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Drift-Free Simulation and Libor Market Models

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Drift-Free Simulation and Libor Market Models PhD. Thesis Drift-Free Simulation and Libor Market Models AUTORA: Marta Pou Bueno DIRECTORES: Jos´eLuis Fern´andezP´erez Carlos V´azquezCend´on TESE PRESENTADA PARA A OBTENCION´ DO T´ITULO DE DOUTOR NA UNIVERSIDADE DA CORUNA~ DEPARTAMENTO DE MATEMATICAS´ FACULTADE DE INFORMATICA,´ A CORUNA~ (SPAIN) JUNIO, 2013 Esta memoria fue defendida por Marta Pou Bueno el d´ıa25 de Junio de 2013 en la Universidade da Coru~naante el Tribunal abajo firmante: PRESIDENTE: VOCALES: SECRETARIO: The undersigned hereby certify that they are supervisors of the Thesis entitled "Drift-Free Simulation and Libor Market Models" developed by Marta Pou Bueno inside the Ph.D Program "Mathematical Modelling and Numerical Methods in Applied Sciences and Engineering" at the Department of Mathe- matics (University of A Coru~na). Dated: June 25, 2013 Ph.D. Advisors: Jos´eLuis Fern´andezP´erez,Carlos V´azquezCend´on Funding This research has been partially funded by the following projects: ∙ Project MTM2007-67596-C02-01 from Ministerio de Educaci´ony Ciencia. ∙ Project 2007/137 from Xunta de Galicia. ∙ Project MTM2010-21135-C02-01 from Ministerio de Ciencia e Innovaci´on. ∙ Project CN2011/004 from Xunta de Galicia. ∙ FEDER funds. A mi abuelo Felipe. ”Defiende tu derecho a pensar, porque incluso pensar de manera err´oneaes mejor que no pensar." Hipat´ıa de Alejandr´ıa. Agradecimientos Lo mejor de terminar esta etapa doctoral es poder pararme a dedicar unas l´ıneasde agradecimiento y recuerdo a ciertas personas que han sido relevantes en dicho peri- odo, tanto a nivel profesional como personal. Espero saber transmitir lo importantes que han sido para m´ı. Quiero agradecer a los directores de esta tesis doctoral, Jos´eLuis Fern´andezP´erez y Carlos V´azquezCend´on,su apoyo en todos y cada uno de los periodos vividos desde que empec´eesta experiencia. Les agradezco de un modo especial haberme brindado esta oportunidad. Era algo que no pensaba posible cuando estudiaba la carrera, y ellos creyeron en que yo podr´ıacon ello antes que yo lo imaginase. Previamente, en la etapa de m´aster,hab´ıanlogrado motivar en m´ıel inter´espor este campo. Una vez embarcada en este proyecto, les agradezco mucho lo que me han aportado cada uno, logrando un equilibrio perfecto entre ambos, que me hizo sentir apoyada en todo mo- mento. Ahora que la etapa finaliza, sonr´ıoal darme cuenta de c´omome marcasteis los tiempos, fren´andome para pensar o haci´endomeavanzar en el instante exacto. Carlos, a t´ıen especial quiero agradecerte tu lucha constante con todos los tr´amites. Siempre he podido confiar en que t´uresolver´ıaslos problemas, pudi´endomecentrar yo en disfrutar del doctorado. Josechu, a t´ıquiero transmitirte que siempre recordar´e las estancias en Madrid, estancias hasta en las que en un caf´ehab´ıatiempo para las matem´aticas.Gracias por acogerme siempre con tanto cari~no. Desde que empec´eel doctorado he tenido muy cerca a Mar´ıaR. Nogueiras. Ella ha puesto a mi disposici´onsus conocimientos y experiencia. Cada minuto a su lado ha sido una gran carga de aprendizaje y una inyecci´onde seguridad y motivaci´on. A los tres os digo que me tomo la licencia de teneros cerca de cara al futuro. En estos a~nos de doctorado, he estado arropada por el Departamento de Matem´aticas de la Universidade da Coru~na.Son muchos los profes que, al coincidir en un congreso o esperando a que salgan las impresiones, se han interesado en c´omo iba. Gracias a ´I~nigo,Ana, Jos´eAntonio, ... Quiero agradecer a mis compa~neros del Labo y de fuera del Labo todos los mo- mentos que hemos ido viviendo juntos, tanto los sencillos caf´esy comidas del d´ıaa d´ıacomo las comidas a la canasta, las ca~nas,las cenas, y dem´asplanes que han ido surgiendo. Algunos ya se han ido yendo pero espero poder acordarme de todos: Eli, Adri, Bru~no,Mar´ıa,Carmi~na,Bolu, Gonzalo, Moli, Raquel, Jes´us,Paula (Asturies), Porta, Miguel´on,Alvarito, Jose, Brais, Xurxo, Manu, las otras dos Paulas... Gracias en especial a los de las finanzas (Mar´ıa,Carmi~na,Alvarito y Jose) por su ayuda y apoyo, y a Mar´ıa,Carmi~nay la Bolu por estar ah´ıpara todo. Y ahora viene el mejor momento. Tener la oportunidad de agradecer a las per- sonas que me han acompa~nadoen mi vida todo lo que han hecho por m´ıen esta etapa. Quiero agradecer a mis amigas del alma (Catu, Mar´ıa, Cris y Marta) y a mis grandes amigos (Manu, Pablo, Santi y Jorge) todo su cari~no,pero sobre todo y m´as importante las risas juntos. Con vosotros todo lo veo de otra manera, desconecto y estoy feliz. Porque para ellos soy "Marta Prou", porque siempre me preguntan por el t´ıtulode la tesis y se est´anriendo o atendiendo a otra cosa antes de que acabe, porque les hace mucha gracia que vaya a ser doctora... Porque son geniales. 14 Y ahora, y teniendo miedo de no poder llegar a transmitir todo el agradecimiento que siento, quiero dedicar las ´ultimasl´ıneasa mi familia. A todos en general por su apoyo incansable, por su inmenso cari~noy por hacerme sentir siempre lo orgullosos que est´ande m´ı.En especial, a mis padres. Ellos siempre est´anah´ı,orgullos´ısimosde sus hijas, y quiero que sepan que hoy soy yo la que estoy orgullos´ısimade ellos, porque el m´eritode que estemos aqu´ıy as´ıes vuestro. Gracias por todo. A mi hermana Eva por su alegr´ıa,ella siempre est´aah´ıalegre, haci´endomereir, habl´andomede mil cosas, interesada por c´omovan los "escol´ısticos”,por qu´ees lo que hago cuando me siento delante del ordenador... Ella se nota que est´aa mi lado. As´ıque a ella y a Diego mil gracias tambi´en. A mis abuelos, a mi gran debilidad, agradecerles ser como son. A mi abuela Julia por quererme de esa manera tan dulce, a mi abuelo Felipe por sus ojos de orgullo infinito ("sigo yendo por la sombra, abuelo!!"), a mi abuela Chicha por creer que siempre era un sobresaliente y a mi abuelo Mart´ınpor ser un ejemplo de entusiasmo y vitalidad. Y a David... David lo es todo, qu´ele voy a decir! Me centro en darle las gracias por acompa~narmeen todo lo que hago, por apoyarme, por respetarme, por valorarme tant´ısimo,por hacerme feliz... Os quiero. A Coru~na,2013. 15 16 Table of Contents Table of Contents i Abstract xi Resumen xiii Resumo xv Abbreviations and Notation xvii Introduction 1 I Drift-Free Simulation Techniques for Market Models 9 1 Market Models and Drift-Free Simulation techniques 11 2 Drift-Free Simulation techniques for Libor Market Model 15 2.1 Introduction . 15 2.2 Libor Market Model and Drif-Free Simulations . 16 2.2.1 Existing DFS methods for LMM . 19 2.2.2 An alternative Parameterized DFS (PDFS) . 25 2.3 Model Calibration . 31 2.4 Numerical Results . 33 2.4.1 Example 1: Flat structure of forward rates and volatilities . 36 2.4.2 Example 2: Uneven structure of forward rates and volatilities 39 3 Drift-Free Simulation techniques for Coterminal Swap Market Model and Generic Market Models 45 3.1 Introduction . 45 3.2 Coterminal Swap Market Model and Drift-Free Simulations . 46 i 3.3 Generic Market Models and Drift-Free Simulations . 55 4 Drift-Free Simulation techniques in a multicurve context 59 4.1 Introduction . 59 4.2 LMM extension to multicurve setting and DFS techniques . 61 N Ml N+1−Ml 4.2.1 Multicurve LMM 1: modeling fFjgj=1 and fLj gj=1 , l = 1; :::; x ............................... 64 N Ml N+1−Ml 4.2.2 Multicurve LMM 2: modeling fFjgj=1 and fSj gj=1 , l = 1; :::; x ............................... 67 Mp N+1−Mp 4.2.3 Multicurve LMM 3: modeling fLj gj=1 and N Ml N+1−Ml fFjgj=N+2−Mp , for some index p, and fSj gj=1 , l = 1; :::; x ............................. 72 N Ml N+1−Ml 4.2.4 Multicurve LMM 4: modeling fFjgj=1 and fJj gj=1 , l = 1; :::; x ............................... 73 II Drift-Free Simulation Techniques for Cross-Market Mod- els 79 5 Cross-Market Models and Drift-Free Simulation techniques 81 6 Drift-Free Simulation techniques for a First Cross-Market Model 87 6.1 Introduction . 87 6.2 Formulation . 88 6.3 DFS technique . 90 6.4 Model calibration . 91 6.5 Dynamics of the remaining of forward exchange rates . 93 6.6 Example of the First Cross-Market Model: Two currencies . 95 6.6.1 Cross-currency derivatives: Quanto Caplets . 95 6.6.2 Numerical results . 98 7 Drift-Free Simulation techniques for a Second Cross-Market Model105 7.1 Introduction . 105 7.2 Formulation . 106 7.3 DFS technique . 107 7.4 Model calibration . 109 7.5 Dynamics of the economy B forward rates . 111 7.6 Example of the Second Cross-Market Model: Commodities . 112 7.6.1 The convenience yield . 113 7.6.2 Commodity derivatives: Spread Options . 117 7.6.3 Numerical results . 118 ii 7.7 Inflation . 122 III Some graphs tools for Market Models 125 8 Connecting graphs with Generic Market Models 127 A An auxiliary result in matrix algebra to proof (3.42) 135 B Number of possible Cross-Market Models 141 Conclusions 145 Resumen extenso 149 Bibliography 158 iii iv List of Tables 2.1 Parameters of the simulation procedure. 33 Sj S 2.2 Swap volatilities. We simplify the notation: l ≡ j;j+l.
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