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Option Pricing under Incompleteness and Stochastic Volatility

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Author Info
N. Hofmann
E. Platen
M. Schweizer

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Abstract

We consider a very general diffusion model for asset prices which allows the description of stochastic and past-dependent volatilities. Since this model typically yields an incomplete market, we show that for the purpose of pricing options, a small investor should use the minimal equivalent martingale measure associated to the underlying stock price process. Then we present stochastic numerical methods permitting the explicit computation of option prices and hedging strategies, and we illustrate our approach by specific examples.

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Publisher Info
Paper provided by University of Bonn, Germany in its series Discussion Paper Serie B with number 209.

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Length: 49 pages
Date of creation: Feb 1992
Date of revision:
Handle: RePEc:bon:bonsfb:209

Contact details of provider:
Postal: Bonn Graduate School of Economics, University of Bonn, Adenauerallee 24 - 26, 53113 Bonn, Germany
Fax: +49 228 73 9221
Web page: http://www.bgse.uni-bonn.de/index.php?id=517

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Related research
Keywords: ption pricing; stochastic volatility; incomplete markets; equivalent martingale measures; stochastic numerical methods;

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Cited by:
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  1. M. E. Mancino & S. Ogawa & S. Sanfelici, 2004. "A numerical study of the smile effect in implied volatilities induced by a nonlinear feedback model," Economics Department Working Papers 2004-ME01, Department of Economics, Parma University (Italy). [Downloadable!]
  2. Frey, Rüdiger, 1997. "Derivative Asset Analysis in Models with Level-Dependent and Stochastic Volatility," Discussion Paper Serie B 401, University of Bonn, Germany. [Downloadable!]
  3. Ashkan Nikeghbali & Eckhard Platen, 2008. "On honest times in financial modeling," Quantitative Finance Papers 0808.2892, arXiv.org. [Downloadable!]
    Other versions:
  4. E. Platen, . "Risk Premia and Financial Modelling Without Measure Transformation," Sonderforschungsbereich 373 2000-92, Humboldt Universitaet Berlin.
    Other versions:
  5. Hans Buhlmann & Eckhard Platen, 2002. "A Discrete Time Benchmark Approach for Finance and Insurance," Research Paper Series 74, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  6. Gatfaoui Hayette & Chauveau Thierry, 2004. "Pricing and Hedging Options in Incomplete Markets: Idiosyncratic Risk, Systematic Risk and Stochastic Volatility," Finance 0404002, EconWPA. [Downloadable!]
    Other versions:
  7. Antonio Mele & Fabio Fornari, 1999. "ARCH Models and Option Pricing: the Continuous-Time Connection," Computing in Economics and Finance 1999 113, Society for Computational Economics. [Downloadable!]
    Other versions:
  8. Martin Schweizer, 1994. "On the Minimal Martingale Measure and the Foellmer- Schweizer Decomposition," Discussion Paper Serie B 284, University of Bonn, Germany. [Downloadable!]
  9. Gatfaoui Hayette, 2004. "Idiosyncratic Risk, Systematic Risk and Stochastic Volatility: An Implementation of Merton’s Credit Risk Valuation," Finance 0404004, EconWPA. [Downloadable!]
  10. T. Kleinow, . "Testing the Diffusion Coefficient," Sonderforschungsbereich 373 2002-38, Humboldt Universitaet Berlin.
  11. Umberto Cherubini, 1997. "Fuzzy measures and asset prices: accounting for information ambiguity," Applied Mathematical Finance, Taylor and Francis Journals, vol. 4(3), pages 135-149, September. [Downloadable!] (restricted)
  12. Eckhard Platen, 2003. "Pricing and Hedging for Incomplete Jump Diffusion Benchmark Models," Research Paper Series 110, Quantitative Finance Research Centre, University of Technology, Sydney. [Downloadable!]
  13. Vicky Henderson & David Hobson, 2001. "Passport options with stochastic volatility," Applied Mathematical Finance, Taylor and Francis Journals, vol. 8(2), pages 97-118, May. [Downloadable!] (restricted)
  14. 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!]
  15. C. Hafner & H. Herwartz, . "Option Pricing under Linear Autoregressive Dynamics, Heteroskedasticity, and Conditional Leptokurtosis," Sonderforschungsbereich 373 1999-58, Humboldt Universitaet Berlin.
    Other versions:
  16. Jan Bergenthum & Ludger Rüschendorf, 2006. "Comparison of Option Prices in Semimartingale Models," Finance and Stochastics, Springer, vol. 10(2), pages 222-249, April. [Downloadable!] (restricted)
  17. K. Ronnie Sircar, George C. Papanicolaou, 1999. "Stochastic volatility, smile & asymptotics," Applied Mathematical Finance, Taylor and Francis Journals, vol. 6(2), pages 107-145, June. [Downloadable!] (restricted)
  18. Leisen, Dietmar P.J., . "Stock Evolution under Stochastic Volatility: A Discrete Approach," Discussion Paper Serie B 407, University of Bonn, Germany, revised May 1999. [Downloadable!]
  19. Eckhard Platen & Wolfgang Runggaldier, 2004. "A Benchmark Approach to Filtering in Finance," Asia-Pacific Financial Markets, Springer, vol. 11(1), pages 79-105, March. [Downloadable!] (restricted)
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  20. Hayette Gatfaoui, 2004. "Idiosyncratic Risk, Systematic Risk and Stochastic Volatility: An Implementation of Merton's Credit Risk Valuation," Research Paper Series 123, Quantitative Finance Research Centre, University of Technology, Sydney.
  21. Nigel Clarke, Kevin Parrott, 1999. "Multigrid for American option pricing with stochastic volatility," Applied Mathematical Finance, Taylor and Francis Journals, vol. 6(3), pages 177-195, September. [Downloadable!] (restricted)
  22. Eckhard Platen & Jason West, 2004. "A Fair Pricing Approach to Weather Derivatives," Asia-Pacific Financial Markets, Springer, vol. 11(1), pages 23-53, March. [Downloadable!] (restricted)
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