  13. David Heath & Eckhard Platen, 2004. "Local Volatility Function Models under a Benchmark Approach," Research Paper Series 124, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
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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. 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. [Downloadable!] (restricted)
    Other versions:
  2. Eckhard Platen, 2004. "A Benchmark Approach to Finance," Research Paper Series 138, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
    Other versions:
  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. [Downloadable!]
    Other versions:
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