Inference for stochastic volatility models using time change transformations. (English) Zbl 1189.91220

Diffusion processes provide a reasonable model for continuous time phenomena. Unfortunately, inference for diffusion processes is difficult. A review of results concerning inference for diffusion processes was compiled by H. Sørensen [Int. Stat. Rev. 72, 337–354 (2004)]. Likelihood-based methods usually approximate the likelihood so that the discretization error becomes arbitrarily small. These methods often depend on the Markov property.
Sequential Monte Carlo techniques can be used to implement parameter estimation for stochastic volatility models. Another approach to estimating parameters for these models arises from Bayesian inference using Markov chain Monte Carlo (MCMC) methods. Initial MCMC schemes were introduced by Jones [Review of Financial Studies 16, 793–843 (2003)]. Unfortunately, the algorithm’s convergence properties degenerate as the number of imputed points increases. G. O. Roberts and O. Stramer overcome this problem in the one-dimensional context with a reparametrization [Biometrika 88, No. 3, 603–621 (2001; Zbl 0985.62066)].
This paper introduces another reparametrization which operates on the time scale of the observed diffusion rather than on its path. This transformation alters the time axis of the diffusion. This allows for the construction of irreducible and efficient MCMC models to accommodate the time change of the diffusion path. This approach also relies upon the Markov property and can be coupled with the approaches presented in the paper by Roberts and Stramer [loc. cit.] and a submitted result by S. Chib, M. K. Pitt, and N. Shephard.
The authors discuss the need for reparametrization in section 2 and introduce the class of transformations based on changing the time scale of the diffusion process in section 3. Section 4 provides the details of the MCMC implementation. Numerical experiments of the method and an application to US Treasury bill rates are also presented. This method compares well with competing models and is not computationally expensive. The computing cost of the authors’ approach is \(O(m)\), compared to \(O(m^2)\) for competing models.


91G60 Numerical methods (including Monte Carlo methods)
65C05 Monte Carlo methods
65C60 Computational problems in statistics (MSC2010)


Zbl 0985.62066
Full Text: DOI arXiv


[1] Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed form approximation approach. Econometrica 70 223-262. · Zbl 1104.62323
[2] Aït-Sahalia, Y. (2008). Closed form likelihood expansions for multivariate diffusions. Ann. Statist. 36 906-937. · Zbl 1246.62180
[3] Andersen, T. G. and Lund, J. (1997). Estimating continuous-time stochastic volatility models of the short term interest rate. J. Econometrics 77 343-377. · Zbl 0925.62529
[4] Beskos, A., Papaspiliopoulos, O., Roberts, G. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333-382. · Zbl 1100.62079
[5] Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422-2444. · Zbl 1101.60060
[6] Bibby, B. and Sorensen, M. (1995). Martingale estimating functions for discretely observed diffusion processes. Bernoulli 1 17-39. · Zbl 0830.62075
[7] Chib, S., Pitt, M. K. and Shephard, N. (2007). Efficient likelihood based inference for observed and partially observed diffusions. Submitted.
[8] Durham, G. B. (2003). Likelihod based specification analysis of continuous time models of the short term interest rate. Journal of Financial Economics 70 463-487.
[9] Durham, G. B. and Gallant, A. R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econom. Statist. 20 297-316. JSTOR:
[10] Elerian, O. S., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed non-linear diffusions. Econometrica 69 959-993. JSTOR: · Zbl 1017.62068
[11] Eraker, B. (2001). Markov chain Monte Carlo analysis of diffusion models with application to finance. J. Bus. Econom. Statist. 19 177-191.
[12] Fearnhead, P., Papaspiliopoulos, O. and Roberts, G. O. (2008). Particle filters for partially observed diffusions. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 755-777. · Zbl 05563368
[13] Gallant, A. R. and Long, J. R. (1997). Estimating stochastic differential equations efficiently by minimum chi-squared. Biometrika 84 125-141. JSTOR: · Zbl 0953.62084
[14] Gallant, A. R. and Tauchen, G. (1996). Which moments to match? Econometric Theory 12 657-681. JSTOR: · Zbl 04534738
[15] Gallant, A. R. and Tauchen, G. (1998). Reprojecting partially observed systems with applications to interest rate diffusions. J. Amer. Statist. Assoc. 93 10-24. · Zbl 0920.62132
[16] Golightly, A. and Wilkinson, D. (2006). Bayesian sequential inference for nonlinear multivariate diffusions. Stat. Comput. 16 323-338.
[17] Golightly, A. and Wilkinson, D. (2007). Bayesian inference for nonlinear multivariate diffusions observed with error. Comput. Statist. Data Anal. 52 1674-1693. · Zbl 1452.62603
[18] Gouriéroux, C., Monfort, A. and Renault, E. (1993). Indirect inference. J. Appl. Econometrics 8 S85-S118. · Zbl 1448.62202
[19] Heston, S. (1993). A closed-form solution for options with stochastic volatility. With applications to bonds and currency options. Review of Financial Studies 6 327-343. · Zbl 1384.35131
[20] Hull, J. C. and White, A. D. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance 42 281-300. · Zbl 1126.91369
[21] Jones, C. S. (1999). Bayesian estimation of continuous-time finance models. Unpublished paper, Simon School of Business, Univ. Rochester.
[22] Jones, C. S. (2003). Nonlinear mean reversion in the short-term interest rate. Review of Financial Studies 16 793-843.
[23] Kalogeropoulos, K. (2007). Likelihood based inference for a class of multidimensional diffusions with unobserved paths. J. Statist. Plann. Inference 137 3092-3102. · Zbl 1114.62080
[24] Kalogeropoulos, K., Dellaportas, P. and Roberts, G. (2007). Likelihood-based inference for correllated diffusions. Submitted. · Zbl 1221.65027
[25] Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice (A. Doucet, J. F. G. De Freitas and N. J. Gordon, eds.). Springer, New York. · Zbl 1056.93583
[26] Oksendal, B. (2000). Stochastic Differential Equations , 5th ed. Springer, Berlin.
[27] Papaspiliopoulos, O. and Roberts, G. (2008). Retrospective MCMC for Dirichlet process hierarchical models. Biometrika 95 169-186. · Zbl 1437.62576
[28] Pedersen, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22 55-71. · Zbl 0827.62087
[29] Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters. J. Amer. Statist. Assoc. 94 590-599. JSTOR: · Zbl 1072.62639
[30] Roberts, G. and Stramer, O. (2001). On inference for partial observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika 88 603-621. JSTOR: · Zbl 0985.62066
[31] Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales, 2, Itô Calculus . Wiley, Chicester. · Zbl 0826.60002
[32] Sørensen, H. (2004). Parametric inference for diffusion processes observed at discrete points in time: A survey. Int. Stat. Rev. 72 337-354.
[33] Stein, E. M. and Stein, J. C. (1991). Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies 4 727-752. · Zbl 1458.62253
[34] Stroud, J. R., Polson, N. G. and Muller, P. (2004). Practical filtering for stochastic volatility models. In State Space and Unobserved Component Models 236-247. Cambridge Univ. Press, Cambridge. · Zbl 05280148
[35] Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation. J. Amer. Statist. Assoc. 82 528-540. JSTOR: · Zbl 0619.62029
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