# Dynamic correlation in a convergence model of interest rates

## Main Article Content

## Abstract

Short rate models are formulated in terms of stochastic differential equations governing the instantaneous interest rate, so called short rate. The bond prices, as well as other derivatives, are then given as a solution to a parabolic partial differential equation with a terminal condition equal to the payoff of the derivative. Convergence models are used to model a situation where a country is going to enter a monetary union and its short rate is affected by the short rate in the monetary union. In addition, Wiener processes which model random shocks in the behaviours of the short rates can be correlated. In this paper we consider a dynamic correlation (i.e., the correlation is a given function of time) in a convergence model with volatilities proportional to powers of the respective short rates. Firstly, we consider a special case with constant volatilities which is analytically tractable. Based on observations made in this case, we propose an approximate analytical solution for the bond prices in the general model and derive order of its accuracy.

## Article Details

How to Cite

Stehlíková, B., & Bučková, Z.
(2020).
Dynamic correlation in a convergence model of interest rates.

*Proceedings Of The Conference Algoritmy,*, 201 - 210. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1584/837
Section

Articles

## References

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[2] K. C. Chan, G. A. Karolyi, F. A. Longstaff, and A. B. Sanders, An empirical comparison of alternative models of the short-term interest rat, The Journal of Finance, 47(3) (1992), pp. 1209–1227.

[3] Y. Choi and T. S. Wirjanto, An analytic approximation formula for pricing zero-coupon bonds, Finance Research Letters, 4(2) (2007), pp. 116–126.

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[7] B. Stehlı́ková, A simple analytic approximation formula for the bond price in the Chan- Karolyi-Longstaff-Sanders model, International Journal of Numerical Analysis and Modeling, 4(3) (2013), pp. 224–234.

[8] L. Teng, M. Ehrhardt, and M. Günther, The pricing of Quanto options under dynamic correlation, Journal of Computational and Applied Mathematics, 275 (2015), pp. 304–310.

[9] L. Teng, M. Ehrhardt, and M. Günther, On the Heston model with stochastic correlation, International Journal of Theoretical and Applied Finance, 19(06) (2016), 1650033.

[10] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5(2) (1977), pp. 177–188.

[11] Z. Zı́ková and B. Stehlı́ková, Convergence model of interest rates of CKLS type, Kybernetika, 48(3) (2012), pp. 567–586.

[2] K. C. Chan, G. A. Karolyi, F. A. Longstaff, and A. B. Sanders, An empirical comparison of alternative models of the short-term interest rat, The Journal of Finance, 47(3) (1992), pp. 1209–1227.

[3] Y. Choi and T. S. Wirjanto, An analytic approximation formula for pricing zero-coupon bonds, Finance Research Letters, 4(2) (2007), pp. 116–126.

[4] T. Corzo Santamaria and E. .S. Schwartz, Convergence within the EU: Evidence from interest rates, Economic Notes, 29(2) (2000), pp. 243–266.

[5] J. C. Cox, J. E. Ingersoll Jr, and S. A. Ross, A theory of the term structure of interest rates Econometrica, 53(2) (1985), pp. 385–407.

[6] Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer, 2008.

[7] B. Stehlı́ková, A simple analytic approximation formula for the bond price in the Chan- Karolyi-Longstaff-Sanders model, International Journal of Numerical Analysis and Modeling, 4(3) (2013), pp. 224–234.

[8] L. Teng, M. Ehrhardt, and M. Günther, The pricing of Quanto options under dynamic correlation, Journal of Computational and Applied Mathematics, 275 (2015), pp. 304–310.

[9] L. Teng, M. Ehrhardt, and M. Günther, On the Heston model with stochastic correlation, International Journal of Theoretical and Applied Finance, 19(06) (2016), 1650033.

[10] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5(2) (1977), pp. 177–188.

[11] Z. Zı́ková and B. Stehlı́ková, Convergence model of interest rates of CKLS type, Kybernetika, 48(3) (2012), pp. 567–586.