Forecasting and testing a non-constant volatility

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Forecasting and testing a non-constant volatility

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Abramov, Vyacheslav
Klebaner, Fima

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Vyacheslav M. Abramov

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.

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 207.

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Date of creation: 06 Jun 2006
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Handle: RePEc:pra:mprapa:207

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Find related papers by JEL classification:
G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

This paper has been announced in the following NEP Reports:

NEP-ALL-2006-10-21 (All new papers)
NEP-ECM-2006-10-21 (Econometrics)
NEP-ETS-2006-10-21 (Econometric Time Series)
NEP-FIN-2006-10-21 (Finance)
NEP-FMK-2006-10-21 (Financial Markets)
NEP-FOR-2006-10-21 (Forecasting)

References listed on IDEAS
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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. [Downloadable!]
Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford. [Downloadable!]
Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," NBER Working Papers 11188, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
Other versions:
- Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania. [Downloadable!]
- Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," CFS Working Paper Series 2005/08, Center for Financial Studies. [Downloadable!]
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. [Downloadable!] (restricted)
Other versions:
- Heather Anderson & Fashid Vahid, 2005. "Forecasting the Volatility of Australian Stock Returns: Do Common Factors Help?," ANUCBE School of Economics Working Papers 2005-451, Australian National University, College of Business and Economics, School of Economics. [Downloadable!]
Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Oxford University Press, vol. 2(1), pages 1-37. [Downloadable!] (restricted)

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