COMENIUS UNIVERSITY IN BRATISLAVA FACULTY OF MATHEMATICS,PHYSICS AND INFORMATICS DEPARTMENT OF APPLIED MATHEMATICS AND STATISTICS UNIVERSITY OF WUPPERTAL FACULTY OF MATHEMATICS AND NATURAL SCIENCES DEPARTMENT OF MATHEMATICS AND INFORMATICS Analytical and Numerical Approximative Methods for solving Multifactor Models for pricing of Financial Derivatives Dissertation Thesis Binational doctoral study Mgr. ZUZANA BUCKOVÁˇ 2016 Die Dissertation kann wie folgt zitiert werden: urn:nbn:de:hbz:468-20170412-150623-2 [http://nbn-resolving.de/urn/resolver.pl?urn=urn%3Anbn%3Ade%3Ahbz%3A468-20170412-150623-2] COMENIUS UNIVERSITY IN BRATISLAVA FACULTY OF MATHEMATICS,PHYSICS AND INFORMATICS DEPARTMENT OF APPLIED MATHEMATICS AND STATISTICS UNIVERSITY OF WUPPERTAL FACULTY OF MATHEMATICS AND NATURAL SCIENCES DEPARTMENT OF MATHEMATICS AND INFORMATICS Analytical and Numerical Approximative Methods for solving Multifactor Models for pricing of Financial Derivatives Dissertation Thesis Binational doctoral study Study program: Economic and Financial Mathematics Field od study: 9.1.9 Applied mathematics Department: Department of Applied Mathematics and Statistics FMFI UK, Mlynská dolina, 842 48 Bratislava, Slovakia Supervisor: Prof. RNDr. Daniel ŠEVCOVIˇ C,ˇ CSc. Consultant: Doc. RNDr. Beáta STEHLÍKOVÁ, PhD. Study program: Applied Mathematics and Numerical Analysis Address Department: Bergische Universität Wuppertal, Gaussstrasse 20, 42119 Wuppertal, Germany Supervisor: Prof. Dr. Matthias EHRHARDT Consultant: Prof. Dr. Michael GÜNTHER Author: Mgr. Zuzana BUCKOVÁˇ Date: 2016 Place: Bratislava, Slovakia; Wuppertal, Germany Analytické a numerické aproximacnéˇ metódy riešenia viacfaktorových modelov ocenovaniaˇ financnýchˇ derivátov Dizertacnᡠpráca na dosiahnutie akademického titulu Philosophiae Doctor (PhD.) na Fakulte matematiky, fyziky a informatiky Univerzity Komenského v Bratislave v rámci binacionálneho doktorandského štúdia (Cotutelle) je predložená od Mgr. ZUZANA BUCKOVÁˇ rod. Zíková narodená 23.09.1987 v Kežmarku Analytische und numerische Approximationsmethoden zum Lösen von Mehrfaktormodellen zur Bewertung von Finanzderivaten Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) dem Fakultät für Mathematik und Naturwissenschaften der Bergischen Universität Wuppertal (BUW) im Rahmen eines gemeinsamen binationalen Promotionsverfahrens (Cotutelle) vorgelegt von Mgr. ZUZANA BUCKOVÁˇ geb. Zíková geboren am 23.09.1987 in Kežmarok, Slowakei 20504548 Univerzita Komenského v Bratislave Fakulta matematiky, fyziky a informatiky ZADANIE ZÁVEREČNEJ PRÁCE Meno a priezvisko študenta: Mgr. Zuzana Bučková Študijný program: aplikovaná matematika (Jednoodborové štúdium, doktorandské III. st., denná forma) Študijný odbor: aplikovaná matematika Typ záverečnej práce: dizertačná Jazyk záverečnej práce: anglický Sekundárny jazyk: slovenský Názov: Analytical and Numerical Approximative Methods for solving Multifactor Models for pricing of Financial Derivatives Analytické a numerické aproximačné metódy riešenia viacfaktorových modelov oceňovania finančných derivátov Cieľ: Cieľom práce je matematická analýza viacfaktorových modelov pre oceňovanie finančných derivátov so zameraním na modelovanie úrokových mier. Školiteľ: prof. RNDr. Daniel Ševčovič, CSc. Školiteľ: prof. Dr. rer. nat. Matthias Ehrhardt Konzultant: prof. Dr. rer. nat. Michael Guenther Konzultant: doc. RNDr. Beáta Stehlíková, PhD. Katedra: FMFI.KAMŠ - Katedra aplikovanej matematiky a štatistiky Vedúci katedry: prof. RNDr. Daniel Ševčovič, CSc. Spôsob sprístupnenia elektronickej verzie práce: bez obmedzenia Dátum zadania: 01.09.2011 Dátum schválenia: 23.02.2011 prof. RNDr. Marek Fila, DrSc. garant študijného programu študent školiteľ Acknowledgement I would like to thank to my supervisor at Comenius University in Bratislava, prof. RNDr. Daniel Ševcoviˇ c,ˇ CSc., for his guidance and support during my PhD. study. I am sincerely thankful to Doc. RNDr. Beáta Stehlíková, PhD. for her professional support, patience, and her enthusiasm in this long-lasting cooperation. At the University of Wuppertal, my thanks go to Prof. Dr. Michael Günther for his guidance and support. I would like to express many thanks for supervision from Prof. Dr. Matthias Ehrhardt and for involving me from the very beginning in the STRIKE project. The numerous discussions we had were always insightful and his detailed and deep corrections continuously improved the quality of my contributions. His sense for networking and cooperation with other groups opened new perspectives throughout my PhD. Thus, I had the opportunity to cooperate with Dr. Daniel Duffy about ADE methods and Prof. Igor Tsukermann about Trefftz methods. I would like to thank them for their time and enthusiasm. Beáta, Matthias, Daniel, and Michael are experts with overview in different fields and it has been my pleasure to learn from them. I am indebted to them for guiding me within my PhD.study and for their valuable comments on this thesis. This 5-year binational PhD. study brought me two different perspectives, two approaches from two groups of colleagues. In the last three years of my PhD. study I was involved in the STRIKE project, Novel Methods in Computational Finance, what is a European mathematical research training network. Experience as an early stage researcher in this project has given me an inspiration and enabled to join a strong community of people working on similar topics. I am very thankful to Matthias and Dr. Jan ter Maten for this life-lasting experience and to all my supervisors for the possibility to join this unique project. I would like to thank my colleagues and friends in Wuppertal and Bratislava and all the people whose I met at the numerous STRIKE events. For the very nice scientific dis- cussion we had, I am thankful to Pedro, Shih-Hau, Igor, José, Christian, Vera, Long, Radoslav. I would like to mention some others of my friends that supported the course of my research work: Janka, Majka, Magda, Darinka, Lara, Dmitry, Guillaume. For the many pieces of advice in connection with an English language that I received, I am thank- ful to Martinka and for the German language ones, I am indebted to Matthias, Daniel, Christoph, and Christian. I would especially like to thank my amazing family for the love, support, and constant encouragement I have received over the years. My deepest thanks go to my kind mum and my father, who passed away, for raising me up, being such good example, and always showing me good directions. I warmly thank my supportive husband Andrej for his love and patience as well his parents and sisters Barborka and Danka. Last, but not least I thank God for health and bleesing me for all the time. 9 Abstract 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 con- vergence model with non-constant volatility which is given by two stochastic differential equations (SDEs). Convergence model describes the evolution of interest rate in connec- tion 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 approxima- tions, 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 Alter- nating 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. Keywords: short rate models, convergence model of interest rate,