Abramov, Vyacheslav and Klebaner, Fima (2006): Forecasting and testing a nonconstant volatility.

PDF
MPRA_paper_207.pdf Download (864kB)  Preview 
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 BlackScholes 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.
Item Type:  MPRA Paper 

Original Title:  Forecasting and testing a nonconstant volatility 
Language:  English 
Keywords:  Nonconstant volatility; approximating and forecasting volatility; BlackScholes formula; best linear predictor 
Subjects:  G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing; Futures Pricing 
Item ID:  207 
Depositing User:  Vyacheslav Abramov 
Date Deposited:  09. Oct 2006 
Last Modified:  18. Feb 2013 17:18 
References:  \begin{thebibliography}{99} \bibitem{Andersen Bollerslev Christoffersen and Diebold 2005}\textsc{Andersen, T.G., Bollerslev, T., Christoffersen, P.F. and Diebold, F.X.} Volatility forecasting. \emph{NBER working paper 11188} (March, 2005). Accessible at: \texttt{http://www.nber.org/papers/w11188} \bibitem{Anderson Vahid 2005}\textsc{Anderson, H.M. and Vahid, F.} Forecasting the volatility of Australian stock returns: Do common factor help? \emph{Working paper 451} (2005). Australian National University, Canberra. \bibitem{BarndorffNielsen and Shephard 2002} \textsc{BarndorffNielsen, O.E. and Shephard, N.} Estimating quadratic variation using realized variance. \emph{Journal of Applied Econometrics}, 17 (2002), 457477. \bibitem{BarndorffNielsen and Shephard 2002*} \textsc{BarndorffNielsen, O.E. and Shephard, N.} Econometric analysis of realised volatility and its use in estimating stochastic volatility models. \emph{Journal of the Royal Statistical Society}, Ser. B, 64 (2002), 253280. \bibitem{BarndorffNielsen and Shephard 2004} \textsc{BarndorffNielsen, O.E. and Shephard, N.} Power and bipower variation with stochastic volatility and jumps. \emph{Journal of Financial Econometrics}, 2 (2004), 137. \bibitem{Black and Scholes 1973} \textsc{Black, F. and Scholes, M.} The pricing of options and corporate liabilities. \emph{Journal of Political Economics}, 81 (1973), 637659. \bibitem{ITSM2000}\textsc{Brockwell, P.J. and Davis, R.A.} \emph{Introduction to Time Series Analysis and Forecasting}, 2nd edn. Springer, New York, 2002. \bibitem{Chow Khasminskii Liptser 1997}\textsc{Chow, P.L., Khasminskii, R. and Liptser, R.} Tracking of signal and its derivatives in Gaussian white noise. \emph{Stochastic Processes and Their Application}, 69 (1997), 259273. \bibitem{FPSS 2003a} \textsc{Fouque, J.P., Papanicolaou, G., Sircar, R. and Solna, K.} Singular perturbation in option prices. \emph{SIAM Journal on Applied Mathematics}, 63 (2003), 16481665. \bibitem{FPSS 2003b} \textsc{Fouque, J.P., Papanicolaou, G., Sircar, R. and Solna, K.} Multiscale stochastic volatility asymptotics. \emph{SIAM Journal for Multiscale Modelling and Simulation}, 2 (2003), 2242. \bibitem{Ghysels Harvey Renault 1996} \textsc{Ghysels, E., Harvey, A.C. and Renault, E.} Stochastic volatility. In: \emph{Statistical Methods in Finance} C.R.Rao and G.S.Maddala (eds) Amsterdam, NorthHolland, 1996, pp. 119191. \bibitem{Goldentayer Klebaner and Liptser} \textsc{Goldentayer, L., Klebaner, F. and Liptser, R.} Tracking volatility, \emph{Problems of Information Transmission}, {41} (2005), 212229. \bibitem{Harvey Ruiz Shephard 1994}\textsc{Harvey, A.C., Ruiz, E. and Shephard, N.} Multivariate stochastic variance models, \emph{Review of Economic Studies}, 61 (1994), 247264. \bibitem{Hillebrand 2003} \textsc{Hillebrand, E.} Overlaying time scales and persistence estimation in GARCH(1,1) models. Econometrics 0301003 EconWPA (2003). \bibitem{Hillebrand 2005} \textsc{Hillebrand, E.} Overlaying time scales in financial volatility. Econometrics 0501015 EconWPA (2005). \bibitem{Hull White 1987}\textsc{Hull, J.C. and White, A.} The pricing of options on asset with stochastic volatilities. \emph{Journal of Finance}, 42 (1987), 281300. \bibitem{Ibragimov Khasminskii 1980}\textsc{Ibragimov, I.A. and Khasminskii, R.Z.} On nonparametric estimation of regression. \emph{Soviet Mathematical Doklads} 21 (1980), 810814. \bibitem{Ibragimov Khasminskii 1981} \textsc{Ibragimov, I.A. and Khasminskii, R.Z.} \emph{Statistical Estimation. Asymptotic Theory.} Springer, New York, 1981. \bibitem{KhL} \textsc{Khasminskii, R and Liptser, R.} Online estimation of a smooth regression function, \emph{Theory of Probability and its Applications}, 47 (2002), 541550. \bibitem{Kim Shephard Chib 1998}\textsc{Kim, S., Shephard, N. and Chib, S.} Stochastic volatility: likelihood inference and comparison with ARCH models, \emph{Review of Economic Studies}, 65 (1998), 363393. \bibitem{Klebaner Le and Liptser 2005}\textsc{Klebaner, F.C., Le, T. and Liptser, R.} Estimating and predicting volatility by using implied and historical volatilities, submitted. \bibitem{Mercurio Spokoiny 2004}\textsc{Mercurio, D. and Spokoiny, V.} Statistical inference for timeinhomogeneous volatility models. \emph{The Annals of Statistics}, 32 (2004), 577602. \bibitem{Merton 1973} \textsc{Merton, R.C.} Theory of rational option pricing. \emph{Bell Journal of Economics and Management Sciences}, 4 (1973), 141183. \bibitem{Polsehl Spokoiny 2000}\textsc{Polsehl, J. and Spokoiny, V.} Adaptive weights smoothing with applications to image restoration. \emph{Journal of the Royal Statistical Society} 62, Ser. B, (2000), 335354. \bibitem{Polsehl Spokoiny 2002}\textsc{Polsehl, J. and Spokoiny, V.} Local likelihood modelling in adaptive weights smoothing. Preprint 787, WIAS, 2002. \bibitem{Polsehl Spokoiny 2003}\textsc{Polsehl, J. and Spokoiny, V.} Varying coefficients regression modelling by adaptive weights smoothing. Preprint 818, WIAS, 2003. \bibitem{Polsehl Spokoiny 2004}\textsc{Polsehl, J. and Spokoiny, V.} Varying coefficient GARCH versus local constant volatility modelling. Comparison of the predictive power. Preprint 977, WIAS, 2004. \bibitem{Shephard 1996}\textsc{Shephard, N.} Statistical aspects of ARCH and stochastic volatility. In: \emph{Time Series Models in Econometrics, Finance and Other Fields} D.R.Cox, D.V.Hinkley and O.E.BarndorffNielsen (eds) London, Chapman and Hall, 1996, pp. 167. \bibitem{Stein and Stein 1991}\textsc{Stein, E. and Stein, J.C.} Stock price distributions with stochastic volatility: An analytic approach. \emph{Review of Financial Studies}, 4 (1991) 727752. \end{thebibliography} 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/207 