POLITECNICO DI MILANO Corso di Laurea Magistrale di Ingegneria Matematica Facolt`a di Ingegneria dei Sistemi Pricing di Derivati del Credito e Credit Valuation Adjustment su multi-GPU Relatore: Professor Daniele Marazzina Tesi di Laurea di: Giorgio G. Re matr. 799260 Anno Accademico 2013-2014 Failure is not an option. Abstract The aim of this work is to show a financial application of calculus on GPUs, i.e. graphic processing units. Using CUDA, a parallel pro- gramming model created by nVIDIA for general purpose computing, we face the pricing problem of some credit derivatives, such as Nth- to-DefaultSwapsandSyntheticCollateralizedDebtObligations,and the calculation of Expected Exposure and Credit Valuation Adjust- ment of an Interest Rate Swap, that is the credit exposure plus a price correction in order to take into account the counterparty credit risk. Forthispurpose,weexploitMonteCarloandQuasiMonteCar- lomethodsforsimulations. Moreoverwemodeljointdefaulttimesof the underlying reference portfolio of Nth-to-Default Swaps and Col- lateralized Debt Obligations, considering Gaussian and Student’s t copulas, and we deal with Vasicek, Hull-White and Cox-Ingersol- Ross interest rate models to evaluate Expected Exposure and Credit Valuation Adjustment. Therefore we created a C++/CUDA library1 which implements and solves the problems introduced. This dissertation is arranged in th- ree parts: at the beginning we formally describe the structure of the derivatives and the models and methods used, then we analyze our source codes and finally we show the obtained results. Keyword: CUDA, multi-GPU, Student’s t Copula, Monte Carlo, Quasi Monte Carlo (QMC), Pricing Credit Derivatives, Counterpar- ty Credit Risk, Expected Exposure, Credit Valuation Adjustment (CVA). 1Thelibraryisfreelyavailabletodownloadfromhttps://github.com/giogio12345/Credit/releases/ latest. 5 Sommario Lo scopo di questo elaborato `e mostrare un’applicazione in ambito finanziario del calcolo su GPU, cio`e sulle unit`a di elaborazione grafi- ca. Utilizzando CUDA, un paradigma di programmazione parallela offerto da nVIDIA che permette appunto di sfruttare le schede gra- fiche, abbiamo affrontato il problema di pricing di alcuni derivati del credito, ovvero Nth-to-Default Swap e Collateralized Debt Obli- gation sintetici, e il calcolo di Expected Exposure e Credit Valuation Adjustment di un Interest Rate Swap, ovvero l’esposizione creditizia e la correzione al prezzo del derivato in modo da tenere conto del rischio di controparte. Per risolvere questi problemi abbiamo esegui- to simulazioni con metodi di Monte Carlo e Quasi Monte Carlo, e abbiamo utilizzato copule gaussiane e t-Student per modellizzare le probabilit`a di default dei titoli sottostanti agli Nth-to-Default Swap e ai Collateralized Debt Obligation, e modelli di Vasicek, Hull-White e Cox-Ingersol-Ross per la simulazione dei tassi di interesse utilizzati per valutare Expected Exposure e Credit Valuation Adjustment. Questo lavoro `e composto da una libreria1 scritta in C++/CUDA che implementa e risolve questi problemi e da questa tesi, che descri- ve prima in maniera formale i titoli finanziari succitati e i modelli e i metodi utilizzati, poi analizza il codice implementato e infine espone i risultati ottenuti. Keyword: CUDA, multi-GPU, Copula t-Student, Monte Carlo, Quasi Monte Carlo (QMC), Pricing di Derivati del Credito, Ri- schio di Controparte, Expected Exposure, Credit Valuation Adjust- ment (CVA). 1La libreria `e scaricabile gratuitamente da https://github.com/giogio12345/Credit/releases/ latest. 7 Indice Introduzione 19 I Cenni Teorici e Strumenti Utilizzati 23 1 Derivati del Credito 25 1.1 Rischio di Credito e Mercati OTC . . . . . . . . . . . . . . . . . . . . . . . 25 1.2 Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3 Interest Rate Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4 Basket Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5 Collateralized Debt Obligation . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.5.1 Synthetic CDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2 Rischio di Controparte e Credit Valuation Adjustment 37 2.1 Rischio di Controparte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Unilateral Credit Valuation Adjustment . . . . . . . . . . . . . . . . . . . . 41 2.3 Bilateral Credit Valuation Adjustment . . . . . . . . . . . . . . . . . . . . . 42 3 Copule e Tempi di Default 45 3.1 Copule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.1 Copula Gaussiana . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.2 Copula t-Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.3 Confronto fra Copula Gaussiana e t-Student . . . . . . . . . . . . . . 54 3.2 Modelli per Tempi di Default . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Modelli di Tasso 61 4.1 Zero Coupon Bond, Tassi forward e short . . . . . . . . . . . . . . . . . . . 61 4.2 Modello di Vasicek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Modello di Hull-White . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4 Modello di Cox-Ingersoll-Ross . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5 Metodi Monte Carlo e Quasi Monte Carlo 69 5.1 Principi dei Metodi Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Numeri Pseudo Casuali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.3 Metodi Quasi Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.4 Sequenza di Sobol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9 Indice 5.5 Inversione e Acceptance-Rejection . . . . . . . . . . . . . . . . . . . . . . . . 77 5.5.1 Inversione . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.5.2 Acceptance-Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 CUDA: Compute Unified Device Architecture 79 6.1 Architettura della GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2 Modello CUDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3 Gestione della memoria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.4 Librerie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.4.1 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.4.2 cuBLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.4.3 cuRAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.5 Compilazione del Codice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.6 Librerie Esterne e Strumenti Utilizzati . . . . . . . . . . . . . . . . . . . . . 90 6.6.1 Eigen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.6.2 OpenMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.6.3 GitHub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.6.4 Profilers e Memory Checkers . . . . . . . . . . . . . . . . . . . . . . 91 II Algoritmi e Implementazione 93 7 Struttura del Codice e Propriet`a del C++ 95 7.1 Struttura del Codice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2 Ereditariet`a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.3 Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.4 CUDA e C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8 Default Time Generator 99 8.1 Struttura delle Classi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.2 Classe DefTimesGen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.3 Classe DefTimesGen GaussianCopula. . . . . . . . . . . . . . . . . . . . . . . 102 8.4 Classe DefTimesGen tCopula . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9 Credit Derivative 111 9.1 Il costruttore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10 Nth-to-Default Swap 117 10.1 La funzione run cpu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 10.2 La funzione run gpu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11 Collateralized Debt Obligation 125 11.1 La funzione run cpu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 11.2 La funzione run gpu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 10