Recursive Estimation for Continuous Time Stochastic Volatility Models Using the Milstein Approximation

Affiliation(s)

Department of Statistics, University of Manitoba, Winnipeg, Canada.

Department of Accounting and Finance, University of Manitoba, Winnipeg, Canada.

Department of Statistics, University of Manitoba, Winnipeg, Canada.

Department of Accounting and Finance, University of Manitoba, Winnipeg, Canada.

ABSTRACT

Optimal as well as recursive parameter estimation for semimartingales had been studied in [1,2]. Recently, there has been a growing interest in modelling volatility of the observed process by nonlinear stochastic processes [3]. In this paper, we study the recursive estimates for various classes of discretely sampled continuous time stochastic volatility models using the Milstein approximation. We provide closed form expressions for the recursive estimates for recently proposed stochastic volatility models. We also give an example of computation of the term structure of zero rates in an incomplete information environment. In this case, learning about an unobserved state variable is done jointly with the valuation procedure.

Cite this paper

T. Koulis, A. Paseka and A. Thavaneswaran, "Recursive Estimation for Continuous Time Stochastic Volatility Models Using the Milstein Approximation,"*Journal of Mathematical Finance*, Vol. 3 No. 3, 2013, pp. 357-365. doi: 10.4236/jmf.2013.33036.

T. Koulis, A. Paseka and A. Thavaneswaran, "Recursive Estimation for Continuous Time Stochastic Volatility Models Using the Milstein Approximation,"

References

[1] A. Thavaneswaran and M. E. Thompson, “A Criterion for Filtering in Semimartingale Models,” Stochastic Processes and Their Applications, Vol. 28, No. 2, 1988, pp. 259-265. doi:10.1016/0304-4149(88)90099-3

[2] A. Thavaneswaran and M. E. Thompson, “Optimal Estimation for Semimartingales,” Journal of Applied Probability, Vol. 23, No. 2, 1986, pp. 409-417. doi:10.2307/3214183

[3] S. Taylor, “Asset Price Dynamics, Volatility, and Prediction,” Princeton University Press, Princeton, 2011.

[4] S. L. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial Studies, Vol. 13, No. 3, 2000, pp. 585-625. doi:10.1093/rfs/13.3.585

[5] H. Kawakatsu, “Specification and Estimation of Discrete time Quadratic Stochastic Volatility Models,” Journal of Empirical Finance, Vol. 14, No. 3, 2007, pp. 424-442. doi:10.1016/j.jempfin.2006.07.001

[6] U. V. Naik-Nimbalkar and M. B. Rajarshi, “Filtering and Smoothing via Estimating Functions,” Journal of the American Statistical Association, Vol. 90, No. 429, 1995, pp. 301-306. doi:10.1080/01621459.1995.10476513

[7] M. E. Thompson and A. Thavaneswaran, “Filtering via Estimating Functions,” Applied Mathematics Letters, Vol. 12, No. 5, 1999, pp. 61-67. doi:10.1016/S0893-9659(99)00058-0

[8] A. Thavaneswaran, Y. Liang and N. Ravishanker, “Inference for Diffusion Processes Using Combined Estimating Functions,” Sri Lankan Journal of Applied Statistics, Vol. 12, No. 1, 2012, pp. 145-160.

[9] T. Koulis and A. Thavaneswaran, “Inference for Interest Rate Models Using Milstein’s Approximation,” Journal of Mathematical Finance, Vol. 3, No. 1, 2013, pp. 110-118. doi:10.4236/jmf.2013.31010

[10] H. Gong and A. Thavaneswaran, “Recursive Estimation for Continuous Time Stochastic Volatility Models,” Applied Mathematics Letters, Vol. 22, No. 11, 2009, pp. 1770-1774. doi:10.1016/j.aml.2009.06.014

[11] M. Jeong and J. Y. Park, “Asymptotic Theory of Maximum Likelihood Estimator for Diffusion Model,” Working Paper, Indiana University, 2010.

[12] P. E. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” Applications of Mathematics, Vol. 23, 1992, in press. doi:10.1007/978-3-662-12616-5

[13] D. Kennedy, “Stochastic Financial Models,” Financial Mathematics Series, Chapman & Hall/CRC, London, 2010.

[14] Y. Ait-Sahalia, “Testing Continuous-Time Models of the Spot Interest Rate,” Review of Financial Studies, Vol. 9, No. 2, 1996, pp. 385-426. doi:10.1093/rfs/9.2.385

[15] F. Klebaner, “Introduction to Stochastic Calculus with Applications,” Imperial College Press, London, 2005. doi:10.1142/p386

[16] F. Black and M. S. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[17] J. C. Hull and A. D. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.1111/j.1540-6261.1987.tb02568.x

[18] P. Balduzzi, S. R. Das and S. Foresi, “The Central Tendency: A Second Factor in Bond Yields,” The Review of Economics and Statistics, Vol. 80, No. 1, 1998, pp. 62-72. doi:10.1162/003465398557339

[19] D. McLeish, “Monte Carlo Simulation and Finance,” Wiley Finance, Wiley, Hoboken, 2005.

[1] A. Thavaneswaran and M. E. Thompson, “A Criterion for Filtering in Semimartingale Models,” Stochastic Processes and Their Applications, Vol. 28, No. 2, 1988, pp. 259-265. doi:10.1016/0304-4149(88)90099-3

[2] A. Thavaneswaran and M. E. Thompson, “Optimal Estimation for Semimartingales,” Journal of Applied Probability, Vol. 23, No. 2, 1986, pp. 409-417. doi:10.2307/3214183

[3] S. Taylor, “Asset Price Dynamics, Volatility, and Prediction,” Princeton University Press, Princeton, 2011.

[4] S. L. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial Studies, Vol. 13, No. 3, 2000, pp. 585-625. doi:10.1093/rfs/13.3.585

[5] H. Kawakatsu, “Specification and Estimation of Discrete time Quadratic Stochastic Volatility Models,” Journal of Empirical Finance, Vol. 14, No. 3, 2007, pp. 424-442. doi:10.1016/j.jempfin.2006.07.001

[6] U. V. Naik-Nimbalkar and M. B. Rajarshi, “Filtering and Smoothing via Estimating Functions,” Journal of the American Statistical Association, Vol. 90, No. 429, 1995, pp. 301-306. doi:10.1080/01621459.1995.10476513

[7] M. E. Thompson and A. Thavaneswaran, “Filtering via Estimating Functions,” Applied Mathematics Letters, Vol. 12, No. 5, 1999, pp. 61-67. doi:10.1016/S0893-9659(99)00058-0

[8] A. Thavaneswaran, Y. Liang and N. Ravishanker, “Inference for Diffusion Processes Using Combined Estimating Functions,” Sri Lankan Journal of Applied Statistics, Vol. 12, No. 1, 2012, pp. 145-160.

[9] T. Koulis and A. Thavaneswaran, “Inference for Interest Rate Models Using Milstein’s Approximation,” Journal of Mathematical Finance, Vol. 3, No. 1, 2013, pp. 110-118. doi:10.4236/jmf.2013.31010

[10] H. Gong and A. Thavaneswaran, “Recursive Estimation for Continuous Time Stochastic Volatility Models,” Applied Mathematics Letters, Vol. 22, No. 11, 2009, pp. 1770-1774. doi:10.1016/j.aml.2009.06.014

[11] M. Jeong and J. Y. Park, “Asymptotic Theory of Maximum Likelihood Estimator for Diffusion Model,” Working Paper, Indiana University, 2010.

[12] P. E. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” Applications of Mathematics, Vol. 23, 1992, in press. doi:10.1007/978-3-662-12616-5

[13] D. Kennedy, “Stochastic Financial Models,” Financial Mathematics Series, Chapman & Hall/CRC, London, 2010.

[14] Y. Ait-Sahalia, “Testing Continuous-Time Models of the Spot Interest Rate,” Review of Financial Studies, Vol. 9, No. 2, 1996, pp. 385-426. doi:10.1093/rfs/9.2.385

[15] F. Klebaner, “Introduction to Stochastic Calculus with Applications,” Imperial College Press, London, 2005. doi:10.1142/p386

[16] F. Black and M. S. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[17] J. C. Hull and A. D. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.1111/j.1540-6261.1987.tb02568.x

[18] P. Balduzzi, S. R. Das and S. Foresi, “The Central Tendency: A Second Factor in Bond Yields,” The Review of Economics and Statistics, Vol. 80, No. 1, 1998, pp. 62-72. doi:10.1162/003465398557339

[19] D. McLeish, “Monte Carlo Simulation and Finance,” Wiley Finance, Wiley, Hoboken, 2005.