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							               Option Pricing in Garch Models


                          Fabio Bellini
                        Lorenzo Mercuri
Dip. Metodi Quantitativi per le Scienze Economiche ed Aziendali
                Università di Milano-Bicocca.



Abstract: The literature on option pricing in Garch models is
characterized by the risk premium specification, in fact this
parameter plays an important role in the choice of an equivalent
martingale measure.
From a theoretical point of view, following the Duan ’s seminal
work, this approach can be justified by equilibrium arguments
based on the maximization of the expected utility function of a
representative agent.
Two cases are well known in literature :
    1. The risk premium is linear in volatility (Duan 1995).
    2. The risk premium is linear in variance (Heston-Nandi
        2000).
Many empirical works show that the difference between the
conditional mean of returns and the risk-free rate under real
measure is more complex than above models (see for example
Engle, Litien and Robins (1987) and recently Bollerslev,
Gibbons and Zhou (2005)). Therefore, using only no-arbitrage
arguments, Christoffersen, Elkami and Jacobs (2005) propose a
more general change of measure that encompasses the
aforementioned formulations.
Moreover in a recent work Siu, Tong and Yang generalise this
result choosing an equivalent martingale measure by means of a
conditional Esscher Transform.
In our work we implement this approach and compare the
goodness of calibration of option prices with the models of
Duan and Heston-Nandi.
References
  1. Bollerslev, T. (1986),       Generalized Autoregresive
      Conditional Heteroskedasticity, Journal of Econometrics,
      31, 307-327.
  2. Bollerslev, T., M. Gibbons, and H. Zhou (2005),
      Dynamic Estimation of volatility Risk Premia and
      Investor Risk Aversion from Option-Implied and
      Realized Volatilities, Manuscript, Duke University.
  3. Bühlmann, H., F. Delbaen, P. Embrechts and A. N.
      Shiryaev (1996), No-Arbitrage, Change of Measure and
      Conditional Esscher Transforms, CWI quarterly 9(4),
      291-317.
  4. Christoffersen, P. And K. Jacobs (2004), Which Garch
      Model for Option Pricing?, Management Science, 50,
      1204-1221.
  5. Christoffersen, P., R. Elkamhi And K. Jacobs (2005),
      No-Arbitrage Valuation of Contingent Claims in
      Discrete Time, Manuscript, McGill University.
  6. Duan, J. - C. (1995), The Garch Option pricing Model,
      Mathematical Finance, 5, 13-32.
  7. Engle,       R.(1982),    Autoregressive      Conditional
      Heteroskedasticity with Estimates of the Variance of
      U.K. Inflation, Econometrica, 50, 987-1008.
  8. Gerber, H. U., E. S. W. Shiu (1994), Option Pricing by
      Esscher Transforms, Transaction of the Society of
      Actuaries, 46, 99-191.
  9. Shiryaev, A. N. (1999), Essential Stochastic Finance:
      Facts, Models, Theory, Singapore: World Scientific.
  10. Siu, T. K., H. Tong and H. Yang (2005), On Pricing
      Derivatives under Garch Models: A Dymnamic Gerber-
      Shiu ’s Approach, Manuscript.