Asia-Pacific Financial Markets

, Volume 22, Issue 2, pp 133–149 | Cite as

Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment

  • Robert J. Elliott
  • Tak Kuen Siu


By utilizing information about prices and trading volumes, we discuss the pricing of European contingent claims in a continuous-time hidden regime-switching environment. Hidden market sentiments described by the states of a continuous-time, finite-state, hidden Markov chain represent a common factor for an asset’s drift and volatility, as well as its trading volumes. Using observations about trading volumes, we present a filtered estimate of the hidden common factor. The asset pricing problem is then considered in a filtered market, where the hidden drift and volatility are replaced by their filtered estimates. We adopt the Esscher transform to select an equivalent martingale measure for pricing and derive a partial-differential integral equation for the option price.


Asset pricing Trading volumes Hidden Markov models Filtering Esscher transform PDIE 


  1. Andersen, T. G. (1996). Return volatility and trading volume: An information flow interpretation of stochastic volatility. Journal of Finance, 51, 169–204.CrossRefGoogle Scholar
  2. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.CrossRefGoogle Scholar
  3. Buffington, J., & Elliott, R. J. (2001). American options with regime switching. International Journal of Theoretical and Applied Finance, 5, 497– 514.Google Scholar
  4. Bühlmann, H., Delbaen, F., Embrechts, P., & Shiryaev, A. N. (1996). No-arbitrage, change of measure and conditional esscher transform. CWI Quarterly, 9(4), 291–317.Google Scholar
  5. Bühlmann, H., Delbaen, F., Embrechts, P., & Shiryaev, A. (1998). On Esscher transforms in discrete finance models. ASTIN Bulletin, 28, 171–186.CrossRefGoogle Scholar
  6. Clark, P. K. (1973). A subordinated stochastic process with finite variance for speculative prices. Econometrica, 41, 135–156.CrossRefGoogle Scholar
  7. Clark, J. M. C. (1978). The design of robust approximations to the stochastic differential equations for nonlinear filtering. In J. K. Skwirzynski (Ed.), Communications systems and random process theory (pp. 721–734). The Netherlands: Sijthoff and Noorhoff.CrossRefGoogle Scholar
  8. Elliott, R. J., Aggoun, L., & Moore, J. B. (1994). Hidden Markov models: Estimation and control. Berlin: Springer.Google Scholar
  9. Elliott, R. J., & Malcolm, W. P. (2005). General smoothing formulas for Markov-modulated poisson observations. IEEE Transactions on Automatic Control, 50(8), 1123–1134.CrossRefGoogle Scholar
  10. Elliott, R. J., Chan, L., & Siu, T. K. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance, 1(4), 423–432.CrossRefGoogle Scholar
  11. Elliott, R. J., & Siu, T. K. (2013) Option pricing and filtering with hidden markov-modulated pure jump processes. Applied Mathematical Finance, 20(1), 1–25.Google Scholar
  12. Epps, T. W. (1975). Security price changes and transaction volumes: Theory and evidence. American Economic Review, 65, 586–597.Google Scholar
  13. Epps, T. W., & Epps, M. L. (1976). The stochastic dependence of security price changes and transaction volumes: Implications for the mixture-of-distributions hypothesis. Econometrica, 44, 305–321.CrossRefGoogle Scholar
  14. Gallant, A. R., Rossi, P. E., & Tauchen, G. (1992). Stock prices and volume. Review of Financial Studies, 5, 199–242.CrossRefGoogle Scholar
  15. Gerber, H. U., & Shiu, E. S. W. (1994). Option pricing by Esscher transforms (with discussions). Transactions of the Society of Actuaries, 46(6), 99–191.Google Scholar
  16. Lo, A., & Wang, J. (2006). Trading volume: Implications of an intertemporal capital asset pricing model. Journal of Finance, 61, 2805–2840.Google Scholar
  17. Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20, 381–408.CrossRefGoogle Scholar
  18. Harrison, J. M., & Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Their Applications, 11, 215–280.CrossRefGoogle Scholar
  19. Harrison, J. M., & Pliska, S. R. (1993). A stochastic calculus model of continuous trading: Complete markets. Stochastic Processes and Their Applications, 15, 313–316.CrossRefGoogle Scholar
  20. Guo, X. (2001) Information and option pricings. Quantitative Finance 1, 38–44.Google Scholar
  21. Merton, R. C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183.CrossRefGoogle Scholar
  22. Ono, S. (2006). Option pricing under stochastic volatility and trading volume. Preprint. University of York.Google Scholar
  23. Shalen, C. T. (1993). Volume, volatility, and the dispersion of beliefs. Review of Financial Studies, 6, 405–434.CrossRefGoogle Scholar
  24. Siu, T. K. (2008). A game theoretic approach to option valuation under Markovian regime-switching models. Insurance: Mathematics and Economics, 42(3), 1146–1158.Google Scholar
  25. Siu, T. K. (2011). Regime switching risk: To price or not to price? International Journal of Stochastic Analysis, Article ID 843246Google Scholar
  26. Siu, T.K. (2014a). 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, pp. 185–209). New York:Springer.Google Scholar
  27. Siu, T.K. (2014b). American option pricing and filtering with a hidden regime-switching jump diffusion. Preprint.Google Scholar
  28. Tauchen, G., & Pitts, M. (1983). The price variability-volume relationship on speculative markets. Econometrica, 51, 485–505.CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  2. 2.Haskayne School of BusinessUniversity of CalgaryCalgaryCanada
  3. 3.Centre for Applied FinanceUniversity of South AustraliaAdelaideAustralia
  4. 4.Department of Applied Finance and Actuarial Studies, Faculty of Business and EconomicsMacquarie UniversitySydneyAustralia
  5. 5.Cass Business SchoolCity University LondonLondonUK

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