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Recovering the probability density function of asset prices using garch as diffusion approximations

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  • Fornari, Fabio
  • Mele, Antonio

Abstract

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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  • Fornari, Fabio & Mele, Antonio, 2001. "Recovering the probability density function of asset prices using garch as diffusion approximations," Journal of Empirical Finance, Elsevier, vol. 8(1), pages 83-110, March.
  • Handle: RePEc:eee:empfin:v:8:y:2001:i:1:p:83-110
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    Cited by:

    1. Conrad, Christian & Karanasos, Menelaos & Zeng, Ning, 2011. "Multivariate fractionally integrated APARCH modeling of stock market volatility: A multi-country study," Journal of Empirical Finance, Elsevier, vol. 18(1), pages 147-159, January.
    2. repec:zbw:rwirep:0018 is not listed on IDEAS
    3. Nicolas Langren'e & Geoffrey Lee & Zili Zhu, 2015. "Switching to non-affine stochastic volatility: A closed-form expansion for the Inverse Gamma model," Papers 1507.02847, arXiv.org, revised Mar 2016.
    4. Antonio Mele, 2003. "Fundamental Properties of Bond Prices in Models of the Short-Term Rate," Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 679-716, July.
    5. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching To Nonaffine Stochastic Volatility: A Closed-Form Expansion For The Inverse Gamma Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-37, August.
    6. Menelaos Karanasos & Stefanie Schurer, 2008. "Is the Relationship between Inflation and Its Uncertainty Linear?," German Economic Review, Verein für Socialpolitik, vol. 9, pages 265-286, August.
    7. Luca Dedola & Eugenio Gaiotti & Luca Silipo, 2001. "Money demand in the euro area: do national differences matter?," Temi di discussione (Economic working papers) 405, Bank of Italy, Economic Research and International Relations Area.
    8. Fabio Fornari, 2002. "The size of the equity premium," Temi di discussione (Economic working papers) 447, Bank of Italy, Economic Research and International Relations Area.
    9. Fornari, Fabio, 2010. "Assessing the compensation for volatility risk implicit in interest rate derivatives," Journal of Empirical Finance, Elsevier, vol. 17(4), pages 722-743, September.
    10. Fornari, Fabio, 2008. "Assessing the compensation for volatility risk implicit in interest rate derivatives," Working Paper Series 859, European Central Bank.
    11. Menelaos Karanasosa & Stefanie Schurer, 2007. "Is the Relationship Between Inflation and its Uncertainty Linear?," Ruhr Economic Papers 0018, Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Ruhr-Universität Bochum, Universität Dortmund, Universität Duisburg-Essen.
    12. Karanasos, Menelaos & Kim, Jinki, 2006. "A re-examination of the asymmetric power ARCH model," Journal of Empirical Finance, Elsevier, vol. 13(1), pages 113-128, January.
    13. Fornari, Fabio & Mele, Antonio, 2006. "Approximating volatility diffusions with CEV-ARCH models," Journal of Economic Dynamics and Control, Elsevier, vol. 30(6), pages 931-966, June.

    More about this item

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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