Abstract
Stochastic volatility (SV) is the main concept used in the fields of financial economics and mathematical finance to deal with the endemic time-varying volatility and codependence found in financial markets. Such dependence has been known for a long time; early commentators include Mandelbrot (1963) and Officer (1973). It was also clear to the founding fathers of modern continuous time finance that homogeneity was an unrealistic if convenient simplification; for example, Black and Scholes (1972, p. 416) wrote, ‘…there is evidence of non-stationarity in the variance. More work must be done to predict variances using the information available.’ Heterogeneity has deep implications for the theory and practice of financial economics and econometrics. In particular, asset pricing theory is dominated by the idea that higher rewards may be expected when we face higher risks, but these risks change through time in complicated ways. Some of the changes in the level of risk can be modelled stochastically, where the level of volatility and degree of codependence between assets is allowed to change over time. Such models allow us to explain, for example, empirically observed departures from Black-Scholes-Merton prices for options and understand why we should expect to see occasional dramatic moves in financial markets.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Andersen, T. and Bollerslev, T. 1998. Answering the skeptics: yes, standard volatility models do provide accurate forecasts. International Economic Review 39, 885–905.
Andersen, T., Bollerslev, T., Diebold, F. and Labys, P. 2001. The distribution of exchange rate volatility. Journal of the American Statistical Association 96, 42–55. Correction published in vol. 98 (2003), p. 501.
Andersen, T. G. and Sørensen, B. 1996. GMM estimation of a stochastic volatility model: a Monte Carlo study. Journal of Business and Economic Statistics 14, 328–52.
Bandi, F. and Russell, J. 2003. Microstructure noise, realized volatility, and optimal sampling. Mimeo. Graduate School of Business, University of Chicago.
Barndorff-Nielsen, O. 2001. Superposition of Ornstein-Uhlenbeck type processes. Theory of Probability and its Applications 45, 175–94.
Barndorff-Nielsen, O. and Shephard, N. 2001. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society, Series B 63, 167–241.
Barndorff-Nielsen, O. and 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, 253–80.
Barndorff-Nielsen, O. and Shephard, N. 2004a. Econometric analysis of realised covariation: high frequency covariance, regression and correlation in financial economics. Econometrica 72, 885–925.
Barndorff-Nielsen, O. and Shephard, N. 2004b. Power and bipower variation with stochastic volatility and jumps (with discussion). Journal of Financial Econometrics 2, 1–48.
Barndorff-Nielsen, O. and Shephard, N. 2006a. Variation, jumps, and high frequency data in financial econometrics. In Advances in Economics and Econometrics: Theory and Applications, vol. 1, ed. R. Blundell, P. Torsten and W. Newey. Cambridge: Cambridge University Press.
Barndorff-Nielsen, O. and Shephard, N. 2006b. Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4, 1–30.
Bates, D. 1996. Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options. Review of Financial Studies 9, 69–107.
Bates, D. 2000. Post-’97 crash fears in the S-&P 500 futures option market. Journal of Econometrics 94, 181–238.
Black, F. 1976. Studies of stock price volatility changes. Proceedings of the Business and Economic Statistics Section, American Statistical Association, 177–81.
Black, F. and Scholes, M. 1972. The valuation of option contracts and a test of market efficiency. Journal of Finance 27, 399–418.
Black, F. and Scholes, M. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–54.
Bochner, S. 1949. Diffusion equation and stochastic processes. Proceedings of the National Academy of Science of the United States of America 85, 369–70.
Bollerslev, T. and Zhou, H. 2002. Estimating stochastic volatility diffusion using conditional moments of integrated volatility. Journal of Econometrics 109, 33–65.
Breidt, F., Crato, N. and de Lima, P. 1998. On the detection and estimation of long memory in stochastic volatility. Journal of Econometrics 83, 325–18.
Chernov, M. and Ghysels, E. 2000. A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation. Journal of Financial Economics 56, 407–58.
Chernov, M., Gallant, A., Ghysels, E. and Tauchen, G. 2003. Alternative models of stock price dynamics. Journal of Econometrics 116, 225–57.
Clark, P. 1973. A subordinated stochastic process model with fixed variance for speculative prices. Econometrica 41, 135–56.
Cochrane, J. 2001. Asset Pricing. Princeton: Princeton University Press.
Comte, F., Coutin, L. and Renault, E. 2003. Affine fractional stochastic volatility models. Mimeo. University of Montreal.
Comte, F. and Renault, E. 1998. Long memory in continuous-time stochastic volatility models. Mathematical Finance 8, 291–323.
Cont, R. and Tankov, P. 2004. Financial Modelling with Jump Processes. London: Chapman and Hall.
Das, S. and Sundaram, R. 1999. Of smiles and smirks: a term structure perspective. Journal of Financial and Quantitative Analysis 34, 211–40.
Diebold, F. and Nerlove, M. 1989. The dynamics of exchange rate volatility: a multivariate latent factor ARCH model. Journal of Applied Econometrics 4, 1–21.
Doob, J. 1953. Stochastic Processes. New York: John Wiley and Sons.
Duffle, D., Filipovic, D. and Schachermayer, W. 2003. Affine processes and applications in finance. Annals of Applied Probability 13, 984–1053.
Duffle, D., Pan, J. and Singleton, K. 2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–76.
Elerian, O., Chib, S. and Shephard, N. 2001. Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69, 959–93.
Eraker, B. 2001. Markov chain Monte Carlo analysis of diffusion models with application to finance. Journal of Business and Economic Statistics 19, 177–91.
Eraker, B., Johannes, M. and Poison, N. 2003. The impact of jumps in returns and volatility. Journal of Finance 53, 1269–300.
Fang, Y. 1996. Volatility modeling and estimation of high-frequency data with Gaussian noise. Ph.D. thesis. Sloan School of Management, MIT.
Fiorentini, G., Sentana, E. and Shephard, N. 2004. Likelihood-based estimation of latent generalised ARCH structures. Econometrica 12, 1481–517.
Gallant, A. and Tauchen, G. 1996. Which moments to match. Econometric Theory 12, 657–81.
Garcia, R., Ghysels, E. and Renault, E. 2006. The econometrics of option pricing. In Handbook of Financial Econometrics, ed. Y. Ait-Sahalia and L. Hansen. Amsterdam: North-Holland.
Genon-Catalot, V., Jeantheau, T. and Larédo, C. 2000. Stochastic volatility as hidden Markov models and statistical applications. Bernoulli 6, 1051–79.
Gourieroux, C, Monfort, A. and Renault, E. 1993. Indirect inference. Journal of Applied Econometrics 6, S85–S118.
Hansen, P. and Lunde, A. 2006. Realized variance and market microstructure noise (with discussion). Journal of Business and Economic Statistics 24, 127–61.
Harvey, A. 1998. Long memory in stochastic volatility. In Forecasting Volatility in Financial Markets, ed. J. Knight and S. Satchell. Oxford: Butterworth-Heinemann.
Harvey, A., Ruiz, E. and Shephard, N. 1994. Multivariate stochastic variance models. Review of Economic Studies 61, 247–64.
Heston, S. 1993. A closed-form solution for options with stochastic volatility, with applications to bond and currency options. Review of Financial Studies 6, 327–13.
Hoffmann, M. 2002. Rate of convergence for parametric estimation in stochastic volatility models. Stochastic Processes and their Application 97, 147–70.
Hull, J. and White, A. 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance 42, 281–300.
Jacod, J. 1994. Limit of random measures associated with the increments of a Brownian semimartingale. Preprint No. 120. Laboratoire de Probabilitiés, Université Pierre et Marie Curie, Paris.
Jacquier, E., Poison, N. and Rossi, P. 1994. Bayesian analysis of stochastic volatility models (with discussion). Journal of Business and Economic Statistics 12, 371–417.
Jacquier, E., Poison, N. and Rossi, P. 2004. Stochastic volatility models: univariate and multivariate extensions. Journal of Econometrics 122, 185–212.
Johnson, H. 1979. Option pricing when the variance rate is changing. Working paper. University of California, Los Angeles.
Johnson, H. and Shanno, D. 1987. Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis 22, 143–51.
Kim, S., Shephard, N. and Chib, S. 1998. Stochastic volatility: likelihood inference and comparison with ARCH models. Review of Economic Studies 65, 361–93.
King, M., Sentana, E. and Wadhwani, S. 1994. Volatility and links between national stock markets. Econometrica 62, 901–33.
Mandelbrot, B. 1963. The variation of certain speculative prices. Journal of Business 36, 394–419.
Meddahi, N. 2001. An eigenfunction approach for volatility modeling. Cahiers de recherche No. 2001–29. Department of Economics, University of Montreal.
Melino, A. and Turnbull, S. 1990. Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239–65.
Merton, R. 1980. On estimating the expected return on the market: an exploratory investigation. Journal of Financial Economics 8, 323–61.
Nelson, D. 1991. Conditional heteroskedasticity in asset pricing: a new approach. Econometrica 59, 347–70.
Nicolato, E. and Venardos, E. 2003. Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Mathematical Finance 13, 445–66.
Officer, R. 1973. The variability of the market factor of the New York stock exchange. Journal of Business 46, 434–53.
Pastorello, S., Patilea, V. and Renault, E. 2003. Iterative and recursive estimation in structural non-adaptive models. Journal of Business and Economic Statistics 21, 449–509.
Phillips, P. and Yu, J. 2005. A two-stage realized volatility approach to the estimation for diffusion processes from discrete observations. Discussion Paper No. 1523. Cowles Foundation, Yale University.
Renault, E. and Touzi, N. 1996. Option hedging and implied volatilities in a stochastic volatility model. Mathematical Finance 6, 279–302.
Roberts, G. and Stramer, O. 2001. On inference for nonlinear diffusion models using the Hastings-Metropolis algorithms. Biometrika 88, 603–21.
Rosenberg, B. 1972. The behaviour of random variables with nonstationary variance and the distribution of security prices. Working paper 11, Graduate School of Business Administration, University of California, Berkeley. Reprinted in N. Shephard (2005).
Shephard, N. 2005. Stochastic Volatility: Selected Readings. Oxford: Oxford University Press.
Smith, A. 1993. Estimating nonlinear time series models using simulated vector autoregressions. Journal of Applied Econometrics 8, S63–S84.
Sørensen, M. 2000. Prediction based estimating equations. Econometrics Journal 3, 123–17.
Stein, E. and Stein, J. 1991. Stock price distributions with stochastic volatility: an analytic approach. Review of Financial Studies 4, 727–52.
Taylor, S. 1982. Financial returns modelled by the product of two stochastic processes — a study of daily sugar prices 1961–79. In Time Series Analysis: Theory and Practice, vol. 1, ed. O. Anderson. Amsterdam: North-Holland.
Wiggins, J. 1987. Option values under stochastic volatilities. Journal of Financial economics 19, 351–72.
Yu, J. 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127, 165–78.
Zhang, L., Mykland, P. and Aït-Sahalia, Y 2005. A tale of two time scales: determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association 100, 1394–411.
Zhou, B. 1996. High-frequency data and volatility in foreign-exchange rates. Journal of Business and Economic Statistics 14, 45–52.
Editor information
Editors and Affiliations
Copyright information
© 2010 Palgrave Macmillan, a division of Macmillan Publishers Limited
About this chapter
Cite this chapter
Shephard, N. (2010). Stochastic volatility models. In: Durlauf, S.N., Blume, L.E. (eds) Macroeconometrics and Time Series Analysis. The New Palgrave Economics Collection. Palgrave Macmillan, London. https://doi.org/10.1057/9780230280830_31
Download citation
DOI: https://doi.org/10.1057/9780230280830_31
Publisher Name: Palgrave Macmillan, London
Print ISBN: 978-0-230-23885-5
Online ISBN: 978-0-230-28083-0
eBook Packages: Palgrave Economics & Finance CollectionEconomics and Finance (R0)