Abstract
There is considerable empirical evidence suggesting that the random walk model for changes in stock prices is not appropriate. One of the reasons is that this model fails to account for economic cycles because increments are independent and identically distributed. A reliable solution for modeling economic cycles consists in modulating the parameters of a basis process, e.g., a Brownian motion by a hidden Markov chain. This approach has received a lot of attention in the recent econometric literature. In this chapter, we aim to introduce its main features in continuous time. The first sections focus on regime switching diffusions with transition jumps. We present their properties and an estimation procedure to fit this process to stock log-returns. Pricing of options is also discussed. We next introduce the switching multifractal model of Calvet and Fisher (J Econom 105:17–58, 2001). This is a switching diffusion with a large number of regimes that are structured in order to limit the number of parameters. This chapter partly serves as introduction to Chap. 2 in which a multivariate extension is estimated by a Monte Carlo Markov Chain method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As \(\tau _{x}^{+}\) and \(\tau _{x}^{-}\) are stopping times, this is a consequence of the strong Markov property of γ t.
References
Al-Anaswah, N., Wilfing, B.: Identification of speculative bubbles using state-space models with Markov-switching. J. Banking Financ. 35(5), 1073–1086 (2011)
Calvet, L., Fisher, A.: Forecasting multifractal volatility. J. Econ. 105, 17–58 (2001)
Calvet, L., Fisher, A.: How to forecast long-run volatility: regime switching and the estimation of multifractal processes. J. Financ. Econ. 2, 49–83 (2004)
Carr, P., Madan, D.: Option valuation using the fast fourier transform. J. Comput. Financ. 2, 61–73 (1999)
Cholette, L., Heinen, A., Valdesogo, A.: Modelling international financial returns with a multivariate regime switching copula. J. Financ Econ. 7(4), 437–480 (2009)
Engle, R.F.: Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–10 (1982)
Guidolin, M., Timmermann, A.: Economic implications of bull and bear regimes in UK stock and bond returns. Econ. J. 115, 11–143 (2005)
Guidolin, M., Timmermann, A.: Asset allocation under multivariate regime switching. J. Econ. Dyn. Control 31(11), 3503–3544 (2007)
Guidolin, M., Timmermann, A.: International asset allocation under regime switching, skew, and kurtosis preferences. Rev. Financ. Stud. 21(2), 889–935 (2008)
Hainaut, D.: A fractal version of the Hull–White interest rate model. Econ. Modell. 31, 323–334 (2013)
Hainaut, D., Deelstra, G.: A self-excited switching jump diffusion (SESJD): properties, calibration and hitting time. Quant. Financ. 19(3), 407–426 (2019)
Hainaut, D., Deelstra, G.: A bivariate mutually-excited switching jump diffusion (BMESJD) for asset prices. Methodol. Comput. Appl. Probab. 21(4), 1337–1375 (2019)
Hainaut, D., MacGilchrist, R.: Strategic asset allocation with switching dependence. Ann. Financ. 8(1), 75–96 (2012)
Hamilton, J.D.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2), 357–384 (1989)
Hardy, M.: A regime-switching model of long-term stock returns. North Amer. Actuar. J. 5(2), 41–53 (2001)
Jiang, Z., Pistorius, M.R.: On perpetual American put valuation and first-passage in a regime-switching model with jumps. Financ. Stochast. 12(2), 331–355 (2008)
Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, London (2009)
Kou, S.G.: A jump diffusion model for option pricing. Manag. Sci. 48, 1086–1101 (2002)
Kou, S.G., Wang, H.: First passage times of a jump diffusion process. Adv. Appl. Probab. 35, 504–531 (2003)
Kou, S.G., Wang, H.: Option pricing under a double exponential jump diffusion model. Manag. Sci. 50, 1178–1192 (2004)
Nakajima, J.: Stochastic volatility model with regime-switching skewness in heavy-tailed errors for exchange rate returns. Stud. Nonlinear Dyn. Econ. 17(5), 499–520 (2013)
Rogers, L.C.G.: Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Probab. 4(2), 390–413 (1994)
Stovall, S.: Standard & Poor’s Sector Investing: How to Buy the Right Stock in the Right Industry at the Right Time. McGraw-Hill Companies, New York (1996)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hainaut, D. (2022). Switching Models: Properties and Estimation. In: Continuous Time Processes for Finance. Bocconi & Springer Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-031-06361-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-031-06361-9_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-06360-2
Online ISBN: 978-3-031-06361-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)