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Forecasting and testing a non-constant volatility

Author

Listed:
  • Abramov, Vyacheslav
  • Klebaner, Fima

Abstract

In this paper we study volatility functions. Our main assumption is that the volatility is deterministic or stochastic but driven by a Brownian motion independent of the stock. We propose a forecasting method and check the consistency with option pricing theory. To estimate the unknown volatility function we use the approach of \cite{Goldentayer Klebaner and Liptser} based on filters for estimation of an unknown function from its noisy observations. One of the main assumptions is that the volatility is a continuous function, with derivative satisfying some smoothness conditions. The two forecasting methods correspond to the the first and second order filters, the first order filter tracks the unknown function and the second order tracks the function and it derivative. Therefore the quality of forecasting depends on the type of the volatility function: if oscillations of volatility around its average are frequent, then the first order filter seems to be appropriate, otherwise the second order filter is better. Further, in deterministic volatility models the price of options is given by the Black-Scholes formula with averaged future volatility \cite{Hull White 1987}, \cite{Stein and Stein 1991}. This enables us to compare the implied volatility with the averaged estimated historical volatility. This comparison is done for five companies and shows that the implied volatility and the historical volatilities are not statistically related.

Suggested Citation

  • Abramov, Vyacheslav & Klebaner, Fima, 2006. "Forecasting and testing a non-constant volatility," MPRA Paper 207, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:207
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    File URL: https://mpra.ub.uni-muenchen.de/207/1/MPRA_paper_207.pdf
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    References listed on IDEAS

    as
    1. Ole E. Barndorff-Nielsen & Neil Shephard, 2002. "Estimating quadratic variation using realized variance," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 457-477.
    2. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2005. "Volatility forecasting," CFS Working Paper Series 2005/08, Center for Financial Studies (CFS).
    3. Anderson, Heather M. & Vahid, Farshid, 2007. "Forecasting the Volatility of Australian Stock Returns: Do Common Factors Help?," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 76-90, January.
    4. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 2(1), pages 1-37.
    5. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    Non-constant volatility; approximating and forecasting volatility; Black-Scholes formula; best linear predictor;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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