Mathematical Analysis of Term Structure Models
Beáta Stehlíková

PhD thesis - Full text    (PDF 760 K)
Synopsis    (PDF 236 K)

Summary: In the thesis we study and analyze several questions and problems that are related to short rate interest rate models. The main goals of the thesis can be summarized as follows:
• Approximate analytical solution for one-factor models. We study the approximate analytical solution for bond prices derived by Choi and Wirjanto. We prove the order of accuracy of their formula and present numerical examples. Afterwards, we provide a new approximation of higher order of accuracy.
• Calibration of one-factor models. We use the approximate analytical solution mentioned above to calibrate one-factor models. We use Nowman's Gaussian estimates to estimate the volatility and the comparison of real term structures with theoretical ones to estimate the drift. Here we also study the question of existence of the estimates, i.e. the existence of maximum of likelihood function. Then we consider different weights when comparing the term structures and we see the differences in estimates caused by different criteria used.
• Averaging in two-factor models. We consider the following two-factor models: two-factor Vasicek, two-factor Cox-Ingersoll-Ross and Fong-Vasicek. In all these models, not all of the factors is are observable on the market. We consider their limiting distribution and compute the distribution of bond priced and interest rates. Afterwards, we compute their averaging, i.e. the expected values with respect to limiting distribution of unobservable factors. The averaged bond prices are functions of maturity and short rate. It is the same dependence as in one-factor models. Hence we study the question, whether there exists a one-factor model, which yields the same bond prices as the averaged values from the two-factor model. In all the models considered, the answer is negative.

Related papers

[1] B. Stehlíková and D. Ševčovič, On the singular limit of solutions to the Cox-Ingersoll-Ross interest rate model with stochastic volatility.
Submitted.
Full text    (PDF 239 K)

[2] B. Stehlíková and D. Ševčovič, On non-existence of a one factor interest rate model for volatility averaged generalized Fong--Vasicek term structures.
To appear in COE Lecture Note Series, Faculty of Mathematics, Kyushu University.
Full text    (PDF 166 K)

[3] B. Stehlíková and D. Ševčovič, Approximate formulae for pricing zero-coupon bonds and their asymptotic analysis.
To appear in International Journal of Numerical Analysis and Modeling.
Full text    (PDF 161 K)

[4] B. Stehlíková, Averaged Bond Prices for Fong-Vasicek and the Generalized Vasicek Interest Rates Models.
Proceeding of MMEI, 2007, pp. 166-175.
Full text    (PDF 167 K)

[5] B. Stehlíková, Averaged Bond Prices in Generalized Cox-Ingersoll-Ross Model of Interest Rates.
Proceedings of 5th Actuarial and Financial Mathematics Day, 2007, pp. 77-87.
Full text    (PDF 80 K)

[6] B. Stehlíková, Fast Mean Reverting Volatility in Fong-Vasicek Model of Interest Rates.
Journal of Electrical Engineering, 57, No.12/s, 2006, pp.65-97.
Full text    (PDF 162 K)

[7] B. Stehlíková, Modeling Volatility Clusters with Application to Two-Factor Interest Rate Models.
Journal of Electrical Engineering, 56, No.12/s, 2005, pp.90-93.
Full text    (PDF 2.3 M)

[8] B. Stehlíková and D. Ševčovič, On a Volatility Averaging in a Two-Factor Interest Rate Model.
Proceedings of Algoritmy 2005, pp. 325-333.
Full text    (PDF 218 K)