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Pricing of Index Options Under a Minimal Market Model with Lognormal Scaling

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Abstract

This paper describes a two-factor model for a diversified market index using the growth optimal portfolio with a stochastic and possibly correlated intrinsic time scale. The index is modeled using a time transformed squared Bessel process of dimension four with a lognormal scaling factor for the time transformation. A consistent pricing and hedging framework is established by using the benchmark approach. here the numeraire is taken to be the growth optimal portfolio. Benchmarked traded prices appear as conditional expectations of future benchmarked prices under the real world probability measure. The proposed minimal market model with lognormal scaling produces the type of implied volatility term structures for European call nd put options typically observed in real markets. In addition, the prices of binary options and their deviations from corresponding Black-Scholes prices are examined.

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File URL: http://www.business.uts.edu.au/qfrc/research/research_papers/rp101.pdf
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Bibliographic Info

Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 101.

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Date of creation: 01 Jun 2003
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Handle: RePEc:uts:rpaper:101

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

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  1. Eckhard Platen, 2001. "A Minimal Financial Market Model," Research Paper Series 48, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. 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.
  3. Sanjiv R. Das & Rangarajan K. Sundaram, 1998. "Of Smiles and Smirks: A Term-Structure Perspective," New York University, Leonard N. Stern School Finance Department Working Paper Seires 98-024, New York University, Leonard N. Stern School of Business-.
  4. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
  5. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, July.
  6. Eckhard Platen, 2003. "Diversified Portfolios in a Benchmark Framework," Research Paper Series 87, Quantitative Finance Research Centre, University of Technology, Sydney.
  7. Schönbucher, Philpp J., . "A Market Model for Stochastic Implied Volatility," Discussion Paper Serie B 453, University of Bonn, Germany, revised May 1999.
  8. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
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Cited by:
  1. 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.
  2. Eckhard Platen, 2004. "Diversified Portfolios with Jumps in a Benchmark Framework," Research Paper Series 129, Quantitative Finance Research Centre, University of Technology, Sydney.
  3. Eckhard Platen, 2003. "A Benchmark Framework for Risk Management," Research Paper Series 113, Quantitative Finance Research Centre, University of Technology, Sydney.
  4. Eckhard Platen, 2003. "Pricing and Hedging for Incomplete Jump Diffusion Benchmark Models," Research Paper Series 110, Quantitative Finance Research Centre, University of Technology, Sydney.

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