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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.
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