Analysis of Short-Term Interest Rate Models with Stochastic Volatility

Mary Avwerosuoghene Ojarikre; Bernard Ojonugwa Anthony; Jonathan Ojarikre Oniore

doi:doi:10.11648/j.ijssam.20251002.11

Research Article |

| Peer-Reviewed

Analysis of Short-Term Interest Rate Models with Stochastic Volatility

Mary Avwerosuoghene Ojarikre^*

, Bernard Ojonugwa Anthony

, Jonathan Ojarikre Oniore

Published in International Journal of Systems Science and Applied Mathematics (Volume 10, Issue 2)

Received: 19 January 2025 Accepted: 7 February 2025 Published: 29 May 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The examination of short-term interest-rate behaviour is of critical importance in financial analysis, risk management, and in formulating monetary policy. Fluctuations in financial markets in recent times have emphasised the need for strong and reliable models that can effectively model behaviors and dynamics involved in short-term interest-rate fluctuations. Conventional approaches, including Vasicek’s, have been universally embraced; yet such techniques often face difficulty in explaining clustering and autocorrelated volatility in real-world data. This study explores short-term interest rate models with stochastic volatility and evaluates their effectiveness in comparison to Autoregressive Conditional Heteroskedasticity (ARCH) and Generalised ARCH (GARCH) models. Using historical data from Nigerian financial instruments, we carried out Ljung-Box Q-statistic and ARCH tests to examine autocorrelation and volatility clustering. Results indicate that the data exhibits strong autocorrelation and significant volatility clustering. The predictive performance of our stochastic volatility model was measured by 10-day ahead volatility forecasts, which reached the sum of squared deviations of 1.3095, while ARCH had 2.0001 and GARCH had 2.1433. Our findings suggest that the stochastic volatility model outperforms the traditional ones, such as ARCH and GARCH, for interest rate change forecasting. Based on the performance realised, observed stochastic volatility models are recommended to better forecast interest rates, particularly for the emerging markets, where financial data could be volatile.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 10, Issue 2)
DOI	10.11648/j.ijssam.20251002.11
Page(s)	12-26
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Interest Rate Models, Stochastic Volatility, ARCH, GARCH

References

[1]	Ahnert, T., Anand, K., & König, P. J. (2023). Real interest rates, bank borrowing, and fragility. Journal of Money Credit and Banking, 56(6), 1545–1571. https://doi.org/10.1111/jmcb.13033
[2]	Alafif, H. (2023) Interest Rate and Some of Its Applications. Journal of Applied Mathematics and Physics, 11, 1557-1569. https://doi.org/10.4236/jamp.2023.116102.
[3]	Andreasen, J. (2010). Stochastic volatility interest rate models. Encyclopedia of Quantitative Finance. 1-5. https://doi.org/10.1002/9780470061602.eqf11019
[4]	Antonov, A., Konikov, M., & Spector, M. (2020). A New Arbitrage-Free Parametric Volatility Surface. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.3403708
[5]	Awoga, O. A. (2017). Implementation of Short-Rate Models - A Case Study of the Black-Derman-Toy Model of Interest Rate. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2951098
[6]	Ballestra, L. V., D’Innocenzo, E., & Guizzardi, A. (2023). A new bivariate approach for modeling the interaction between stock volatility and interest rate: An application to S&P500 returns and options. European Journal of Operational Research, 314(3), 1185–1194. https://doi.org/10.1016/j.ejor.2023.11.049
[7]	Bank of England. (2020, September 21). What are interest rates? Retrieved November 10, 2020, from https://www.bankofengland.co.uk/knowledgebank/what-are-interest-rates
[8]	Bevans, R. (2020, March 26). An introduction to the Akaike information criterion. Retrieved October 9, 2021, from https://www.scribbr.com/statistics/akaike-information-criterion/.
[9]	Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1–2), 167–179. https://doi.org/10.1016/0304-405x(76)90024-6
[10]	Black, F., & Karasinski, P. (1991) Bond and Option Pricing when Short Rates Are Lognormal. Financial Analysts Journal, 47, 52-59. https://doi.org/10.2469/faj.v47.n4.52
[11]	Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654. http://www.jstor.org/stable/1831029
[12]	Black, F., Derman, E., & Toy, W. (1990). A One-Factor Model of Interest Rates and its Application to Treasury Bond Options. Financial Analysts Journal, 46, 33-39. http://dx.doi.org/10.2469/faj.v46.n1.33
[13]	Brigo D., & Mercurio F. (2001) One-factor short-rate models. In Interest Rate Models Theory and Practice. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34604-3
[14]	Brigo, D., & Mercurio, F. (2003). Analytical pricing of the smile in a forward LIBOR market model. Quantitative Finance, 3(1), 15–27. https://doi.org/10.1080/713666156
[15]	Brigo, D., & Mercurio, F. (2006). Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit (2nd ed.). Berlin: Springer.
[16]	Cairns, A. J. G. (2004). Interest Rate Models: An Introduction. Princeton University Press. http://www.jstor.org/stable/j.ctv3hh51w
[17]	Chan, K., Karolyi, G., Longstaff, F., & Sanders, A. (1992). An Empirical Comparison of Alternative Models of the Short-Term Interest Rate. The Journal of Finance, 47(3), 1209-1227. https://doi.org/10.2307/2328983
[18]	Chang, H., & Rong, X. (2013). An Investment and Consumption Problem with CIR Interest Rate and Stochastic Volatility. Abstract and Applied Analysis, 2013, 1-12. https://doi.org/10.1155/2013/219397
[19]	Chang, Y., & Wang, Y. (2020). Option Pricing under Double Stochastic Volatility Model with Stochastic Interest Rates and Double Exponential Jumps with Stochastic Intensity. Mathematical Problems in Engineering, 2020, 1-13. https://doi.org/10.1155/2020/2743676
[20]	Cheng, B., Sklibosios, C. N., & Schlogl, E. (2016). Pricing of Long-Dated Commodity Derivatives with Stochastic Volatility and Stochastic Interest Rates. SSRN Electronic Journal, 1(1), 1-18. https://doi.org/10.2139/ssrn.2712025
[21]	Choudhry, M., Joannas, D., Landuyt, G., Pereira, R., & Pienaar, R. (2010). Interest Rate Modelling. In Capital Market Instruments (pp. 159-175). London: Palgrave Macmillan.
[22]	Coculescu, D., & Nikeghbali, A. (2010). Filtrations. Encyclopedia of Quantitative Finance. https://doi.org/10.1002/9780470061602.eqf02011
[23]	Cotton, P., Fouque, J., Papanicolaou, G., & Sircar, R. (2004). Stochastic Volatility Corrections for Interest Rate Derivatives. Mathematical Finance, 14(2), 173-200. h ttps://doi.org/10.1111/j.0960-1627.2004.00188.x
[24]	Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A Theory of the Term Structure of Interest Rates. Econometrica, 53(2), 385–407. https://doi.org/10.2307/1911242
[25]	Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4), 987–1007. https://doi.org/10.2307/1912773
[26]	Hall, B. H., & Li, W. C. Y. (2018). Depreciation of Business R&D Capital. Review of Income and Wealth, 66(1), 161–180. https://doi.org/10.1111/roiw.12380
[27]	Harrison, J. M., & Pliska, S. R. (1983). A Stochastic Calculus Model of Continuous Trading: Complete Markets. Stochastic Processes and their Applications, 15(3), 313-316. https://doi.org/10.1016/0304-4149(83)90038-8
[28]	Hautsch, N., & Ou, Y. (2012). Analyzing interest rate risk: Stochastic volatility in the term structure of government bond yields. Journal of Banking & Finance, 36(11), 2988–3007. https://doi.org/10.1016/j.jbankfin.2012.06.020
[29]	Hull, J., & White, A. (1990). Valuing derivative securities using the explicit finite difference method. Journal of Financial and Quantitative Analysis, 25(1), 87-100. https://doi.org/10.2307/2330889
[30]	Hull, J., & White, A. (1990). Pricing Interest-Rate-Derivative Securities. The Review of Financial Studies, 3(4), 573–592. http://www.jstor.org/stable/2962116
[31]	Hull, J., & White, A. (1993). One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities. The Journal of Financial and Quantitative Analysis, 28(2), 235-254. https://doi.org/10.2307/2331288
[32]	Kenton, W. (2020, November 12). Hull-White Model. Retrieved December 18, 2020, from https://www.investopedia.com/terms/h/hullwhite-model.asp
[33]	Khramov, V. (2013). Estimating Parameters of Short-Term Real Interest Rate Models. IMF Working Papers, 13(212), 1-26. https://doi.org/10.5089/9781475594645.001
[34]	Kim, M., Kim, J., & Yoon, J. (2015). Optimal Portfolio Selection Under Stochastic Volatility and Stochastic Interest Rates. Journal of the Korea Society for Industrial and Applied Mathematics, 19(4), 417-428. https://doi.org/10.12941/jksiam.2015.19.417
[35]	Krzyżanowski, G., Mordecki, E., & Sosa, A. (2019). Zero Black-Derman-Toy interest rate model. arXiv (Cornell University). https://doi.org/10.48550/arxiv.1908.04401
[36]	Lavoie, M., & Reissl, S. (2019). Further insights on endogenous money and the liquidity preference theory of interest. Journal of Post Keynesian Economics, 42(4), 503-526. https://doi.org/10.1080/01603477.2018.1548286
[37]	Lesniewski, A., Heng, S., & Wu, Q. (2016). A Dual-Curve Short Rate Model with Multi-Factor Stochastic Volatility: I. Asymptotic Analysis. SSRN Electronic Journal, 1(1), 2-30. https://doi.org/10.2139/ssrn.2634313
[38]	Mackevičius, V. (2016). Models of Interest Rates. Stochastic Models of Financial Mathematics, 83-115. https://doi.org/10.1016/b978-1-78548-198-7.50003-8
[39]	Merton, R. C. (1974). On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. The Journal of Finance, 29(2), 449-470. https://doi.org/10.1111/j.1540-6261.1974.tb03058.x
[40]	Rendleman, R. J., & Bartter, B. J. (1980). The pricing of options on debt securities. Journal of Financial and Quantitative Analysis, 15(1), 11-24. https://doi.org/10.2307/2979016
[41]	Rosa, R. (2024). Interest rates and the central bank. In Routledge eBooks (pp. 216–242). https://doi.org/10.4324/9781003547150-13
[42]	Salah, M. B., & Abid, F. (2014). An Empirical Comparison of the Short Term Interest Rate Models. SSRN Electronic Journal, 1-11. https://doi.org/10.2139/ssrn.2400433
[43]	Shafiu, I. A. (2018). Nigerian economy: business, governance and investment in period of crisis. Munich Personal RePEc Archive (MPRA), 91074, 1-176. https://mpra.ub.uni-muenchen.de/91074/
[44]	Simone, A. D. (2010). Pricing Interest Rate Derivatives Under Different Interest Rate Modeling: A Critical and Empirical Comparison. Investment Management and Financial Innovations, 7(2), 49-58. https://ssrn.com/abstract=1753730
[45]	Song, N., Siu, T. K., Fard, F. A., Ching, W., & Fung, E. S. (2012). Risk measures and behaviors for bonds under stochastic interest rate models. Mathematical and Computer Modelling, 56(9-10), 204-217. https://doi.org/10.1016/j.mcm.2011.11.070
[46]	Svoboda S. (2004). The Black and Karasinski Model. In Interest Rate Modelling (pp. 135-139). London: Palgrave Macmillan. https://doi.org/10.1057/9781403946027_9
[47]	Trolle, A. B., & Schwartz, E. S. (2009). A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives. The Review of Financial Studies, 22(5), 2007-2057. https://doi.org/10.2139/ssrn.970537
[48]	Vaaler, L. J., Harper, S. K., & Daniel, J. W. (2019). Mathematical Interest Theory. MAA Press, an imprint of the American Mathematical Society.
[49]	Vasicek, O. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics, 5(2), 177-188. http://dx.doi.org/10.1016/0304-405X(77)90016-2
[50]	Verschuren, R. M. (2020). Stochastic Interest Rate Modelling using a Single or Multiple Curves: An Empirical Performance Analysis of the Lévy Forward Price Model. Quantitative Finance, 20(7), 1123-1148. https://doi.org/10.1080/14697688.2020.1722318
[51]	Wray, L. (1992). Alternative Approaches to Money and Interest Rates. Journal of Economic Issues, 26(4), 1145-1178. https://www.jstor.org/stable/4226624

Cite This Article

Plain Text BibTeX RIS

APA Style

Ojarikre, M. A., Anthony, B. O., Oniore, J. O. (2025). Analysis of Short-Term Interest Rate Models with Stochastic Volatility. International Journal of Systems Science and Applied Mathematics, 10(2), 12-26. https://doi.org/10.11648/j.ijssam.20251002.11

Copy | Download

ACS Style

Ojarikre, M. A.; Anthony, B. O.; Oniore, J. O. Analysis of Short-Term Interest Rate Models with Stochastic Volatility. Int. J. Syst. Sci. Appl. Math. 2025, 10(2), 12-26. doi: 10.11648/j.ijssam.20251002.11

Copy | Download

AMA Style

Ojarikre MA, Anthony BO, Oniore JO. Analysis of Short-Term Interest Rate Models with Stochastic Volatility. Int J Syst Sci Appl Math. 2025;10(2):12-26. doi: 10.11648/j.ijssam.20251002.11

Copy | Download

@article{10.11648/j.ijssam.20251002.11,
  author = {Mary Avwerosuoghene Ojarikre and Bernard Ojonugwa Anthony and Jonathan Ojarikre Oniore},
  title = {Analysis of Short-Term Interest Rate Models with Stochastic Volatility
},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {10},
  number = {2},
  pages = {12-26},
  doi = {10.11648/j.ijssam.20251002.11},
  url = {https://doi.org/10.11648/j.ijssam.20251002.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20251002.11},
  abstract = {The examination of short-term interest-rate behaviour is of critical importance in financial analysis, risk management, and in formulating monetary policy. Fluctuations in financial markets in recent times have emphasised the need for strong and reliable models that can effectively model behaviors and dynamics involved in short-term interest-rate fluctuations. Conventional approaches, including Vasicek’s, have been universally embraced; yet such techniques often face difficulty in explaining clustering and autocorrelated volatility in real-world data. This study explores short-term interest rate models with stochastic volatility and evaluates their effectiveness in comparison to Autoregressive Conditional Heteroskedasticity (ARCH) and Generalised ARCH (GARCH) models. Using historical data from Nigerian financial instruments, we carried out Ljung-Box Q-statistic and ARCH tests to examine autocorrelation and volatility clustering. Results indicate that the data exhibits strong autocorrelation and significant volatility clustering. The predictive performance of our stochastic volatility model was measured by 10-day ahead volatility forecasts, which reached the sum of squared deviations of 1.3095, while ARCH had 2.0001 and GARCH had 2.1433. Our findings suggest that the stochastic volatility model outperforms the traditional ones, such as ARCH and GARCH, for interest rate change forecasting. Based on the performance realised, observed stochastic volatility models are recommended to better forecast interest rates, particularly for the emerging markets, where financial data could be volatile.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Analysis of Short-Term Interest Rate Models with Stochastic Volatility

AU  - Mary Avwerosuoghene Ojarikre
AU  - Bernard Ojonugwa Anthony
AU  - Jonathan Ojarikre Oniore
Y1  - 2025/05/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20251002.11
DO  - 10.11648/j.ijssam.20251002.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 12
EP  - 26
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20251002.11
AB  - The examination of short-term interest-rate behaviour is of critical importance in financial analysis, risk management, and in formulating monetary policy. Fluctuations in financial markets in recent times have emphasised the need for strong and reliable models that can effectively model behaviors and dynamics involved in short-term interest-rate fluctuations. Conventional approaches, including Vasicek’s, have been universally embraced; yet such techniques often face difficulty in explaining clustering and autocorrelated volatility in real-world data. This study explores short-term interest rate models with stochastic volatility and evaluates their effectiveness in comparison to Autoregressive Conditional Heteroskedasticity (ARCH) and Generalised ARCH (GARCH) models. Using historical data from Nigerian financial instruments, we carried out Ljung-Box Q-statistic and ARCH tests to examine autocorrelation and volatility clustering. Results indicate that the data exhibits strong autocorrelation and significant volatility clustering. The predictive performance of our stochastic volatility model was measured by 10-day ahead volatility forecasts, which reached the sum of squared deviations of 1.3095, while ARCH had 2.0001 and GARCH had 2.1433. Our findings suggest that the stochastic volatility model outperforms the traditional ones, such as ARCH and GARCH, for interest rate change forecasting. Based on the performance realised, observed stochastic volatility models are recommended to better forecast interest rates, particularly for the emerging markets, where financial data could be volatile.

VL  - 10
IS  - 2
ER  -

Copy | Download

Author Information

Mary Avwerosuoghene Ojarikre

Department of Mathematics, University of Abuja, Abuja, Nigeria

Contact Email

http://orcid.org/0009-0006-3309-4295
Bernard Ojonugwa Anthony

Department of Economics and Statistics, Kampala International University, Kampala, Uganda

http://orcid.org/0000-0002-4429-9599
Jonathan Ojarikre Oniore

Department of Economics, Faculty of Social Sciences, Bingham University, Karu, Nigeria

http://orcid.org/0000-0002-6526-722X

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Ojarikre, M. A., Anthony, B. O., Oniore, J. O. (2025). Analysis of Short-Term Interest Rate Models with Stochastic Volatility. International Journal of Systems Science and Applied Mathematics, 10(2), 12-26. https://doi.org/10.11648/j.ijssam.20251002.11

Copy | Download

ACS Style

Ojarikre, M. A.; Anthony, B. O.; Oniore, J. O. Analysis of Short-Term Interest Rate Models with Stochastic Volatility. Int. J. Syst. Sci. Appl. Math. 2025, 10(2), 12-26. doi: 10.11648/j.ijssam.20251002.11

Copy | Download

AMA Style

Ojarikre MA, Anthony BO, Oniore JO. Analysis of Short-Term Interest Rate Models with Stochastic Volatility. Int J Syst Sci Appl Math. 2025;10(2):12-26. doi: 10.11648/j.ijssam.20251002.11

Copy | Download

@article{10.11648/j.ijssam.20251002.11,
  author = {Mary Avwerosuoghene Ojarikre and Bernard Ojonugwa Anthony and Jonathan Ojarikre Oniore},
  title = {Analysis of Short-Term Interest Rate Models with Stochastic Volatility
},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {10},
  number = {2},
  pages = {12-26},
  doi = {10.11648/j.ijssam.20251002.11},
  url = {https://doi.org/10.11648/j.ijssam.20251002.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20251002.11},
  abstract = {The examination of short-term interest-rate behaviour is of critical importance in financial analysis, risk management, and in formulating monetary policy. Fluctuations in financial markets in recent times have emphasised the need for strong and reliable models that can effectively model behaviors and dynamics involved in short-term interest-rate fluctuations. Conventional approaches, including Vasicek’s, have been universally embraced; yet such techniques often face difficulty in explaining clustering and autocorrelated volatility in real-world data. This study explores short-term interest rate models with stochastic volatility and evaluates their effectiveness in comparison to Autoregressive Conditional Heteroskedasticity (ARCH) and Generalised ARCH (GARCH) models. Using historical data from Nigerian financial instruments, we carried out Ljung-Box Q-statistic and ARCH tests to examine autocorrelation and volatility clustering. Results indicate that the data exhibits strong autocorrelation and significant volatility clustering. The predictive performance of our stochastic volatility model was measured by 10-day ahead volatility forecasts, which reached the sum of squared deviations of 1.3095, while ARCH had 2.0001 and GARCH had 2.1433. Our findings suggest that the stochastic volatility model outperforms the traditional ones, such as ARCH and GARCH, for interest rate change forecasting. Based on the performance realised, observed stochastic volatility models are recommended to better forecast interest rates, particularly for the emerging markets, where financial data could be volatile.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Analysis of Short-Term Interest Rate Models with Stochastic Volatility

AU  - Mary Avwerosuoghene Ojarikre
AU  - Bernard Ojonugwa Anthony
AU  - Jonathan Ojarikre Oniore
Y1  - 2025/05/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20251002.11
DO  - 10.11648/j.ijssam.20251002.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 12
EP  - 26
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20251002.11
AB  - The examination of short-term interest-rate behaviour is of critical importance in financial analysis, risk management, and in formulating monetary policy. Fluctuations in financial markets in recent times have emphasised the need for strong and reliable models that can effectively model behaviors and dynamics involved in short-term interest-rate fluctuations. Conventional approaches, including Vasicek’s, have been universally embraced; yet such techniques often face difficulty in explaining clustering and autocorrelated volatility in real-world data. This study explores short-term interest rate models with stochastic volatility and evaluates their effectiveness in comparison to Autoregressive Conditional Heteroskedasticity (ARCH) and Generalised ARCH (GARCH) models. Using historical data from Nigerian financial instruments, we carried out Ljung-Box Q-statistic and ARCH tests to examine autocorrelation and volatility clustering. Results indicate that the data exhibits strong autocorrelation and significant volatility clustering. The predictive performance of our stochastic volatility model was measured by 10-day ahead volatility forecasts, which reached the sum of squared deviations of 1.3095, while ARCH had 2.0001 and GARCH had 2.1433. Our findings suggest that the stochastic volatility model outperforms the traditional ones, such as ARCH and GARCH, for interest rate change forecasting. Based on the performance realised, observed stochastic volatility models are recommended to better forecast interest rates, particularly for the emerging markets, where financial data could be volatile.

VL  - 10
IS  - 2
ER  -

Copy | Download