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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 |
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.
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. Google Scholar |
[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 |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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. Google Scholar |
[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 |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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