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Option Pricing Using Variance Gamma Markov Chains

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Abstract

This paper proposes a Markov Chain between homogeneous Lévy processesas a candidate class of processes for the statistical and risk neutral dynamicsof financial asset prices. The method is illustrated using the variance gammaprocess. Closed forms for the characteristic function are developed and thisrenders feasible, series and option prices respectively. It is observed inthe statistical and risk neutral process is fit to data on time period of4 to 6 months in a state while this reduces to month for indices. Risk neutrallythere is generally a low probability of a move to a state with higher moments.In some cases this is reversed.

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References

  • Andersen, T. G. (1994). “Stochastic Autoregressive Volatility: A Framework for Volatility Modeling,” Mathematical Finance 4, 75–102.

    Google Scholar 

  • Andé, T., and H. Geman. (2000). “Order Flow, Transaction Clock and Normality of Asset Returns,” Journal of Finance 55, 2259–2284.

    Google Scholar 

  • Bakshi, G., C. Cao, and Z. Chen. (1997). “Empirical Performance of Altrernative Option Pricing Models,” Journal of Finance 52, 2003–2049.

    Google Scholar 

  • Barndorff-Nielsen, O. (1998). “Processes of Normal Inverse Gaussian Type,” Finance and Stochastics 2, 41–68.

    Google Scholar 

  • Barndorff-Nielsen, O., and N. Shephard. (1999). “Non-Gaussian OU Based Models and Some of Their Uses in Financial Economics,” Working Paper No. 37, Center for Analytical Finance, University of Aarhus.

  • Bates, D. (1996). “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutschemark Options,” Review of Financial Studies 9, 69–108.

    Google Scholar 

  • Black, F., and M. Scholes. (1973). “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81, 637–654.

    Google Scholar 

  • Carr, P., H. Geman, D. Madan, and M. Yor. (2000). “The Fine Structure of Asset Returns: An Empirical Investigation,” forthcoming in the Journal of Business.

  • Carr, P., H. Geman, D. Madan, and M. Yor. (2001). “Stochastic Volatility for Lévy Processes,” Working Paper, University of Maryland.

  • Carr, P., and D. B. Madan. (1999). “Option Valuation Using the Fast Fourier Transform,” Journal of Computational Finance 1, 61–73.

    Google Scholar 

  • Clark, P. K. (1973). “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices,” Econometrica 41, 135–156.

    Google Scholar 

  • Derman, E., and I. Kani. (1998). “Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility,” International Journal of Theoretical and Applied Finance 3, 7–22.

    Google Scholar 

  • Dupire, B. (1994). “Pricing With a Smile,” Risk 7,1, 18–20.

    Google Scholar 

  • Duan, J, I. Popova, and P. Ritchken. (1999). “Option Pricing under Regime Switching,” Working Paper, Case Western Reserve University.

  • Eberlein, E., U. Keller, and K. Prause. (1998). “New Insights into Smile, Mispricing and Value at Risk: The Hyperbolic Model,” Journal of Business 71, 371–406.

    Google Scholar 

  • Elliott, Robert J., Lakhdar Aggoun, and John B. Moore. (1995). Hidden Markov Models: Estimation and Control (Applications of Mathematics, Vol. 29). Berlin: Springer-Verlag.

    Google Scholar 

  • Engle, R. F. (1982). “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 61, 929–952.

    Google Scholar 

  • Geman, H., D. Madan, and M. Yor. (2001). “Time Changes for Lévy Processes,” Mathematical Finance 11, 79–96.

    Google Scholar 

  • Heston, S. L. (1993). “A Closed Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies 6, 327–343.

    Google Scholar 

  • Madan, D. B., and E. Seneta. (1990). “The Variance Gamma (VG) Model for Share Market Returns,” Journal of Business 63,4, 511–524.

    Google Scholar 

  • Madan, D. B., P. P. Carr, and E. C. Chang. (1998). “The Variance Gamma Process and Option Pricing,” European Finance Review 2, 79–105.

    Google Scholar 

  • Merton, R. C. (1973). “Theory of Rational Option Pricing,” Bell Journal of Economics and Management 4, 141–183.

    Google Scholar 

  • Merton, R. C. (1976). “Option Pricing when Underlying Stock Returns are Discontinuous,” Journal of Financial Economics 3, 125–144

    Google Scholar 

  • Nelson, D. (1991). “Conditional Heteroskedasticity in Asset Returns: A NewApproach,” Econometrica 59, 347–370.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742

    Mikhail Konikov

  2. Robert H. Smith School of Business, University of Maryland, College Park, MD, 20742

    Dilip B. Madan

Authors
  1. Mikhail Konikov
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  2. Dilip B. Madan
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Konikov, M., Madan, D.B. Option Pricing Using Variance Gamma Markov Chains. Review of Derivatives Research 5, 81–115 (2002). https://doi.org/10.1023/A:1013816400834

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