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Recovering the Probability Density Function of Asset Prices using Garch as Diffusion Approximations

  • F. Fornari
  • A. Mele

This paper uses Garch models to estimate the objective and risk-neutral density functions of financial asset prices and, by comparing their shapes, recover detailed information on economic agents' attitudes toward risk. It differs from recent papers investigating analogous issues because it uses Nelson's (1990) result that Garch schemes are approximations of the kind of differential equations typically employed in finance to describe the evolution of asset prices. This feature of Garch schemes usually has been overshadowed by their well-known role as simple econometric tools providing reliable estimates of unobserved conditional variances. We show instead that the diffusion approximation property of Garch gives good results and can be extended to situations with i) non-standard distributions for the innovations of a conditional mean equation of asset price changes and ii) volatility concepts different from the variance. The objective PDF of the asset price is recovered from the estimation of a nonlinear Garch fitted to the historical path of the asset price. The risk-neutral PDF is extracted from crosssections of bond option prices, after introducing a volatility risk premium function. The direct comparison of the shapes of the two PDFS reveals the price attached by economic agents to the different states of nature. Applications are carried out with regard to the futures written on the Italian 10-year bond.

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Paper provided by THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise in its series THEMA Working Papers with number 2000-12.

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Date of creation: 2000
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Handle: RePEc:ema:worpap:2000-12
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  1. Yacine Ait-Sahalia & Andrew W. Lo, 2000. "Nonparametric Risk Management and Implied Risk Aversion," NBER Working Papers 6130, National Bureau of Economic Research, Inc.
  2. BROZE, Laurence & SCAILLET, Olivier & ZAKOIAN, Jean-Michel, 1995. "Quasi Indirect Inference for Diffusion Processes," CORE Discussion Papers 1995005, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Joshua Rosenberg & Robert F. Engle, 2000. "Empirical Pricing Kernels," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-014, New York University, Leonard N. Stern School of Business-.
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  7. Corradi, Valentina, 2000. "Reconsidering the continuous time limit of the GARCH(1, 1) process," Journal of Econometrics, Elsevier, vol. 96(1), pages 145-153, May.
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  10. Nelson, Daniel B., 1992. "Filtering and forecasting with misspecified ARCH models I : Getting the right variance with the wrong model," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 61-90.
  11. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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  16. Ait-Sahalia, Yacine, 1996. "Testing Continuous-Time Models of the Spot Interest Rate," Review of Financial Studies, Society for Financial Studies, vol. 9(2), pages 385-426.
  17. Werker, B.J.M. & Drost, F.C., 1996. "Closing the GARCH gap : Continuous time GARCH modeling," Other publications TiSEM c3d29817-403a-4ad1-9295-8, Tilburg University, School of Economics and Management.
  18. Drost, F.C. & Nijman, T.E., 1993. "Temporal aggregation of GARCH processes," Other publications TiSEM 0642fb61-c7f4-4281-b484-4, Tilburg University, School of Economics and Management.
  19. Yacine Ait-Sahalia, 1995. "Nonparametric Pricing of Interest Rate Derivative Securities," NBER Working Papers 5345, National Bureau of Economic Research, Inc.
  20. Fabio Fornari & Antonio Mele, 1997. "Weak convergence and distributional assumptions for a general class of nonliner arch models," Econometric Reviews, Taylor & Francis Journals, vol. 16(2), pages 205-227.
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  23. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
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  25. Ait-Sahalia, Yacine & Wang, Yubo & Yared, Francis, 2001. "Do option markets correctly price the probabilities of movement of the underlying asset?," Journal of Econometrics, Elsevier, vol. 102(1), pages 67-110, May.
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  29. repec:ner:tilbur:urn:nbn:nl:ui:12-72561 is not listed on IDEAS
  30. repec:ner:tilbur:urn:nbn:nl:ui:12-153273 is not listed on IDEAS
  31. Chernov, Mikhail & Ghysels, Eric, 2000. "A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation," Journal of Financial Economics, Elsevier, vol. 56(3), pages 407-458, June.
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  33. F. Fornari & A. Mele, 2000. "An Equilibrium Model of the Term Structure with Stochastic Volatility," THEMA Working Papers 2000-13, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
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