Computational Economics

, Volume 53, Issue 2, pp 555–586 | Cite as

Option Pricing Under a Stochastic Interest Rate and Volatility Model with Hidden Markovian Regime-Switching

  • Dong-Mei Zhu
  • Jiejun Lu
  • Wai-Ki ChingEmail author
  • Tak-Kuen Siu


In this paper we discuss an option pricing problem in a hidden Markovian regime-switching model with a stochastic interest rate and volatility. Regime switches are attributed to structural changes in an hidden economic environment and are described by a continuous-time, finite-state, unobservable Markov chain. The model is then applied to the valuation of a standard European option. By means of the standard separation principle, filtering and option valuation problems are separated. Robust filters for the hidden states of the economy and their robust filtered estimates of unknown parameters from the expectation maximization algorithm are presented based on standard techniques in filtering theory. Then an explicit expression of a conditional characteristic function relevant to option pricing is presented and the valuation of the option is discussed using the inverse Fourier transformation approach. Using the limiting behavior of the conditional characteristic function, an efficient implementation of the transform inversion integral is considered. Numerical experiments are given to illustrate the flexibility of filtering algorithms and the significance of regime-switching in option pricing.


Option pricing Hidden Markov model (HMM) Regime-switching Characteristic function Fourier transformation 



A two-page abstract of the paper has been accepted for presentation in 2016 International Congress on Banking, Economics, Finance, and Business, 24–26 June 2016, Sapporo, Japan. This research work was supported by Research Grants Council of Hong Kong under Grant Number 17301214, HKU Strategic Theme on Computation and Information and National Natural Science Foundation of China under Grant Number 71601044.


  1. Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. The Journal of Finance, 52, 2003–2049.Google Scholar
  2. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.Google Scholar
  3. Buffington, J., & Elliott, R. J. (2002). American options with regime switching. International Journal of Theoretical and Applied Finance, 5, 497–514.Google Scholar
  4. Carr, P., & Madan, D. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2, 61–73.Google Scholar
  5. Clark, J. (1978). The design of robust approximations to the stochastic differential equations of nonlinear filtering. Communication Systems and Random Process Theory, 25, 721–734.Google Scholar
  6. Costabile, M., Leccadito, A., Massab, I., & Russo, E. (2014). Option pricing under regime-switching jump-diffusion models. Journal of Computational and Applied Mathematics, 256, 152–167.Google Scholar
  7. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–407.Google Scholar
  8. Elliott, R. J., Aggoun, L., & Moore, J. B. (1995). Hidden Markov models: Estimation and control. Berlin: Springer.Google Scholar
  9. Elliott, R. J., Chan, L., & Siu, T. K. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance, 1, 423–432.Google Scholar
  10. Elliott, R. J., & Siu, T. K. (2013). Option pricing and filtering with hidden Markov-modulated pure jump processes. Applied Mathematical Finance, 20, 1–25.Google Scholar
  11. Elliott, R. J., & Siu, T. K. (2015). Asset pricing using trading volumes in a hidden regime-switching environment. Asia Pacific Financial Markets, 22, 133–149.Google Scholar
  12. Elliott, R. J., Siu, T. K., & Badescu, A. (2010). On mean-variance portfolio selection under a hidden Markovian regime-switching model. Economic Modelling, 27, 678–686.Google Scholar
  13. Elliott, R. J., Siu, T. K., & Chan, L. (2014). On pricing barrier options with regime switching. Journal of Computational and Applied Mathematics, 256, 196–210.Google Scholar
  14. Fan, K., Shen, Y., Siu, T. K., & Wang, R. (2017). An FFT approach for option pricing under a regime-switching stochastic interest rate model. Communications in Statistics-Theory and Methods, 46(11), 5292–5310.Google Scholar
  15. Goldfeld, S. M., & Quandt, R. E. (1973). A Markov model for switching regressions. Journal of Econometrics, 1, 3–15.Google Scholar
  16. Guo, X. (2001). Information and option pricings. Quantitative Finance, 1, 38–44.Google Scholar
  17. Hamilton, J. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica: Journal of the Econometric Society, 57, 357–384.Google Scholar
  18. Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6, 327–343.Google Scholar
  19. Ho, T., & Lee, S. (1986). Term structure movements and pricing interest rate contingent claims. The Journal of Finance, 41, 1011–1029.Google Scholar
  20. Huang, J., Zhu, W., & Ruan, X. (2014). Option pricing using the fast Fourier transform under the double exponential jump model with stochastic volatility and stochastic intensity. Journal of Computational and Applied Mathematics, 263, 152–159.Google Scholar
  21. Hull, J., & White, A. (1990). Pricing interest rate derivative securities. The Review of Financial Studies, 3, 573–592.Google Scholar
  22. Ishijima, H., Kihara, T. Option pricing with hidden markov models. In 2005 Daiwa International Workshop on Financial Engineering (p. 117).
  23. Karatzas, I., & Zhao, X. (1998). Bayesian adaptive portfolio optimization. New York: Columbia University.Google Scholar
  24. Korn, R., Siu, T. K., & Zhang, A. (2011). Asset allocation for a DC pension fund under regime switching environment. European Actuarial Journal, 1, 361–377.Google Scholar
  25. Lewis, A. (2001). A simple option formula for general jump-diffusion and other exponential Lvy processes.
  26. Liew, C. C., & Siu, T. K. (2010). A hidden Markov regime-switching model for option valuation. Insurance: Mathematics and Economics, 47, 374–384.Google Scholar
  27. Liptser, R., & Shiryaev, A. N. (2013). Statistics of random processes: I. General theory. New York: Springer.Google Scholar
  28. Lord, R., Lord, R., Kahl, C., & Kahl, C. (2006). Optimal Fourier inversion in semi-analytical option pricing. Working Paper, 10, 1–21.Google Scholar
  29. Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4, 141–183.Google Scholar
  30. Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. The Journal of Finance, 29, 449–470.Google Scholar
  31. Naik, V. (1993). Option valuation and hedging strategies with jumps in the volatility of asset returns. The Journal of Finance, 48, 1969–1984.Google Scholar
  32. Quandt, R. E. (1958). The estimation of the parameters of a linear regression system obeying two separate regimes. Journal of American Statistical Association, 53, 873–880.Google Scholar
  33. Schobel, R., & Zhu, J. (1999). Stochastic volatility with an Ornstein--Uhlenbeck process: An extension. Review of Finance, 3, 23–46.Google Scholar
  34. Shen, Y., & Siu, T. K. (2013a). Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching. Operations Research Letters, 41, 180–187.Google Scholar
  35. Shen, Y., & Siu, T. K. (2013b). Pricing bond options under a Markovian regime-switching Hull--White model. Economic Modelling, 30, 933–940.Google Scholar
  36. Siu, T. K. (2004). A hidden Markov-modulated jump diffusion model for european option pricing. In R. Mamon & R. J. Elliott (Eds.), Hidden Markov models in finance vol. 2 monograph (Vol. 2, pp. 185–209). New York: Springer.Google Scholar
  37. Siu, T. K. (2005). Fair valuation of participating policies with surrender options and regime switching. Insurance: Mathematics and Economics, 37, 533–552.Google Scholar
  38. Siu, T. K. (2011). Long-term strategic asset allocation with inflation risk and regime switching. Quantitative Finance, 11, 1565–1580.Google Scholar
  39. Siu, T. K. (2012). A BSDE approach to risk-based asset allocation of pension funds with regime switching. Annals of Operations Research, 201, 449–473.Google Scholar
  40. Stein, E. M., & Stein, J. C. (1991). Stock price distributions with stochastic volatility: An analytic approach. The Review of Financial Studies, 4, 727–752.Google Scholar
  41. Siu, T. K. (2013) American option pricing and filtering in a hidden regime-switching jump-diffusion market, Submitted.Google Scholar
  42. van Haastrecht, A., Lord, R., Pelsser, A., & Schrager, D. (2009). Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility. Insurance: Mathematics and Economics, 45, 436–448.Google Scholar
  43. Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Dong-Mei Zhu
    • 1
  • Jiejun Lu
    • 2
  • Wai-Ki Ching
    • 2
    • 3
    • 4
    Email author
  • Tak-Kuen Siu
    • 5
  1. 1.School of Management and EconomicsSoutheast UniversityNanjingChina
  2. 2.Advanced Modeling and Applied Computing Laboratory, Department of MathematicsThe University of Hong KongPokfulamHong Kong
  3. 3.Hughes HallCambridgeUK
  4. 4.School of Economics and ManagementBeijing University of Chemical TechnologyBeijingChina
  5. 5.Department of Applied Finance and Actuarial Studies, Faculty of Business and EconomicsMacquarie UniversitySydneyAustralia

Personalised recommendations