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PhD thesis - Full text

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**Summary**

The thesis covers different approaches used in current modern computational finance. Analytical and numerical approximative methods are studied and discussed. Effective algorithms for solving multi-factor models for pricing of financial derivatives have been developed.

The first part of the thesis is dealing with modeling of aspects and focuses on analytical approximations in short rate models for bond pricing. We deal with a two-factor convergence model with non-constant volatility which is given by two stochastic differential equations (SDEs). Convergence model describes the evolution of interest rate in connection with the adoption of the Euro currency. From the SDE it is possible to derive the PDE for bond price. The solution of the PDE for bond price is known in closed form only in special cases, e.g. Vasicek or CIR model with zero correlation. In other cases we derived the approximation of the solution based on the idea of substitution of constant volatilities, in solution of Vasicek, by non-constant volatilities. To improve the quality in fitting exact yield curves by their estimates, we proposed a few changes in models. The first one is based on estimating the short rate from the term structures in the Vasicek model. We consider the short rate in the European model for unobservable variable and we estimate it together with other model parameters. The second way to improve a model is to define the European short rate as a sum of two unobservable factors. In this way, we obtain a three-factor convergence model. We derived the accuracy for these approximaions, proposed calibration algorithms and we tested them on simulated and real market data, as well.

The second part of the thesis focuses on the numerical methods. Firstly we study Fichera theory which describes proper treatment of defining the boundary condition. It is useful for partial differential equation which degenerates on the boundary. The derivation of the Fichera function for short rate models is presented. The core of this part is based on Alternating direction explicit methods (ADE) which belong to not well studied finite difference methods from 60s years of the 20th century. There is not a lot of literature regarding this topic. We provide numerical analysis, studying stability and consistency for convection-diffusion-reactions equations in the one-dimensional case. We implement ADE methods for two-dimensional call option and three-dimensional spread option model. Extensions for higher dimensional Black-Scholes models are suggested. We end up this part of the thesis with an alternative numerical approach called Trefftz methods which belong to Flexible Local Approximation MEthods (FLAME). We briefly outline the usage in com- putational finance.

**Related papers **

[1] Z. Bučková (Zíková), B. Stehlíková, Convergence model of interest rates of CKLS type,

Kybernetika 48(3), 2012, 567-586

[2] J. Halgašová, B. Stehlíková, Z. Bučková (Zíková): Estimating the short rate from the term

structures in the Vasicek model, Tatra Mountains Mathematical Publications 61: 87-103,

2014

[3] Z. Bučková, J. Halgašová, B. Stehlíková: Short rate as a sum of CKLS-type processes,

accepted for publication in Proceedings of Numerical analysis and applications conference,

Springer Verlag in LNCS, 2016.

[4] B. Stehlíková, Z. Bučková (Zíková): A three-factor convergence model of interest rates.

Proceedings of Algoritmy 2012, pp. 95-104.

[5] Z. Bučková, M. Ehrhardt, M. Günther: Fichera theory and its application to finance, Pro-

ceedings ECMI 2014, Taormina, Sicily, Italy, 2016

[6] Z. Bučková, M. Ehrhardt, M. Günther: Alternating Direction Explicit Methods for Convec-

tion Diffusion Equations, Acta Math. Univ. Comenianae, Vol. LXXXI: 309–325, 2015

[7] Z. Bučková, P. Pólvora, M. Ehrhardt, M. Günther: Implementation of Alternating Direction

Explicit Methods to higher dimensional Black-Scholes Equation, AIP Conf. Proc. 1773,

030001; 2016

[8] Z. Bučková, B. Stehlíková, D. Ševčovič: Numerical and analytical methods for bond pricing

in short rate convergence models of interest rates, book chapter in the book Advances in

Mathematics Research. Volume 21, 2016