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Stochastic Volatility Libor Modeling and Efficient Algorithms for Optimal Stopping Problems

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Stochastic Volatility Libor Modeling and Efficient Algorithms for Optimal Stopping Problems Stochastic volatility Libor modeling and efficient algorithms for optimal stopping problems DISSERTATION zur Erlangung des akademischen Grades Dr. rer. nat. im Fach Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät Humboldt-Universität zu Berlin von Dipl.-Math. Marcel Ladkau Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaftlichen Fakultät: Prof. Dr. Elmar Kulke Gutachter: 1. Dr. John Schoenmakers 2. Prof. Dr. Denis Belomestny 3. Prof. Dr. Dirk Becherer eingereicht am: 20.01.2015 Tag der mündlichen Prüfung: 23.07.2015 Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit verschiedenen Aspekten der Finanzmathema- tik. Ein allgemeiner Rahmen wird erarbeitet in welchem stochastische Prozesse existieren mit deren Hilfe Finanzinstrumente modelliert werden können. Die risikoneutrale Bewer- tung solcher Instrumente wird erläutert, wobei der Schwerpunkt auf Zinsvorwärtsraten, sogenannten Liboren liegt. Danach wird ein erweitertes Libor Markt Modell betrachtet, welches genug Flexibilität bietet um akkurat an verschiedene Marktdaten von Caplets und Swaptions zu kalibrie- ren. Dafür werden analytische Preisformeln für solche Instrumente für eine effiziente Kalibrierung benötigt. In diesem Modell erhalten wir solche Preisformeln nach einer Ap- proximation unter Zuhilfenahme des Konzeptes der Schnellen Fourier Transformierten. Weiterhin sind wir interessiert an der Bewertung komplexerer Finanzderivate, zum Beispiel durch Simulation. In einem Modell mit hohen Dimensionen können solche Si- mulationen sehr zeitaufwendig sein. Wir zeigen mögliche Verbesserungen bezüglich der Komplexität der Simulation auf. Faktorreduktion identifiziert die Haupfaktoren des Mo- dells und kann zu einer signifikanten Komplexitätsverringerung führen. Zusätzlich er- weitern wir ein Simulationsschema, eingeführt in Andersen[2007], von einer auf mehrere Dimensionen, wobei wir das Konzept des „Momentmatchings“ zur Approximation des Volaprozesses in einem Heston Modell nutzen. Daraus resultiert eine verbesserte Kon- vergenz des Gesamtprozesses und somit eine verringerte Komplexität bei der simulati- onsbasierten Bewertung von Optionen. Als Nächstes beschäftigen wir uns mit der Bewertung sogenannter Amerikanischer Op- tionen. Diese können als Stoppproblem interpretiert werden. In höheren Dimensionen, wie sie zum Beispiel bei Basketauszahlungsprofilen auftreten, ist die simulationsbasier- te Bewertung meist die einzig praktikable Lösung, da diese eine dimensionsunabhängige Konvergenz gewährleistet. Eine neue Methode der Varianzreduktion, die Multilevel-Idee, wird hier auf die simulationsbasierte Optionsbewertung angewandt. Wir leiten eine un- tere Preisschranke unter zu Hilfenahme der Methode der „policy iteration“ her. Dafür werden Konvergenzraten für die Simulation des Optionspreises erarbeitet und eine de- taillierte Komplexitätsanalyse dargestellt. Abschließend wird das Preisen von Amerikanischen Optionen unter Modellunsicherheit behandelt, wodurch die Restriktion, nur ein bestimmtes Wahrscheinlichkeitsmodell zu betrachten, entfällt. Verschiedene Modelle können plausibel sein und zu verschiedenen Optionswerten führen. Dieser Ansatz führt zu einem nichtlinearen, verallgemeinerten Erwartungsfunktional. Wir erhalten eine verallgemeinerte Snell’sche Einhüllende und leiten das Bellman Prinzip her. Dadurch kann eine Lösung durch Rückwärtsrekursion erhalten werden. Unser numerischer Algorithmus liefert untere und obere Preisschranken. ii Abstract The work presented here deals with several aspects of financial mathematics. A general framework is established in which the existence of stochastic processes can be guaranteed which can be used to model financial instruments. A risk neutral evaluation ofsuch instruments is explained, while focusing on the modeling of interest forward rates, so called Libors. These allow to work without the usual assumption of the existence of a risk neutral financial instrument. Next we deal with an extended Libor market model offering enough flexibility to ac- curately calibrate to various market data for caplets and swaptions. To compete with standard models with regard to calibration, analytical price formulas for European op- tions within this model are called for. Those are obtained after an approximation using the concept of Fast Fourier Transform. Further we are interested in the evaluation of more complex financial derivatives for instance by simulation. Due to the typically high dimension of the model involved such simulations can be very time consuming. We show possible improvements regarding the complexity of the simulation. Factor reduction, a method known from statistics, identifies the main driving factors of a model and may lead to a significant reduction of complexity. In addition we extend a known simulation scheme, established in Andersen[2007], from one to multiple dimensions using the concept of moment matching for the approximation of the vola process in a Heston model. This results in an improved convergence of the whole process thus reducing the complexity of a simulation based evaluation of options. Next we address the problem of evaluating so called American options. These vastly traded options can be interpreted as a stopping problem. An efficient evaluation of these options, particularly in high dimensions, is a delicate problem. For example when dealing with basket cash flows, a simulation based approach offering dimension independent convergence often happens to be the only practicable solution. A new method of variance reduction given by the multilevel idea is applied to this approach. Starting with the ideas of Belomestny and Schoenmakers[2011] a lower bound for the option price is obtained using “multilevel policy iteration” method. Convergence rates for the simulation of the option price are obtained and a detailed complexity analysis is presented. Finally we deal with the valuation of American options under model uncertainty. This lifts the restriction of considering one particular probabilistic model only. Different mo- dels might be plausible and may lead to different option values. This approach leads to a non-linear expectation functional, calling for a generalization of the standard expec- tation case. We obtain a generalized Snell envelope, enabling a backward recursion via Bellman principle. We then provide a numerical algorithm to valuate American options under ambiguity, obtaining lower and upper price bounds. iii Acknowledgement Many people have participated, directly or indirectly throughout my Ph. D. work and I would like to name and thank a few of them for their valuable contributions. I thank PD Dr. John Schoenmakers and Prof. Dr. Vladimir Spokoiny for giving me the opportunity to work in a very stimulating atmosphere at the Weierstrass Institute. Further I thank PD Dr. John Schoenmakers and Prof. Dr. Denis Belomestny for intro- ducing me to several fields and mathematical tools of financial mathematics. PDDr. John Schoenmakers is a remarkable supervisor and I thank him for his patience, taking the time for long discussions and his detailed advices. I am grateful for working and discussing several aspects of mathematics and beyond with friends and colleagues of the research group “Stochastic Algorithms and Nonparametric Statistics” at the Weierstrass Institute. Thanks to all of you for a most enjoyable time and all the best for the future. Special thanks go to Dr. Jianing Zhang for fruitful and joyful discussions. Further I thank Nikolai Strogies and Ulrich Wilbrandt, discussions with you always offered a new perspective different from the stochastic point of view. Financial support by Matheon is gratefully acknowledged. Last but not least I thank my family for supporting and encouraging me along my educational way and my girlfriend Lisa who gifted me my beloved daughter Lara Pauline, encouraged me and gave me the opportunity to step back from time to time. iv Contents Background and overview1 1. Modeling and Pricing of Interest Rates under No-Arbitrage Assumption 17 1.1. Arbitrage-free Pricing............................. 17 1.2. Measure Change................................ 26 1.3. Libor Rate Process and Libor Market Model................ 28 1.4. Swap Rate Process and Swap Market Model................. 37 2. A new multi-factor stochastic volatility model with displacement 40 2.1. Libor modeling in a new setting........................ 40 2.1.1. Instantaneous correlations....................... 41 2.1.2. Discussion of the Wu-Zhang model as a special case........ 42 2.2. Approximate caplet pricing and calibration................. 43 2.2.1. Caplet pricing via characteristic function.............. 44 2.2.2. Carr & Madan inversion formula................... 47 2.2.3. Putting the caplet approximation to the test............ 48 2.2.4. Further structuring and calibration.................. 49 2.2.5. Calibration to caplet volatility-strike-maturity........... 51 2.3. Swap rate dynamics and approximate swaption pricing........... 51 2.3.1. Swap contracts and dynamics under swap measures........ 51 2.3.2. Approximate affine swap rate dynamics............... 54 2.3.3. Fourier based swaption pricing.................... 56 2.3.4. Putting the swaption approximation to the test........... 57 2.4. Advanced calibration.............................. 58 2.5. Outlook..................................... 60 3. Simulation and related optimization ideas 64 3.1. Simulation.................................... 64 3.2.
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