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Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching


  • Robert Elliott
  • Tak Kuen Siu
  • Leunglung Chan


A model is developed for pricing volatility derivatives, such as variance swaps and volatility swaps under a continuous-time Markov-modulated version of the stochastic volatility (SV) model developed by Heston. In particular, it is supposed that the parameters of this version of Heston's SV model depend on the states of a continuous-time observable Markov chain process, which can be interpreted as the states of an observable macroeconomic factor. The market considered is incomplete in general, and hence, there is more than one equivalent martingale pricing measure. The regime switching Esscher transform used by Elliott et al. is adopted to determine a martingale pricing measure for the valuation of variance and volatility swaps in this incomplete market. Both probabilistic and partial differential equation (PDE) approaches are considered for the valuation of volatility derivatives.

Suggested Citation

  • Robert Elliott & Tak Kuen Siu & Leunglung Chan, 2007. "Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(1), pages 41-62.
  • Handle: RePEc:taf:apmtfi:v:14:y:2007:i:1:p:41-62
    DOI: 10.1080/13504860600659222

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    References listed on IDEAS

    1. Steven L. Heston & Saikat Nandi, 2000. "Derivatives on volatility: some simple solutions based on observables," FRB Atlanta Working Paper 2000-20, Federal Reserve Bank of Atlanta.
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    Cited by:

    1. repec:eee:dyncon:v:78:y:2017:i:c:p:149-163 is not listed on IDEAS
    2. Zhengjun Jiang & Martijn Pistorius, 2008. "Optimal dividend distribution under Markov-regime switching," Papers 0812.4978,, revised Apr 2011.
    3. Qi-min, Zhang, 2011. "Convergence of numerical solutions for a class of stochastic age-dependent capital system with Markovian switching," Economic Modelling, Elsevier, vol. 28(3), pages 1195-1201, May.
    4. Robert J. Elliott & Katsumasa Nishide & Carlton‐James U. Osakwe, 2016. "Heston‐Type Stochastic Volatility with a Markov Switching Regime," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 36(9), pages 902-919, September.
    5. Mengzhe Zhang & Leunglung Chan, 2016. "Saddlepoint approximations to option price in a regime-switching model," Annals of Finance, Springer, vol. 12(1), pages 55-69, February.
    6. Daniela Neykova & Marcos Escobar & Rudi Zagst, 2015. "Optimal investment in multidimensional Markov-modulated affine models," Annals of Finance, Springer, vol. 11(3), pages 503-530, November.
    7. Mengzhe Zhang & Leunglung Chan, 2016. "Pricing volatility swaps in the Heston’s stochastic volatility model with regime switching: A saddlepoint approximation method," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-20, December.
    8. Gao, Huan & Mamon, Rogemar & Liu, Xiaoming & Tenyakov, Anton, 2015. "Mortality modelling with regime-switching for the valuation of a guaranteed annuity option," Insurance: Mathematics and Economics, Elsevier, vol. 63(C), pages 108-120.
    9. Jerome Detemple & Yerkin Kitapbayev, 2016. "On American VIX options under the generalized 3/2 and 1/2 models," Papers 1606.00530,, revised Jul 2017.
    10. Jiling Cao & Teh Raihana Nazirah Roslan & Wenjun Zhang, 2016. "Pricing variance swaps in a hybrid model of stochastic volatility and interest rate with regime-switching," Papers 1603.08289,
    11. Lux, Thomas, 2013. "Exact solutions for the transient densities of continuous-time Markov switching models: With an application to the poisson multifractal model," Kiel Working Papers 1871, Kiel Institute for the World Economy (IfW).
    12. Lian, Guanghua & Chiarella, Carl & Kalev, Petko S., 2014. "Volatility swaps and volatility options on discretely sampled realized variance," Journal of Economic Dynamics and Control, Elsevier, vol. 47(C), pages 239-262.
    13. Calvet, Laurent E. & Fearnley, Marcus & Fisher, Adlai J. & Leippold, Markus, 2015. "What is beneath the surface? Option pricing with multifrequency latent states," Journal of Econometrics, Elsevier, vol. 187(2), pages 498-511.
    14. Siu, Tak Kuen, 2008. "A game theoretic approach to option valuation under Markovian regime-switching models," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1146-1158, June.
    15. Lorenzo Torricelli, 2012. "Valuation of asset and volatility derivatives using decoupled time-changed L\'evy processes," Papers 1210.5479,, revised Jan 2015.
    16. Stéphane Goutte & Amine Ismail & Huyên Pham, 2017. "Regime-switching Stochastic Volatility Model : Estimation and Calibration to VIX options," Working Papers hal-01212018, HAL.
    17. Marcos Escobar & Daniela Neykova & Rudi Zagst, 2015. "Portfolio Optimization In Affine Models With Markov Switching," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-46.
    18. Ben-zhang Yang & Jia Yue & Nan-jing Huang, 2017. "Variance swaps under L\'{e}vy process with stochastic volatility and stochastic interest rate in incomplete markets," Papers 1712.10105,, revised Mar 2018.
    19. Andrew Papanicolaou & Ronnie Sircar, 2014. "A regime-switching Heston model for VIX and S&P 500 implied volatilities," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1811-1827, October.
    20. repec:eee:apmaco:v:250:y:2015:i:c:p:920-933 is not listed on IDEAS
    21. repec:eee:apmaco:v:277:y:2016:i:c:p:72-81 is not listed on IDEAS
    22. Leunglung Chan & Eckhard Platen, 2010. "Exact Pricing and Hedging Formulas of Long Dated Variance Swaps under a $3/2$ Volatility Model," Papers 1007.2968,, revised Jan 2011.


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