Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-08T19:27:14.982Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted least-squares estimation for the subcritical Heston process

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  26 July 2018

M. du Roy de Chaumaray
M. du Roy de Chaumaray*
Affiliation:
Institut de Mathématiques de Bordeaux
*
* Current address: ENSAI, Campus de Ker Lann, Rue Blaise Pascal, BP 37203, 35172 Bruz Cedex, France. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We simultaneously estimate the four parameters of a subcritical Heston process. We do not restrict ourselves to the case where the stochastic volatility process never reaches zero. In order to avoid the use of unmanageable stopping times and a natural but intractable estimator, we use a weighted least-squares estimator. We establish strong consistency and asymptotic normality for this estimator. Numerical simulations are also provided, illustrating the favorable performance of our estimation procedure.

Keywords

Squared radial Ornstein–Uhlenbeck processHeston processparameter estimationweighted least-squares estimateconsistencyasymptotic normality

MSC classification

Primary: 62M05: Markov processes
Secondary: 60G20: Generalized stochastic processes 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 543 - 558
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. J. Financial Econom. 83, 413–452. Google Scholar
[2]Alfonsi, A. (2010). High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comput. 79, 209–237. Google Scholar
[3]Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model. J. Comput. Finance 11, 1–42. Google Scholar
[4]Azencott, R. and Gadhyan, Y. (2009). Accurate parameter estimation for coupled stochastic dynamics. Discrete Contin. Dyn. Syst. 2009, 44–53. Google Scholar
[5]Barczy, M. and Pap, G. (2016). Asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations. Statistics 50, 389–417. Google Scholar
[6]Ben Alaya, M. and Kebaier, A. (2012). Parameter estimation for the square-root diffusions: ergodic and nonergodic cases. Stoch. Models 28, 609–634. Google Scholar
[7]Ben Alaya, M. and Kebaier, A. (2013). Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square-root diffusions. Stoch. Anal. Appl. 31, 552–573. Google Scholar
[8]Cox, J. C., IngersollJ. E., Jr. J. E., Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407. Google Scholar
[9]Du Roy de Chaumaray, M. (2017). Large deviations for the squared radial Ornstein–Uhlenbeck process. Theory Prob. Appl. 61, 408–441. Google Scholar
[10]Feller, W. (1951). Two singular diffusion problems. Ann. Math. (2) 54, 173–182. Google Scholar
[11]Forde, M. and Jacquier, A. (2011). The large-maturity smile for the Heston model. Finance Stoch. 15, 755–780. Google Scholar
[12]Forde, M., Jacquier, A. and Lee, R. (2012). The small-time smile and term structure of implied volatility under the Heston model. SIAM J. Financial Math. 3, 690–708. Google Scholar
[13]Fournié, E. and Talay, D. (1991). Application de la statistique des diffusions à un modèle de taux d'intérêt. Finance 12, 79–111. Google Scholar
[14]Gao, F. and Jiang, H. (2009). Moderate deviations for squared Ornstein–Uhlenbeck process. Statist. Prob. Lett. 79, 1378–1386. Google Scholar
[15]Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. John Wiley, Hoboken, NJ. Google Scholar
[16]Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series, and Products. Academic Press, New York. Google Scholar
[17]Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343. Google Scholar
[18]Jacquier, A. and Roome, P. (2016). Large-maturity regimes of the Heston forward smile. Stoch. Process. Appl. 126, 1087–1123. Google Scholar
[19]Janek, A., Kluge, T., Weron, R. and Wystup, U. (2011). FX smile in the Heston model. In Statistical Tools for Finance and Insurance, Springer, Heidelberg, pp. 133–162. Google Scholar
[20]Lamberton, D. and Lapeyre, B. (1997). Introduction au Calcul Stochastique Appliqué à la Finance, 2nd edn. Ellipses Édition Marketing, Paris. Google Scholar
[21]Lee, R. W. (2004). Option pricing by transform methods: extensions, unification and error control. J. Comput. Finance 7, 51–86. Google Scholar
[22]Lewis, A. L. (2000). Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach, CA. Google Scholar
[23]Luke, Y. L. (1969). The Special Functions and Their Approximations, Vol. II. Academic Press, New York. Google Scholar
[24]Overbeck, L. (1998). Estimation for continuous branching processes. Scand. J. Statist. 25, 111–126. Google Scholar
[25]Stein, E. M. and Stein, J. C. (1991). Stock price distributions with stochastic volatility: an analytic approach. Rev. Financial Studies 4, 727–752. Google Scholar
[26]Stein, J. (1989). Overreactions in the options market. J. Finance 44, 1011–1023. Google Scholar
[27]Wei, C. Z. and Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Ann. Statist. 18, 1757–1773. Google Scholar
[28]Zani, M. (2002). Large deviations for squared radial Ornstein–Uhlenbeck processes. Stoch. Process. Appl. 102, 25–42. Google Scholar