Skip to main content

Continuous-Time Markov Chain and Regime Switching Approximations with Applications to Options Pricing

Part of the The IMA Volumes in Mathematics and its Applications book series (IMA,volume 164)

Abstract

In this chapter, we present recent developments in using the tools of continuous-time Markov chains for the valuation of European and path-dependent financial derivatives. We also survey results on a newly proposed regime switching approximation to stochastic volatility, and stochastic local volatility models. The presented framework is part of an exciting recent stream of literature on numerical option pricing, and offers a new perspective that combines the theory of diffusion processes, Markov chains, and Fourier techniques. It is also elegantly connected to partial differential equation (PDE) approaches.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-25498-8_6
  • Chapter length: 32 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   139.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-25498-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   179.99
Price excludes VAT (USA)
Hardcover Book
USD   179.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abate, Joseph, and Ward Whitt. ”The Fourier-series method for inverting transforms of probability distributions.” Queueing Systems 10.1-2 (1992): 5-87.

    MathSciNet  MATH  CrossRef  Google Scholar 

  2. Ackerer, Damien, Damir Filipovic, and Sergio Pulido. “The Jacobi stochastic volatility model.” Finance and Stochastics (2017): 1-34.

    Google Scholar 

  3. Ait-Sahalia, Yacine. “Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach.” Econometrica 70, no. 1 (2002): 223-262.

    MathSciNet  MATH  CrossRef  Google Scholar 

  4. Ang, Andrew, and Geert Bekaert. “Regime switches in interest rates.” Journal of Business & Economic Statistics 20, no. 2 (2002): 163-182.

    Google Scholar 

  5. Antonov, Alexandre, Michael Konikov, and Michael Spector. “The free boundary SABR: natural extension to negative rates.” Preprint, ssrn 2557046 (2015).

    Google Scholar 

  6. Bangia, Anil, Francis X. Diebold, Andr Kronimus, Christian Schagen, and Til Schuermann. “Ratings migration and the business cycle, with application to credit portfolio stress testing.” Journal of Banking and Finance 26, no. 2-3 (2002): 445-474.

    CrossRef  Google Scholar 

  7. Buffington, John, and Robert J. Elliott. “American options with regime switching.” International Journal of Theoretical and Applied Finance 5, no. 05 (2002): 497-514.

    MathSciNet  MATH  CrossRef  Google Scholar 

  8. Cai, Ning, Yingda Song, and Steven Kou. ”A general framework for pricing Asian options under Markov processes.” Operations Research 63, no. 3 (2015): 540-554.

    MathSciNet  MATH  CrossRef  Google Scholar 

  9. Chourdakis, Kyriakos, “Continuous Time Regime Switching Models and Applications in Estimating Processes with Stochastic Volatility and Jumps (November 2002)”. U of London Queen Mary Economics Working Paper No. 464. Available at SSRN: https://ssrn.com/abstract=358244 or http://dx.doi.org/10.2139/ssrn.358244

  10. Chatterjee, Rupak, Zhenyu Cui, Jiacheng Fan, and Mingzhe Liu. “An efficient and stable method for short maturity Asian options.” Journal of Futures Markets 38 (12) (2018): 1470-1486.

    CrossRef  Google Scholar 

  11. Corsaro, Stefania, Ioannis Kyriakou, Daniele Marazzina, and Zelda Marino. “A general framework for pricing Asian options under stochastic volatility on parallel architectures.” European Journal of Operational Research, 272(3) (2019): 1082-1095.

    MathSciNet  MATH  CrossRef  Google Scholar 

  12. Cui, Z, J. Lars Kirkby, and Duy Nguyen. “Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps.” Insurance: Mathematics and Economics 74 (2017): 46-62.

    MathSciNet  MATH  Google Scholar 

  13. Cui, Z, J. Lars Kirkby, and Duy Nguyen. “A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps.” European Journal of Operational Research 262(1) (2017): 381-400.

    MathSciNet  MATH  CrossRef  Google Scholar 

  14. Cui, Z, J. Lars Kirkby, and Nguyen, Duy. “A general valuation framework for SABR and stochastic local volatility models.” SIAM Journal on Financial Mathematics 9(2) (2018): 520-563.

    MathSciNet  MATH  CrossRef  Google Scholar 

  15. Cui, Z., J. Lars Kirkby and Nguyen, Duy. “A general framework time-changed Markov processes and applications.” European Journal of Operational Research, 273(2) (2018):785-800.

    Google Scholar 

  16. Cui, Z., J. Lars Kirkby and Nguyen, Duy. “Full-fledged SABR through Markov Chains.” Working paper (2017).

    Google Scholar 

  17. Cui, Z., J. Lars Kirkby and Nguyen, Duy. “Efficient simulation of stochastic differential equations based on Markov Chain approximations with applications.” Working paper (2018).

    Google Scholar 

  18. Cui, Z, C. Lee, and Y. Liu. “Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes.” European Journal of Operational Research 266, no. 3 (2018): 1134-1139.

    MathSciNet  MATH  CrossRef  Google Scholar 

  19. Duan, Jin-Chuan, and Jean-Guy Simonato. “American option pricing under GARCH by a Markov chain approximation.” Journal of Economic Dynamics and Control 25, no. 11 (2001): 1689-1718.

    MathSciNet  MATH  CrossRef  Google Scholar 

  20. Duan, Jin-Chuan, Evan Dudley, Genevive Gauthier, and J. Simonato. “Pricing discretely monitored barrier options by a Markov chain.” Journal of Derivatives 10 (2003).

    CrossRef  Google Scholar 

  21. Duffie, Darrell, Jun Pan, and Kenneth Singleton. “Transform analysis and asset pricing for affine jump diffusions.” Econometrica 68, no. 6 (2000): 1343-1376.

    MathSciNet  MATH  CrossRef  Google Scholar 

  22. Durham, Garland B., and A. Ronald Gallant. “Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes.” Journal of Business & Economic Statistics 20, no. 3 (2002): 297-338.

    MathSciNet  CrossRef  Google Scholar 

  23. Da Fonseca, Jose, and Claude Martini. “The α-hypergeometric stochastic volatility model.” Stochastic Processes and their Applications 126.5 (2016): 1472-1502.

    Google Scholar 

  24. Ethier, Stewart N., and Thomas G. Kurtz. Markov processes: characterization and convergence. Vol. 282. John Wiley & Sons, (2009).

    Google Scholar 

  25. Fusai, Gianluca, and Ioannis Kyriakou. “General optimized lower and upper bounds for discrete and continuous arithmetic Asian options.” Mathematics of Operations Research 41, no. 2 (2016): 531-559.

    MathSciNet  MATH  CrossRef  Google Scholar 

  26. Gihman, Iosif IlIch, and Anatoli Vladimirovich Skorohod. ”Stochastic differential equations.” The Theory of Stochastic Processes III. Springer, New York, NY, 1979. 113-219.

    MATH  CrossRef  Google Scholar 

  27. Grasselli, Martino. “The 4/2 stochastic volatility model: A unified approach for the Heston and the 3/2 model.” Mathematical Finance 27.4 (2017): 1013-1034.

    MathSciNet  MATH  CrossRef  Google Scholar 

  28. Hagan, Patrick S., et al. ”Managing smile risk.” The Best of Wilmott 1 (2002): 249-296.

    Google Scholar 

  29. Hamilton, James D. “Analysis of time series subject to changes in regime.” Journal of Econometrics 45, no. 1-2 (1990): 39-70.

    MathSciNet  MATH  CrossRef  Google Scholar 

  30. Henry-Labordere, Pierre, “A General Asymptotic Implied Volatility for Stochastic Volatility Models (April 2005)”. Available at SSRN: https://ssrn.com/abstract=698601 or http://dx.doi.org/10.2139/ssrn.698601.

  31. Heston, Steven L. “A closed-form solution for options with stochastic volatility with applications to bond and currency options”. The Review of Financial Studies 6.2 (1993): 327-343.

    Google Scholar 

  32. Hull, John, and Alan White. “The pricing of options on assets with stochastic volatilities.” The Journal of Finance 42.2 (1987): 281-300.

    MATH  CrossRef  Google Scholar 

  33. Ikeda, Nobuyuki, and Shinzo Watanabe. “Stochastic differential equations and diffusion processes”. Vol. 24. Elsevier, (2014).

    Google Scholar 

  34. Jiang, Jiuxin, R. H. Liu, and D. Nguyen. “A recombining tree method for option pricing with state-dependent switching rates.” International Journal of Theoretical and Applied Finance 19.02 (2016): 1650012.

    MathSciNet  MATH  CrossRef  Google Scholar 

  35. Higham, Desmond J., Xuerong Mao, and Andrew M. Stuart. “Strong convergence of Eulertype methods for nonlinear stochastic differential equations.” SIAM Journal on Numerical Analysis 40, no. 3 (2002): 1041-1063.

    MathSciNet  MATH  CrossRef  Google Scholar 

  36. Jacod, Jean, and Philip Protter. “Discretization of processes”. Vol. 67. Springer Science & Business Media, 2011.

    Google Scholar 

  37. Kahale, Nabil. ”General multilevel Monte Carlo methods for pricing discretely monitored Asian options.” arXiv preprint arXiv:1805.09427 (2018).

  38. Karatzas, Ioannis, and Steven Shreve. “Brownian motion and stochastic calculus”. Vol. 113. Springer Science & Business Media, (2012).

    Google Scholar 

  39. Kim, Chang-Jin, and Charles R. Nelson. “Business cycle turning points, a new coincident index, and tests of duration dependence based on a dynamic factor model with regime switching.” Review of Economics and Statistics 80, no. 2 (1998): 188-201.

    CrossRef  Google Scholar 

  40. Kirkby, J. Lars. “Efficient Option Pricing by Frame Duality with the Fast Fourier Transform”. SIAM J. Financial Mathematics Vol. 6, no.1 (2015): 713-747.

    MathSciNet  MATH  CrossRef  Google Scholar 

  41. Kirkby, J. Lars. “An Efficient Transform Method for Asian Option Pricing”. SIAM J. Financial Mathematics Vol. 7, no.1 (2016): 845-892.

    MathSciNet  MATH  CrossRef  Google Scholar 

  42. Kirkby, J. L., and D. Nguyen. “Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models”. Working paper, (2016).

    Google Scholar 

  43. Kirkby, J. Lars, Duy Nguyen, and Zhenyu Cui. “A unified approach to Bermudan and barrier options under stochastic volatility models with jumps.” Journal of Economic Dynamics and Control 80 (2017): 75-100.

    MathSciNet  MATH  CrossRef  Google Scholar 

  44. Kushner, Harold, and Paul G. Dupuis. “Numerical methods for stochastic control problems in continuous time”. Vol. 24. Springer Science & Business Media, (2013).

    Google Scholar 

  45. Lewis, Alan L. “Option Valuation Under Stochastic Volatility II”. Finance Press, Newport Beach, CA, 2009.

    Google Scholar 

  46. Li, Chenxu, and Xiaocheng Li. “A closed-form expansion approach for pricing discretely monitored variance swaps.” Operations Research Letters 43, no. 4 (2015): 450-455.

    MathSciNet  MATH  CrossRef  Google Scholar 

  47. Li, Lingfei, and Gongqiu Zhang. “Error analysis of finite difference and Markov chain approximations for option pricing.” Mathematical Finance 28.3 (2018): 877-919.

    MathSciNet  MATH  CrossRef  Google Scholar 

  48. Lipton, A. (2002). The volatility smile problem. Risk Magazine. 15(2), 61-65.

    Google Scholar 

  49. Lo, Chia Chun, and Konstantinos Skindilias. “An improved Markov chain approximation methodology: Derivatives pricing and model calibration.” International Journal of Theoretical and Applied Finance 17.07 (2014): 1450047.

    MathSciNet  MATH  CrossRef  Google Scholar 

  50. Liu, R. H. “Regime-switching recombining tree for option pricing.” International Journal of Theoretical and Applied Finance 13.03 (2010): 479-499.

    MathSciNet  MATH  CrossRef  Google Scholar 

  51. Liu, R. H. ”A new tree method for pricing financial derivatives in a regime-switching meanreverting model.” Nonlinear Analysis: Real World Applications 13.6 (2012): 2609-2621.

    MathSciNet  MATH  CrossRef  Google Scholar 

  52. Lord, Roger, Remmert Koekkoek, and Dick Van Dijk. “A comparison of biased simulation schemes for stochastic volatility models.” Quantitative Finance 10, no. 2 (2010): 177-194.

    MathSciNet  MATH  CrossRef  Google Scholar 

  53. Ma. J, W. Yang and Z. Cui. “Convergence rate analysis for the continuous-time Markov chain approximation of occupation time derivatives and Asian option Greeks.” Working paper (2018).

    Google Scholar 

  54. Mijatovic, Aleksandar, and Martijn Pistorius. ”Continuously monitored barrier options under Markov processes.” Mathematical Finance 23 (1),1–38 (2013).

    MathSciNet  MATH  CrossRef  Google Scholar 

  55. Munk, Claus. “The Markov chain approximation approach for numerical solution of stochastic control problems: experiences from Merton’s problem.” Applied Mathematics and Computation 136, no. 1 (2003): 47-77.

    MathSciNet  MATH  CrossRef  Google Scholar 

  56. Nguyen, Duy. “A hybrid Markov chain-tree valuation framework for stochastic volatility jump diffusion models.” International Journal of Financial Engineering Vol. 05, No. 04, 1850039 (2018).

    MathSciNet  CrossRef  Google Scholar 

  57. Ramponi, Alessandro. “Fourier transform methods for regime-switching jump-diffusions and the pricing of forward starting options.” International Journal of Theoretical and Applied Finance 15.05 (2012): 1250037.

    MathSciNet  MATH  CrossRef  Google Scholar 

  58. Schoebel, Rainer, and Jianwei Zhu. “Stochastic volatility with an Ornstein-Uhlenbeck process: an extension.” Review of Finance 3, no. 1 (1999): 23-46.

    MATH  CrossRef  Google Scholar 

  59. Scott, Louis O. “Option pricing when the variance changes randomly: Theory, estimation, and an application.” Journal of Financial and Quantitative analysis 22.4 (1987): 419-438.

    MathSciNet  CrossRef  Google Scholar 

  60. Song, Yingda, Ning Cai, and Steven Kou. “A Unified Framework for Options Pricing Under Regime Switching Models.” Working paper (2016).

    Google Scholar 

  61. Song, Yingda, Ning Cai, and Steven Kou. “Computable Error Bounds of Laplace Inversion for Pricing Asian Options.” INFORMS Journal on Computing 30.4 (2018): 634-645.

    MathSciNet  CrossRef  Google Scholar 

  62. Stein, Elias M., and Jeremy C. Stein. “Stock price distributions with stochastic volatility: an analytic approach.”The review of financial studies 4.4 (1991): 727-752.

    MATH  CrossRef  Google Scholar 

  63. Tavella, Domingo, and Curt Randall. Pricing Financial Instruments: The Finite Difference Method (Wiley Series in Financial Engineering). New York: Wiley, 2000.

    Google Scholar 

  64. Van der Stoep, Anthonie W., Lech A. Grzelak, and Cornelis W. Oosterlee. “The Heston stochastic-local volatility model: Efficient Monte Carlo simulation.” International Journal of Theoretical and Applied Finance 17.07 (2014): 1450045.

    MathSciNet  MATH  CrossRef  Google Scholar 

  65. Yao, David D., Qing Zhang, and Xun Yu Zhou. “A regime-switching model for European options.” Stochastic processes, optimization, and control theory: applications in financial engineering, queueing networks, and manufacturing systems. Springer, Boston, MA, 2006. 281-300.

    Google Scholar 

  66. Yin, G. George, and Qing Zhang. Continuous-time Markov chains and applications: A twotime-scale approach. Vol. 37. Springer Science & Business Media, 2012.

    Google Scholar 

  67. Yin, G. George, and Qing Zhang. Discrete-time Markov chains: two-time-scale methods and applications. Vol. 55. Springer Science & Business Media, 2006.

    Google Scholar 

  68. Yin, George, and Chao Zhu. Hybrid switching diffusions: properties and applications. Vol. 63. New York: Springer, 2010.

    Google Scholar 

  69. Yuen, Fei Lung, and Hailiang Yang. “Option pricing with regime switching by trinomial tree method.” Journal of Computational and Applied Mathematics 233.8 (2010): 1821-1833.

    MathSciNet  MATH  CrossRef  Google Scholar 

  70. Zhang Gongqiu, and Lingfei Li. “Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior.” Operations Research. Forthcoming (2018).

    Google Scholar 

  71. Zhang Gongqiu, and Lingfei Li. “A general method for the valuation of drawdown risk under Markovian models.” Working paper (2018).

    Google Scholar 

  72. Zhang Gongqiu, and Lingfei Li. “A unified approach for the analysis of Parisian stopping times and its applications in finance and insurance.” Working paper (2018).

    Google Scholar 

  73. Zhang Gongqiu, and Lingfei Li. “A general approach for the analysis of occupation times and its applications in finance.” Working paper (2018).

    Google Scholar 

  74. Zhang, Qing. “Stock trading: An optimal selling rule.” SIAM Journal on Control and Optimization 40.1 (2001): 64-87.

    MathSciNet  MATH  CrossRef  Google Scholar 

  75. Zhang, Qing, and Xin Guo. “Closed-form solutions for perpetual American put options with regime switching.” SIAM Journal on Applied Mathematics 64, no. 6 (2004): 2034-2049.

    MathSciNet  MATH  CrossRef  Google Scholar 

  76. Zhou, Xun Yu, and George Yin. “Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model.” SIAM Journal on Control and Optimization 42, no. 4 (2003): 1466-1482.

    MathSciNet  MATH  CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Duy Nguyen .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Cui, Z., Lars Kirkby, J., Nguyen, D. (2019). Continuous-Time Markov Chain and Regime Switching Approximations with Applications to Options Pricing. In: Yin, G., Zhang, Q. (eds) Modeling, Stochastic Control, Optimization, and Applications. The IMA Volumes in Mathematics and its Applications, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-030-25498-8_6

Download citation