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NONPARAMETRIC ESTIMATION OF THE DIFFUSION COEFFICIENT OF STOCHASTIC VOLATILITY MODELS

Published online by Cambridge University Press:  14 May 2008

Roberto Renò
Roberto Renò*
Affiliation:
Università di Siena
*
Address correspondence to Roberto Renò, Dipartimento di Economia Politica, Piazza S. Francesco 7, 53100 Siena, Italy; e-mail: reno@unisi.it
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Abstract

In this paper, new fully nonparametric estimators of the diffusion coefficient of continuous time models are introduced. The estimators are based on Fourier analysis of the state variable trajectory observed and on the estimation of quadratic variation between observations by means of realized volatility. The estimators proposed are shown to be consistent and asymptotically normally distributed. Moreover, the Fourier estimator can be iterated to get a fully nonparametric estimate of the diffusion coefficient in a bivariate model in which one state variable is the volatility of the other. The estimators are shown to be unbiased in small samples using Monte Carlo simulations and are used to estimate univariate and bivariate models for interest rates.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 5 , October 2008 , pp. 1174 - 1206
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Aït-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527–560.CrossRefGoogle Scholar
Aït-Sahalia, Y., Mykland, P., & Zhang, L. (2005) How often to sample a continuous-time process in the presence of microstructure noise. Review of Financial Studies 18, 351–416.CrossRefGoogle Scholar
Andersen, T., Benzoni, L., & Lund, J. (2004) Stochastic Volatility, Mean Drift and Jumps in the Short-Term Interest Rate. Working paper, Northwestern University.Google Scholar
Andersen, T., Bollerslev, T., & Diebold, F.X. (2003) Parametric and Nonparametric Volatility Measurement. Working paper. Forthcoming in L.P. Hansen & Y. Aït-Sahalia (eds.), Handbook of Financial Econometrics. North-Holland.Google Scholar
Andersen, T., Bollerslev, T., Diebold, F., & Labys, P. (2003) Modeling and forecasting realized volatility. Econometrica 71, 579–625.CrossRefGoogle Scholar
Bandi, F. (2002) Short-term interest rate dynamics: A spatial approach. Journal of Financial Economics 65, 73–110.CrossRefGoogle Scholar
Bandi, F. & Moloche, G. (2004) On the Functional Estimation of Multivariate Diffusion Processes. Working paper, University of Chicago and MIT.Google Scholar
Bandi, F. & Phillips, P. (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–283.CrossRefGoogle Scholar
Bandi, F. & Russell, J. (2006) Separating microstructure noise from volatility. Journal of Financial Economics 79, 655–692.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2002) Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B 64, 253–280.CrossRefGoogle Scholar
Barucci, E. & Renò, R. (2002a) On measuring volatility and the GARCH forecasting performance. Journal of International Financial Markets, Institutions and Money 12, 183–200.CrossRefGoogle Scholar
Barucci, E. & Renò, R. (2002b) On measuring volatility of diffusion processes with high frequency data. Economics Letters 74, 371–378.CrossRefGoogle Scholar
Chan, K., Karolyi, A., Longstaff, F., & Sanders, A. (1992) An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227.Google Scholar
Chapman, D., Long, J., & Pearson, N. (1999) Using proxies for the short rate: When are three months like an instant? Review of Financial Studies 12, 763–806.CrossRefGoogle Scholar
Cox, J., Ingersoll, J., & Ross, S. (1985) A theory of the term structure of interest rates. Econometrica 53, 385–406.CrossRefGoogle Scholar
Duffee, G. (1996) Idiosyncratic variation of Treasury bill yield spread. Journal of Finance 51, 527–552.CrossRefGoogle Scholar
Fan, J. (2005) A selective overview of nonparametric methods in finance. Statistical Science 20, 317–337.Google Scholar
Fan, J. & Zhang, C. (2003) A re-examination of Stanton's diffusion estimations with applications to financial model validation. Journal of the American Statistical Association 98, 118–134.CrossRefGoogle Scholar
Florens-Zmirou, D. (1993) On estimating the diffusion coefficient from discrete observations. Journal of Applied Probability 30, 790–804.CrossRefGoogle Scholar
Föllmer, H. (1981) Calcul d'Itô sans probabilités. Seminaire de Probabilites Strasbourg XV, Lecture Notes in Mathematics 850, pp. 143–150. Springer.Google Scholar
Hansen, P. & Lunde, A. (2006) Consistent ranking of volatility models. Journal of Econometrics 131, 97–121.CrossRefGoogle Scholar
Hansen, P., Lunde, A., & Nason, J.M. (2003) Choosing the best volatility models: The model confidence set approach. Oxford Bulletin of Economics and Statistics 65, 939–961.CrossRefGoogle Scholar
Härdle, W. (1990) Applied Nonparametric Regression. Cambridge University Press.CrossRefGoogle Scholar
Hoffmann, M. (1999) L p estimation of the diffusion coefficient. Bernoulli 5, 447–481.CrossRefGoogle Scholar
Hull, J. & White, A. (1987) The pricing of options on assets with stochastic volatility. Journal of Finance 42, 281–300.CrossRefGoogle Scholar
Jacod, J. (1999) Non-parametric kernel estimation of the coefficient of a diffusion. Scandinavian Journal of Statistics 27, 83–96.CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag.CrossRefGoogle Scholar
Jeffrey, A., Kristensen, D., Linton, O., Nguyen, T., & Phillips, P.C.B. (2004) Nonparametric estimation of a multifactor Heath-Jarrow-Morton model: An integrated approach. Journal of Financial Econometrics 2, 251–289.CrossRefGoogle Scholar
Jiang, G. (1998) Nonparametric modeling of U.S. interest rate term structure dynamics and implications on the prices of derivative securities. Journal of Financial and Quantitative Analysts 33, 465–497.CrossRefGoogle Scholar
Jiang, G. & Knight, J. (1997) A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model. Econometric Theory 13, 615–645.CrossRefGoogle Scholar
Karatzas, I. & Shreve, E. (1988) Brownian Motion and Stochastic Calculus. Springer-Verlag.CrossRefGoogle Scholar
Knight, F.B. (1970) A reduction of continuous square integrable martingale to Brownian motion. In Dinges, H. (ed.), Lecture Notes in Mathematics, Volume 190, pp. 19–31. Springer-Verlag.Google Scholar
Malliavin, P. & Mancino, M. (2002) Fourier series method for measurement of multivariate volatilities. Finance and Stochastics 6, 49–61.CrossRefGoogle Scholar
Malliavin, P. & Thalmaier, A. (2005) Stochastic Calculus of Variations in Mathematical Finance. Springer Finance series. Springer.Google Scholar
Mancini, C. & Renò, R. (2006) Threshold Estimation of Jump-Diffusion Models and Interest Rate Modeling. Manuscript, University of Florence and University of Siena.Google Scholar
Meddahi, N. (2002) A theoretical comparison between integrated and realized volatility. Journal of Applied Econometrics 17, 479–508.CrossRefGoogle Scholar
Nielsen, M.O. & Frederiksen, P.H. (2008) Finite sample accuracy and choice of sampling frequency in integrated volatility estimation. Journal of Empirical Finance 15, 265–286.CrossRefGoogle Scholar
Priestley, M. (1979) Spectral Time Series Analysis. Wiley.Google Scholar
Renò, R. (2006) Nonparametric estimation of stochastic volatility models. Economics Letters 90, 390–395.CrossRefGoogle Scholar
Renò, R., Roma, A., & Schaefer, S. (2006) A comparison of alternative nonparametric estimators of the diffusion coefficient. Economic Notes 35, 227–252.CrossRefGoogle Scholar
Revuz, D. & Yor, M. (1998) Continuous Martingales and Brownian Motion. Springer-Verlag.Google Scholar
Scott, D. (1992) Multivariate Density Estimation. Wiley.CrossRefGoogle Scholar
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall/CRC.Google Scholar
Stanton, R. (1997) A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance 52, 1973–2002.CrossRefGoogle Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.CrossRefGoogle Scholar
Wand, M.P. & Jones, M.C. (1995) Kernel Smoothing. Chapman and Hall.CrossRefGoogle Scholar
Wiener, N. (1924) The quadratic variation of a function and its Fourier coefficients. Massachusetts Journal of Mathematics 3, 72–94.Google Scholar