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Option Pricing When the Regime-Switching Risk is Priced

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  • Tak Kuen Siu
  • Hailiang Yang Unim
  • John W Lau

Abstract

Recently, there has been considerable interest in investigating option valuation problem in the context of regime-switching models. However, most of the literature consider the case that the risk due to switching regimes is not priced. Relatively little attention has been paid to investigate the impact of switching regimes on the option price when this source of risk is priced. In this paper, we shall articulate this important problem and consider the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset switch over time according to the state of an economy, which is modeled by a continuous-time hidden Markov chain. We shall develop a two-stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. The latter is called a min-max entropy problem. We shall conduct numerical experiments to illustrate the effect of pricing regime-switching risk. The results of the numerical experiments reveal that the impact of pricing regime-switching risk on the option prices is significant.

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Bibliographic Info

Paper provided by Centre for Research into Industry, Enterprise, Finance and the Firm in its series CRIEFF Discussion Papers with number 0713.

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Date of creation: Nov 2007
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Handle: RePEc:san:crieff:0713

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Keywords: Option valuation; Regime-switching risk; Two-stage pricing procedure; Esscher trans- form; Martingale restriction; Min-max entropy problem.;

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References

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  1. Robert Elliott & Carlton-James Osakwe, 2006. "Option Pricing for Pure Jump Processes with Markov Switching Compensators," Finance and Stochastics, Springer, vol. 10(2), pages 250-275, April.
  2. X. Guo, 2001. "Information and option pricings," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 38-44.
  3. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
  4. Robert J. Elliott & John van der Hoek, 1997. "An application of hidden Markov models to asset allocation problems (*)," Finance and Stochastics, Springer, vol. 1(3), pages 229-238.
  5. Elliott, R. J. & Malcolm, W. P. & Tsoi, Allanus H., 2003. "Robust parameter estimation for asset price models with Markov modulated volatilities," Journal of Economic Dynamics and Control, Elsevier, vol. 27(8), pages 1391-1409, June.
  6. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-84, March.
  7. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  8. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  9. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52.
  10. Robert J. Elliott & Leunglung Chan & Tak Kuen Siu, 2005. "Option pricing and Esscher transform under regime switching," Annals of Finance, Springer, vol. 1(4), pages 423-432, October.
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