Document Type : Research Article
Author
Faculty of Economics, University of East Sarajevo, Lukavica, Bosnia and Herzegovina.
10.22054/jmmf.2025.88162.1212
Abstract
This paper investigates multifactor affine models of the term structure of interest rates, focusing on those that admit closed-form solutions for zero-coupon bond prices. In particular, we study multifactor Vasicek and Cox–Ingersoll–Ross (CIR) models and their hybrid combinations, which integrate Gaussian and square-root dynamics within a single affine framework. By providing a unified analytical treatment, the paper clarifies the economic interpretation of model parameters and explores how they shape the spot and forward rate curves. The hybrid approach enhances the flexibility of term structure modeling, allowing one to capture level, slope, and curvature of the yield curve more accurately than single-class models. These results are directly applicable in practice for yield curve estimation, empirical calibration, risk management, and the pricing of interest rate derivatives.
Keywords
[1] Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Review of Financial
Studies, 9(2)(1996), 385-426. http://www.jstor.org/stable/2962210
[2] S.H. Babbs, B. Nowman, Kalman Filtering of Generalized Vasicek Term Structure Models,
Journal of Financial and Quantitative Analysis, 34(1999), 115-130. http://www.jstor.org/
stable/2676248
[3] F. Black, P. Karasinski, Bond and Option Pricing When Short Rates Are Lognormal, Financial
Analysts Journal, 48(4)(1991), 52-59. http://www.jstor.org/stable/4479456
[4] D.J. Bolder, Affine Term-Structure Models: Theory and Implementation, Bank of Canada,
Working Paper 2001-15, Financial Markets Department, 2001. https://www.bankofcanada.
ca/wp-content/uploads/2010/02/wp01-15a.pdf
[5] M. Brennan, E. Schwartz, An equilibrium model of bond pricing and a test of market efficiency,
Journal of Financial and Quantitative Analysis, 17(1982), 301-329. http://www.jstor.org/
stable/2330832
[6] M. Brennan, E. Schwartz, Analyzing Convertible Bonds, Journal of Financial and Quantitative
Analysis, 15(1980), 907-929. http://www.jstor.org/stable/2330567
[7] M. Brennan, E. Schwartz, The pricing of equity-linked life insurance policies with an
asset value guarantee, Journal of Financial Economics, 3(1976), 195-213. https://www.
sciencedirect.com/science/article/pii/0304405X76900039
[8] J.Y. Campbell, A defence of tradditional hyphoteses about the term structure of interest rates.,
Journal of Finance, 441(1986), 183-193. http://www.jstor.org/stable/2328351
[9] E. Canabarro, Where do One-Factor Interest Rate Models Fail?, Journal of Fixed Income,
5(2)(1995), 31-52. doi: 10.3905/jfi.1995.408145
[10] G. Chaplin, K. Sharp, Analytic solutions for bonds and bond options under n correlated
stochastic processes., Research Report 93-16, University of Waterloo, (1995)
[11] R.R. Chen, L. Scott, Interest Rate Options in Multifactor Cox-Ingersoll-Ross Models of the
Term Structure, Journal of Fixed Income, 5(1)(1995), 14–31.
[12] R.R. Chen, Interest Rate Dynamics, Derivatives Pricing, and Risk Management, Springer,
1996. Zbl 0862.90015
[13] R.R. Chen, L. Scott, Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model, The Journal of Real Estate Finance
and Economics, 27(2)(2003), 143-172. https://link.springer.com/article/10.1023/A:
1024736903090
[14] G. Courtadon, The Pricing of Options on Default-Free Bonds, Journal of Financial and
Quantitative Analysis, 17(1)(1982), 75-100. http://www.jstor.org/stable/2330930
[15] J.C. Cox, J.E. Jr. Ingersoll, S.A. Ross, An Analysis of Variable Rate Loan Contracts, Journal
of Finance, 35(2)(1980), 389-403. doi:10.1111/j.1540-6261.1980.tb02169.x
[16] J.C. Cox, J.E. Jr. Ingersoll, S.A. Ross, A Theory of the Term Structure of Interest Rates,
Econometrica, 53(2)(1985), 385-407. http://www.jstor.org/stable/1911242
[17] Q. Dai and K. J. Singleton, Specification analysis of affine term structure models, Journal
of Finance, 55(5)(2000), 1943–1978.
[18] L.U. Dothan, On the term structure of interest rates, Journal of Financial Economics,
6(1)(1978), 59-69. https://ideas.repec.org/a/eee/jfinec/v6y1978i1p59-69.html
[19] J.C. Duan, J.G. Simonato, Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter, Review of Quantitative Finance and Accounting, 13(1999), 111-135. doi:
10.1023/A:1008304625054
[20] D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance,
6(4)(1996), 379–406.
[21] Y. Esmaeelzade Aghdam, A. Neisy, A. Adl, Simulating and Pricing CAT Bonds Using the
Spectral Method Based on Chebyshev Basis, Computational Economics, 63(2024), 423–435.
doi: 10.1007/s10614-023-10423-7
[22] R.M. Galisteo, Dinamica de la estructura temporal de tipos de interes: modelo de tres factores, Universitat de Barcelona. Departament d’Econometria, Estad´ıstica i Economia Espanyola, 2000. https://www.tdx.cat/handle/10803/1463#page=1
[23] L. G´omez del Valle, J. Mart´ınez Rodr´ıguez, Pricing zero-coupon bonds with differenct market
prices of risk, 2nd Italian-Spanish Conference of Financial Mathematics, Napoli, Italia, (1999).
[24] T.S.Y. Ho, S.B. Lee, Term Structure Movements and Pricing Interest Rate Contingent Claims, The Journal of Finance, 41(5)(1986), 1011-1029. doi: 10.1111/j.1540-
6261.1986.tb02528.x
[25] J. Hull, A. White, Pricing Interest-Rate-Derivative Securities, The Review of Financial
Studies, 3(4)(1990), 573-592. Zbl 1386.91152
[26] J. Hull, A. White, A, Numerical Procedures for Implementing Term Structure
Models I: Single-Factor Models, The Journal of Derivatives, 2(1)(1994), 7-16.
doi:10.3905/jod.1994.407902
[27] F.A. Longstaff, E.S. Schwartz, Interest Rate Volatility and the Term Structure: A
Two-Factor General Equilibrium Model, Journal of Finance, 47(4)(1992), 1259-1282.
doi:10.1111/j.1540-6261.1992.tb04657.x
[28] J. Lund, Econometric analysis of continuous-time arbitrage-free models of the term structure
of interest rates, Working paper, The Aarthus School of Business, 1995. https://www.econbiz.
de/Record/working-paper/10011913908#tabnav
[29] F. Mercurio, J.M. Morelada, An Analytically Tractable Interest Rate Model with Humped
Volatility, European Journal of Operational Research, 120 (1)(2000), 205-214. Zbl 1028.91589
[30] H. Mesgarani, A. Adl, Y. Esmaeelzade Aghdam, The Convergence Investigation of a Numerical Scheme for the Tempered Fractional Black–Scholes Model Arising European Double
Barrier Option, Computational Methods for Differential Equations, 11(2)(2023), 385–398.
doi: 10.22034/cmde.2022.53942.2219
[31] H. Mesgarani, A. Adl, Y. Esmaeelzade Aghdam, Approximate Price of the Option Under
Discretization by Applying Fractional Quadratic Interpolation, Computational Methods for
Differential Equations, 10(4)(2022), 1075–1085. doi: 10.22034/cmde.2021.45656.1918
[32] M. Moreno, Asset pricing under a two-factor model of the term structure of interest rates,
Mathematica de las Operaciones Financieras, IV Congreso MOF, Barcelona, (1997), 609-658.
Zbl 0362.46043.
[33] S.F. Richard, An arbitrage model of the term structure of interest rates, Journal of Financial
Economics, 6(1)(1978), 33-57. doi: 10.1016/0304-405X(78)90019-3
[34] S.M. Schaefer, E.S. Schwartz, A Two-Factor Model of the Term Structure: An Approximate
Analytical Solution, Journal of Financial and Quantitative Analysis, 19(4)(1984), 413-424.
doi:10.2307/2330783
[35] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5(2)(1977), 177-188. Zbl 1372.91113
[36] A. White, A.D. Hull, One-Factor Interest-Rate Models and the Valuation of Interest-Rate
Derivative Securities,Journal of Financial and Quantative Anaysis, 28 (2)(1993), 235-254. doi:
10.2307/2331288
[37] S. Zeytun, A. Gupta, A Comparative Study of the Vasicek and the CIR Model of the
Short Rate, Institut Techno- und Wirtschaftsmathematik, 2007. https://kluedo.ub.rptu.de/
frontdoor/index/index/docId/1979
Studies, 9(2)(1996), 385-426. http://www.jstor.org/stable/2962210
[2] S.H. Babbs, B. Nowman, Kalman Filtering of Generalized Vasicek Term Structure Models,
Journal of Financial and Quantitative Analysis, 34(1999), 115-130. http://www.jstor.org/
stable/2676248
[3] F. Black, P. Karasinski, Bond and Option Pricing When Short Rates Are Lognormal, Financial
Analysts Journal, 48(4)(1991), 52-59. http://www.jstor.org/stable/4479456
[4] D.J. Bolder, Affine Term-Structure Models: Theory and Implementation, Bank of Canada,
Working Paper 2001-15, Financial Markets Department, 2001. https://www.bankofcanada.
ca/wp-content/uploads/2010/02/wp01-15a.pdf
[5] M. Brennan, E. Schwartz, An equilibrium model of bond pricing and a test of market efficiency,
Journal of Financial and Quantitative Analysis, 17(1982), 301-329. http://www.jstor.org/
stable/2330832
[6] M. Brennan, E. Schwartz, Analyzing Convertible Bonds, Journal of Financial and Quantitative
Analysis, 15(1980), 907-929. http://www.jstor.org/stable/2330567
[7] M. Brennan, E. Schwartz, The pricing of equity-linked life insurance policies with an
asset value guarantee, Journal of Financial Economics, 3(1976), 195-213. https://www.
sciencedirect.com/science/article/pii/0304405X76900039
[8] J.Y. Campbell, A defence of tradditional hyphoteses about the term structure of interest rates.,
Journal of Finance, 441(1986), 183-193. http://www.jstor.org/stable/2328351
[9] E. Canabarro, Where do One-Factor Interest Rate Models Fail?, Journal of Fixed Income,
5(2)(1995), 31-52. doi: 10.3905/jfi.1995.408145
[10] G. Chaplin, K. Sharp, Analytic solutions for bonds and bond options under n correlated
stochastic processes., Research Report 93-16, University of Waterloo, (1995)
[11] R.R. Chen, L. Scott, Interest Rate Options in Multifactor Cox-Ingersoll-Ross Models of the
Term Structure, Journal of Fixed Income, 5(1)(1995), 14–31.
[12] R.R. Chen, Interest Rate Dynamics, Derivatives Pricing, and Risk Management, Springer,
1996. Zbl 0862.90015
[13] R.R. Chen, L. Scott, Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model, The Journal of Real Estate Finance
and Economics, 27(2)(2003), 143-172. https://link.springer.com/article/10.1023/A:
1024736903090
[14] G. Courtadon, The Pricing of Options on Default-Free Bonds, Journal of Financial and
Quantitative Analysis, 17(1)(1982), 75-100. http://www.jstor.org/stable/2330930
[15] J.C. Cox, J.E. Jr. Ingersoll, S.A. Ross, An Analysis of Variable Rate Loan Contracts, Journal
of Finance, 35(2)(1980), 389-403. doi:10.1111/j.1540-6261.1980.tb02169.x
[16] J.C. Cox, J.E. Jr. Ingersoll, S.A. Ross, A Theory of the Term Structure of Interest Rates,
Econometrica, 53(2)(1985), 385-407. http://www.jstor.org/stable/1911242
[17] Q. Dai and K. J. Singleton, Specification analysis of affine term structure models, Journal
of Finance, 55(5)(2000), 1943–1978.
[18] L.U. Dothan, On the term structure of interest rates, Journal of Financial Economics,
6(1)(1978), 59-69. https://ideas.repec.org/a/eee/jfinec/v6y1978i1p59-69.html
[19] J.C. Duan, J.G. Simonato, Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter, Review of Quantitative Finance and Accounting, 13(1999), 111-135. doi:
10.1023/A:1008304625054
[20] D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance,
6(4)(1996), 379–406.
[21] Y. Esmaeelzade Aghdam, A. Neisy, A. Adl, Simulating and Pricing CAT Bonds Using the
Spectral Method Based on Chebyshev Basis, Computational Economics, 63(2024), 423–435.
doi: 10.1007/s10614-023-10423-7
[22] R.M. Galisteo, Dinamica de la estructura temporal de tipos de interes: modelo de tres factores, Universitat de Barcelona. Departament d’Econometria, Estad´ıstica i Economia Espanyola, 2000. https://www.tdx.cat/handle/10803/1463#page=1
[23] L. G´omez del Valle, J. Mart´ınez Rodr´ıguez, Pricing zero-coupon bonds with differenct market
prices of risk, 2nd Italian-Spanish Conference of Financial Mathematics, Napoli, Italia, (1999).
[24] T.S.Y. Ho, S.B. Lee, Term Structure Movements and Pricing Interest Rate Contingent Claims, The Journal of Finance, 41(5)(1986), 1011-1029. doi: 10.1111/j.1540-
6261.1986.tb02528.x
[25] J. Hull, A. White, Pricing Interest-Rate-Derivative Securities, The Review of Financial
Studies, 3(4)(1990), 573-592. Zbl 1386.91152
[26] J. Hull, A. White, A, Numerical Procedures for Implementing Term Structure
Models I: Single-Factor Models, The Journal of Derivatives, 2(1)(1994), 7-16.
doi:10.3905/jod.1994.407902
[27] F.A. Longstaff, E.S. Schwartz, Interest Rate Volatility and the Term Structure: A
Two-Factor General Equilibrium Model, Journal of Finance, 47(4)(1992), 1259-1282.
doi:10.1111/j.1540-6261.1992.tb04657.x
[28] J. Lund, Econometric analysis of continuous-time arbitrage-free models of the term structure
of interest rates, Working paper, The Aarthus School of Business, 1995. https://www.econbiz.
de/Record/working-paper/10011913908#tabnav
[29] F. Mercurio, J.M. Morelada, An Analytically Tractable Interest Rate Model with Humped
Volatility, European Journal of Operational Research, 120 (1)(2000), 205-214. Zbl 1028.91589
[30] H. Mesgarani, A. Adl, Y. Esmaeelzade Aghdam, The Convergence Investigation of a Numerical Scheme for the Tempered Fractional Black–Scholes Model Arising European Double
Barrier Option, Computational Methods for Differential Equations, 11(2)(2023), 385–398.
doi: 10.22034/cmde.2022.53942.2219
[31] H. Mesgarani, A. Adl, Y. Esmaeelzade Aghdam, Approximate Price of the Option Under
Discretization by Applying Fractional Quadratic Interpolation, Computational Methods for
Differential Equations, 10(4)(2022), 1075–1085. doi: 10.22034/cmde.2021.45656.1918
[32] M. Moreno, Asset pricing under a two-factor model of the term structure of interest rates,
Mathematica de las Operaciones Financieras, IV Congreso MOF, Barcelona, (1997), 609-658.
Zbl 0362.46043.
[33] S.F. Richard, An arbitrage model of the term structure of interest rates, Journal of Financial
Economics, 6(1)(1978), 33-57. doi: 10.1016/0304-405X(78)90019-3
[34] S.M. Schaefer, E.S. Schwartz, A Two-Factor Model of the Term Structure: An Approximate
Analytical Solution, Journal of Financial and Quantitative Analysis, 19(4)(1984), 413-424.
doi:10.2307/2330783
[35] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5(2)(1977), 177-188. Zbl 1372.91113
[36] A. White, A.D. Hull, One-Factor Interest-Rate Models and the Valuation of Interest-Rate
Derivative Securities,Journal of Financial and Quantative Anaysis, 28 (2)(1993), 235-254. doi:
10.2307/2331288
[37] S. Zeytun, A. Gupta, A Comparative Study of the Vasicek and the CIR Model of the
Short Rate, Institut Techno- und Wirtschaftsmathematik, 2007. https://kluedo.ub.rptu.de/
frontdoor/index/index/docId/1979
)