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

Bond pricing formulas for Markov-modulated affine term structure models

Marianito R. Rodrigo 1, and  Rogemar S. Mamon 2,,

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.

Keywords: term structure modeling, affine processes, Markov chain, regime-switching approach, bond valuation.
Mathematics Subject Classification: Primary: 91B28, 35A22; Secondary: 65R20, 35K05.
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. 

show all references

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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