# American Institute of Mathematical Sciences

September  2021, 17(5): 2685-2702. doi: 10.3934/jimo.2020089

## Bond pricing formulas for Markov-modulated affine term structure models

 1 School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales, Australia 2 Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada, and, Division of Physical Sciences and Mathematics, University of the Philippines Visayas, Miag-ao, Iloilo, Philippines

* Corresponding author: Rogemar S. Mamon, Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada. E-mail: rmamon@stats.uwo.ca

Received  October 2019 Revised  February 2020 Published  September 2021 Early access  April 2020

This article provides new developments in characterizing the class of regime-switching exponential affine interest rate processes in the context of pricing a zero-coupon bond. A finite-state Markov chain in continuous time dictates the random switching of time-dependent parameters of such processes. We present exact and approximate bond pricing formulas by solving a system of partial differential equations and minimizing an error functional. The bond price expression exhibits a representation that shows how it is explicitly impacted by the rate matrix and the time-dependent coefficient functions of the short rate models. We validate the bond pricing formulas numerically by examining a regime-switching Vasicek model.

Citation: Marianito R. Rodrigo, Rogemar S. Mamon. Bond pricing formulas for Markov-modulated affine term structure models. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2685-2702. doi: 10.3934/jimo.2020089
##### References:
 [1] J.-M. Beacco, C. Lubochincky, M. Brière, A. Monfort and C. Hillairet, The challenges imposed by low interest rates, Journal of Asset Management, 20 (2019), 413-420. [2] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [3] R. Criego and A. Swishchuk, A Black-Scholes formula for a market in a random environment, Theory of Probability and Mathematical Statistics, 62 (2000), 9-18. [4] C. Cuchiero, D. Filipović, E. Mayerhofer and J. Teichmann, Affine processes on positive semidefinite matrices, Annals of Applied Probability, 21 (2011), 397-463.  doi: 10.1214/10-AAP710. [5] D. Duffie, D. Filipović and W. Schachermayer, Affine processes and applications in finance, Annals of Applied Probability, 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833. [6] D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance, 6 (1996), 379-406.  doi: 10.1111/j.1467-9965.1996.tb00123.x. [7] Z. Eksi and D. Filipović, Pricing and hedging of inflation-indexed bonds in an affine framework, Journal of Computational and Applied Mathematics, 259 (2014), 452-463.  doi: 10.1016/j.cam.2013.10.023. [8] R. Elliott, Stochastic Calculus and Applications, Applications of Mathematics 18, Springer-Verlag, Berlin-Heidelberg-New York, 1982. [9] R. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics 29, Springer-Verlag, Berlin-Heidelberg-New York, 1995. [10] R. Elliott, P. Fischer and E. Platen, Filtering and parameter estimation for a mean-reverting interest-rate model, Canadian Applied Mathematics Quarterly, 7 (1999), 381-400. [11] R. Elliott and R. Mamon, An interest rate model with a Markovian mean-reverting level, Quantitative Finance, 2 (2002), 454-458.  doi: 10.1080/14697688.2002.0000012. [12] R. Elliott and T. Siu, On Markov-modulated exponential-affine bond price formulae, Applied Mathematical Finance, 16 (2009), 1-15.  doi: 10.1080/13504860802015744. [13] R. Elliott, T. Siu and A. Badescu, Bond valuation under a discrete-time regime-switching term structure model and its continuous-time extension, Managerial Finance, 37 (2011), 1025-1047.  doi: 10.1108/03074351111167929. [14] R. Elliott and J. van der Hoek, Stochastic flows and the forward measure, Finance and Stochastics, 5 (2011), 511-525.  doi: 10.1007/s007800000039. [15] R. Elliott and C. Wilson, The term structure of interest rates in a hidden Markov setting, in Hidden Markov Models in Finance (eds. R. Mamon and R. Elliott), Springer, New York, 104 (2007), 15–30. doi: 10.1007/0-387-71163-5_2. [16] C. Erlwein and R. Mamon, An on-line estimation scheme for a Hull-White model with HMM-driven parameters, Statistical Methods and Applications, 18 (2009), 87-107.  doi: 10.1007/s10260-007-0082-4. [17] K. Fan, Y. Shen, T. Siu and R. Wang, Pricing dynamic fund protection under hidden Markov models, IMA Journal of Management Mathematics, 29 (2018), 99-117.  doi: 10.1093/imaman/dpw014. [18] D. Filipovi, Time-inhomogeneous affine processes, Stochastic Processes and Their Applications, 115 (2005), 639–659. doi: 10.1016/j.spa.2004.11.006. [19] H. Gao, R. Mamon and X. Liu, Pricing a guaranteed annuity option under correlated and regime-switching risk factors, European Actuarial Journal, 5 (2015), 309-326.  doi: 10.1007/s13385-015-0118-3. [20] H. Gao, R. Mamon, X. Liu and A. Tenyakov, Mortality modelling with regime-switching for the valuation of a guaranteed annuity option, Insurance: Mathematics and Economics, 63 (2015), 108-120.  doi: 10.1016/j.insmatheco.2015.03.018. [21] L. Gonon and J. Teichmann, Linearised filtering of affine processes using stochastic Ricatti equations, Stochastic Processes and their Applications, 130 (2020), 394-430.  doi: 10.1016/j.spa.2019.03.016. [22] S. Grimm, C. Erlwein-Sayer and R. Mamon, Discrete-time implementation of continuous-time filters with applications to regime-switching dynamics estimation, Nonlinear Analysis: Hybrid Systems, 35 (2020), 100814, 20 pp. doi: 10.1016/j.nahs.2019.08.001. [23] J. Hlouskova and L. Sögner, GMM estimation of affine term structure models, Econometrics and Statistics, 13 (2020), 2-15.  doi: 10.1016/j.ecosta.2019.10.001. [24] J. Hull and A. White, Numerical procedures for implementing term structure models II: Two factor models, Journal of Derivatives, 2 (1994), 37-48.  doi: 10.3905/jod.1994.407908. [25] C. Landén, Bond pricing in a hidden Markov model of the short rate, Finance and Stochastics, 4 (2000), 371-389.  doi: 10.1007/PL00013526. [26] G. Last and A. Brandt, Marked Point Processes on the Real Line: The Dynamical Approach, Springer-Verlag, New York, 1995. [27] R. Mamon, On the interface of probabilistic and PDE methods in a multi-factor term structure theory, International Journal of Mathematical Education in Science and Technology, 35 (2004), 661-668.  doi: 10.1080/00207390410001714902. [28] M. R. Rodrigo and R. S. Mamon, A unified approach to explicit bond price solutions under a time-dependent affine term structure modelling framework, Quantitative Finance, 11 (2011), 487-493.  doi: 10.1080/14697680903341798. [29] M. R. Rodrigo and R. S. Mamon, An alternative approach to the calibration of the Vasicek and CIR interest rate models via generating functions, Quantitative Finance, 14 (2014), 1961-1970.  doi: 10.1080/14697688.2013.765062. [30] K. Singleton, Empirical Dynamic Asset Pricing: Model Specification and Econometric Assessment, Princeton University Press, Princeton, 2006. [31] A. Tenyakov, R. Mamon and M. Davison, Filtering of a discrete-time HMM-driven multivariate Ornstein-Uhlenbeck model with application to forecasting market liquidity regimes, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 994-1005.  doi: 10.1109/JSTSP.2016.2549499. [32] O. Vasicek, An equilibrium characterisation of the term structure, Journal of Financial Economics, 5 (1977), 177-188. [33] M. van Beek, M. Mandjes, P. Spreij and E. Winands, Markov switching affine processes and applications to pricing, Actuarial and Financial Mathematics Conference, Interplay between Finance and Insurance: February 6–7, 2014 (eds. M. Vanmaele, G. Deelstra, A. De Schepper, J. Dhaene, W. Schoutens, S. Vanduffel and D. Vyncke), Brussels, België: Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten, (2014), 97–102. [34] S. Wu and Y. Zeng, An econometric model of the term structure of interest rates under regime-switching risk, Hidden Markov Models in Finance: Further Developments and Applications (eds. R. Mamon and R. Elliott), Springer, New York, 209 (2014), 55–83. doi: 10.1007/978-1-4899-7442-6_3. [35] X. Xi and R. S. Mamon, Capturing the regime-switching and memory properties of interest rates, Computational Economics, 44 (2014), 307-337.  doi: 10.1007/s10614-013-9396-5. [36] X. Xi, M. R. Rodrigo and R. S. Mamon, Parameter estimation of a regime-switching model using an inverse Stieltjes moment approach, Stochastic Processes, Finance and Control (eds. S. Cohen, D. Madan, T. Siu and H. Yang), World Scientific, Singapore, 1 (2012), 549–569. doi: 10.1142/9789814383318_0022. [37] Y. Zhao and R. Mamon, Annuity contract valuation under dependent risks, Japan Journal of Industrial and Applied Mathematics, 37 (2020), 1-23.  doi: 10.1007/s13160-019-00366-2. [38] Y. Zhao, R. Mamon and H. Gao, A two-decrement model for the valuation and risk measurement of a guaranteed annuity option, Econometrics and Statistics, 8 (2018), 231-249.  doi: 10.1016/j.ecosta.2018.06.004. [39] N. Zhou and R. Mamon, An accessible implementation of interest rate models with regime-switching, Expert Systems with Applications, 9 (2012), 4679-4689. [40] D.-M. Zhu, J. Lu, W.-K. Ching and T.-K. Siu, Option pricing under a stochastic interest rate and volatility model with hidden Markovian regime-switching, Computational Economics, 53 (2019), 555-586.

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##### References:
 [1] J.-M. Beacco, C. Lubochincky, M. Brière, A. Monfort and C. Hillairet, The challenges imposed by low interest rates, Journal of Asset Management, 20 (2019), 413-420. [2] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [3] R. Criego and A. Swishchuk, A Black-Scholes formula for a market in a random environment, Theory of Probability and Mathematical Statistics, 62 (2000), 9-18. [4] C. Cuchiero, D. Filipović, E. Mayerhofer and J. Teichmann, Affine processes on positive semidefinite matrices, Annals of Applied Probability, 21 (2011), 397-463.  doi: 10.1214/10-AAP710. [5] D. Duffie, D. Filipović and W. Schachermayer, Affine processes and applications in finance, Annals of Applied Probability, 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833. [6] D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance, 6 (1996), 379-406.  doi: 10.1111/j.1467-9965.1996.tb00123.x. [7] Z. Eksi and D. Filipović, Pricing and hedging of inflation-indexed bonds in an affine framework, Journal of Computational and Applied Mathematics, 259 (2014), 452-463.  doi: 10.1016/j.cam.2013.10.023. [8] R. Elliott, Stochastic Calculus and Applications, Applications of Mathematics 18, Springer-Verlag, Berlin-Heidelberg-New York, 1982. [9] R. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics 29, Springer-Verlag, Berlin-Heidelberg-New York, 1995. [10] R. Elliott, P. Fischer and E. Platen, Filtering and parameter estimation for a mean-reverting interest-rate model, Canadian Applied Mathematics Quarterly, 7 (1999), 381-400. [11] R. Elliott and R. Mamon, An interest rate model with a Markovian mean-reverting level, Quantitative Finance, 2 (2002), 454-458.  doi: 10.1080/14697688.2002.0000012. [12] R. Elliott and T. Siu, On Markov-modulated exponential-affine bond price formulae, Applied Mathematical Finance, 16 (2009), 1-15.  doi: 10.1080/13504860802015744. [13] R. Elliott, T. Siu and A. Badescu, Bond valuation under a discrete-time regime-switching term structure model and its continuous-time extension, Managerial Finance, 37 (2011), 1025-1047.  doi: 10.1108/03074351111167929. [14] R. Elliott and J. van der Hoek, Stochastic flows and the forward measure, Finance and Stochastics, 5 (2011), 511-525.  doi: 10.1007/s007800000039. [15] R. Elliott and C. Wilson, The term structure of interest rates in a hidden Markov setting, in Hidden Markov Models in Finance (eds. R. Mamon and R. Elliott), Springer, New York, 104 (2007), 15–30. doi: 10.1007/0-387-71163-5_2. [16] C. Erlwein and R. Mamon, An on-line estimation scheme for a Hull-White model with HMM-driven parameters, Statistical Methods and Applications, 18 (2009), 87-107.  doi: 10.1007/s10260-007-0082-4. [17] K. Fan, Y. Shen, T. Siu and R. Wang, Pricing dynamic fund protection under hidden Markov models, IMA Journal of Management Mathematics, 29 (2018), 99-117.  doi: 10.1093/imaman/dpw014. [18] D. Filipovi, Time-inhomogeneous affine processes, Stochastic Processes and Their Applications, 115 (2005), 639–659. doi: 10.1016/j.spa.2004.11.006. [19] H. Gao, R. Mamon and X. Liu, Pricing a guaranteed annuity option under correlated and regime-switching risk factors, European Actuarial Journal, 5 (2015), 309-326.  doi: 10.1007/s13385-015-0118-3. [20] H. Gao, R. Mamon, X. Liu and A. Tenyakov, Mortality modelling with regime-switching for the valuation of a guaranteed annuity option, Insurance: Mathematics and Economics, 63 (2015), 108-120.  doi: 10.1016/j.insmatheco.2015.03.018. [21] L. Gonon and J. Teichmann, Linearised filtering of affine processes using stochastic Ricatti equations, Stochastic Processes and their Applications, 130 (2020), 394-430.  doi: 10.1016/j.spa.2019.03.016. [22] S. Grimm, C. Erlwein-Sayer and R. Mamon, Discrete-time implementation of continuous-time filters with applications to regime-switching dynamics estimation, Nonlinear Analysis: Hybrid Systems, 35 (2020), 100814, 20 pp. doi: 10.1016/j.nahs.2019.08.001. [23] J. Hlouskova and L. Sögner, GMM estimation of affine term structure models, Econometrics and Statistics, 13 (2020), 2-15.  doi: 10.1016/j.ecosta.2019.10.001. [24] J. Hull and A. White, Numerical procedures for implementing term structure models II: Two factor models, Journal of Derivatives, 2 (1994), 37-48.  doi: 10.3905/jod.1994.407908. [25] C. Landén, Bond pricing in a hidden Markov model of the short rate, Finance and Stochastics, 4 (2000), 371-389.  doi: 10.1007/PL00013526. [26] G. Last and A. Brandt, Marked Point Processes on the Real Line: The Dynamical Approach, Springer-Verlag, New York, 1995. [27] R. Mamon, On the interface of probabilistic and PDE methods in a multi-factor term structure theory, International Journal of Mathematical Education in Science and Technology, 35 (2004), 661-668.  doi: 10.1080/00207390410001714902. [28] M. R. Rodrigo and R. S. Mamon, A unified approach to explicit bond price solutions under a time-dependent affine term structure modelling framework, Quantitative Finance, 11 (2011), 487-493.  doi: 10.1080/14697680903341798. [29] M. R. Rodrigo and R. S. Mamon, An alternative approach to the calibration of the Vasicek and CIR interest rate models via generating functions, Quantitative Finance, 14 (2014), 1961-1970.  doi: 10.1080/14697688.2013.765062. [30] K. Singleton, Empirical Dynamic Asset Pricing: Model Specification and Econometric Assessment, Princeton University Press, Princeton, 2006. [31] A. Tenyakov, R. Mamon and M. Davison, Filtering of a discrete-time HMM-driven multivariate Ornstein-Uhlenbeck model with application to forecasting market liquidity regimes, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 994-1005.  doi: 10.1109/JSTSP.2016.2549499. [32] O. Vasicek, An equilibrium characterisation of the term structure, Journal of Financial Economics, 5 (1977), 177-188. [33] M. van Beek, M. Mandjes, P. Spreij and E. Winands, Markov switching affine processes and applications to pricing, Actuarial and Financial Mathematics Conference, Interplay between Finance and Insurance: February 6–7, 2014 (eds. M. Vanmaele, G. Deelstra, A. De Schepper, J. Dhaene, W. Schoutens, S. Vanduffel and D. Vyncke), Brussels, België: Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten, (2014), 97–102. [34] S. Wu and Y. Zeng, An econometric model of the term structure of interest rates under regime-switching risk, Hidden Markov Models in Finance: Further Developments and Applications (eds. R. Mamon and R. Elliott), Springer, New York, 209 (2014), 55–83. doi: 10.1007/978-1-4899-7442-6_3. [35] X. Xi and R. S. Mamon, Capturing the regime-switching and memory properties of interest rates, Computational Economics, 44 (2014), 307-337.  doi: 10.1007/s10614-013-9396-5. [36] X. Xi, M. R. Rodrigo and R. S. Mamon, Parameter estimation of a regime-switching model using an inverse Stieltjes moment approach, Stochastic Processes, Finance and Control (eds. S. Cohen, D. Madan, T. Siu and H. Yang), World Scientific, Singapore, 1 (2012), 549–569. doi: 10.1142/9789814383318_0022. [37] Y. Zhao and R. Mamon, Annuity contract valuation under dependent risks, Japan Journal of Industrial and Applied Mathematics, 37 (2020), 1-23.  doi: 10.1007/s13160-019-00366-2. [38] Y. Zhao, R. Mamon and H. Gao, A two-decrement model for the valuation and risk measurement of a guaranteed annuity option, Econometrics and Statistics, 8 (2018), 231-249.  doi: 10.1016/j.ecosta.2018.06.004. [39] N. Zhou and R. Mamon, An accessible implementation of interest rate models with regime-switching, Expert Systems with Applications, 9 (2012), 4679-4689. [40] D.-M. Zhu, J. Lu, W.-K. Ching and T.-K. Siu, Option pricing under a stochastic interest rate and volatility model with hidden Markovian regime-switching, Computational Economics, 53 (2019), 555-586.
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