First Passage Times of Diffusion Processes and Their Applications to Finance A thesis presented for the degree of Doctor of Philosophy Luting Li Department of Statistics The London School of Economics and Political Science United Kingdom April 2019 Declaration I certify that the thesis I have presented for examination for the Ph.D. degree of the London School of Economics and Political Science is solely my own work other than where I have clearly indicated that it is the work of others (in which case the extent of any work carried out jointly by me and any other person is clearly identified in it). The copyright of this thesis rests with the author. Quotation from it is permitted, provided that full acknowledgement is made. This thesis may not be reproduced without the prior written consent of the author. I warrant that this authorisation does not, to the best of my belief, infringe the rights of any third party. Statement of conjoint work A version of parts of Chapters 3-6 has been submitted for publication in two articles jointly authored with A. Dassios. A version of Chapter 7 has been submitted for publication jointly authored with H. Xing. i Acknowledgements Just before writing down the words appearing on this page, I sit at my desk on the 7th floor of the Columbia House, recalling those countless yet enjoyable days and nights that I have spent on pursuing knowledge and exploring the world. To me, the marvellous beauty of this journey is once in a lifetime. I would like to, first and foremost, express my sincere gratitude to my supervisors Angelos Dassios and Hao Xing, for providing me the opportunity of carrying out my research under their supervisions, for their constant support and the freedom they granted me to conduct my work according to my preferences. Whenever I was struggling on the research, it was Angelos who always pointed me towards the right direction, guided me out of puzzles, and taught me that persistence is the key to success. I am grateful to him for not only offering invaluable academic advice, but also spending time with me having coffees, talking travels, and sharing experiences, which broadened my horizons outside the academic world. I am indebted to Hao as my secondary supervisor for inviting me to the FRTB project which enriched this thesis in the dimension of quantitative risk management. He was more like a friend than a teacher to inspire and encourage me to excel in my work. Nevertheless, without the enormous help and precious comments from many experts, this thesis would not be presented in its current complete format. I am honoured for Goran Peskir and Luciano Campi being my examiners to give me invaluable suggestions and help in finalising this thesis. In addition, I would like to thank Jos´eA. Scheinkman and Michael Schatz for providing insightful advice on the financial bubble topic. I am grateful to Paul Embrechts, Demetris Lappas, Dirk Tasche, and Ruodu Wang, for their comments on the capital allocation project. I would also like to express my gratitude to the referees and editors from the Applied Probability Journals, the International Journal of Theoretical and Applied Finance, and the Risk Magazine, for their helpful comments and advice on three papers in this thesis. ii Great thanks go to Citigroup for providing the funding which allowed me to undertake my research. Particularly, I would like to express my sincere gratitude to Demetris Lappas, Wei Zhu, and Kevin Jian, for providing me with the opportunity of joining the Citi-LSE programme and gaining industrial experience alongside my research. My special thanks go to Damien Quinn for sharing his knowledge and introducing me a real financial world. I would also like to extend my thanks to Pierre-Yves Casteill, and George Dimitropoulos, for beautiful birthday gifts, offering me all those unforgettable and relaxing times of having drinks, dinners, and going for hiking, etc. Moreover, I express my sincere gratitude to all the staff and colleagues from the De- partment of Statistics at LSE for providing such a pleasurable environment. Particularly, I would like to thank my neighbour Jos´eM. Pedraza for his constant support in both academic and non-academic aspects. I gratefully acknowledge the helpful discussions from Yan Qu and Junyi Zhang about the research. And I would like to thank Ian Marshall and Penny Montague for their administrative support. I would like to express my deep friendship to Longjie Jia, Phoenix H. Feng, Cong Liu, Yusong Li, and Yupeng Jiang to share with me all those joy and sadness moments in the past four years in London. I wish to express my gratitude to Shuren Tan, for providing me timely help in Edinburgh. Also, I would like to give my special thanks to my car, for always driving me to office and journey, no matter sunny, raining, or snowing, and never abandoning me on its halfway. Naturally, I should not forget my teachers who gave me knowledge prior to my doctoral training and those friends who accompanied me though not in this country. I would like to thank my junior high school teacher Haiying Zhou who inspired me to realise the beauty of mathematics; my senior high school teacher Chengsheng Ma who encouraged me to always follow my heart; and my teacher Goran Peskir at the University of Manchester who introduced the stochastic calculus world to me. I would like to express my gratitude to Honghua Qiao, Chao Zhang, and Xuheng Lu, for their encouragement throughout these years. Given it is the last paragraph, I would like to express my deepest and the most heartfelt gratitude to my parents Chenghong Li and Peilan Ding, for their endless love and ultimate support which surround me all the time. My sincere thanks also go to all my family members, my dear friends, and to whom that have walked with me in this once in a lifetime journey. iii Abstract This thesis consists of three submitted papers and one working paper. It begins with the study of asymptotic solutions for the first passage time densities of various diffusion processes, and the thesis ends up with an application of such findings in the area of systematic trading. In between, financial bubbles and the regulatory risk management for the banking industry are studied additionally. The purpose of this thesis is to, by combining probability theory with financial practice, provide quantitative tools for investment decision and risk management. Chapters 3-5 are reorganised from the first passage time paper [31]. Our research method is mainly based on the potential theory and the perturbation theory. In Chapter 3, a unified recursive framework for finding first passage time asymptotic densities has been proposed. Besides, we prove the convergence of our framework and provide an error estimation formula. Examples related to the Ornstein-Uhlenbeck and the Bessel processes are demonstrated in Chapters 4 and 5, respectively. The second paper [30] is documented in Chapter 6. It introduces a new diffusion process which is relevant to financial bubbles. During the study of the first passage time, we occa- sionally found that the sample path of the new process coincides with log-price features of bubble assets. In Chapter 6, we show that the new model is a power-exponential transform of the Shiryaev process [116, 117]; and we prove that the model itself, indeed, satisfies var- ious technical requirements for defining a financial bubble [107]. Furthermore, by using our previous framework, we solve the closed-form asymptotic for the model's first passage time; and according to which, we have made predictions to the burst time of BitCoin. Chapter 7 is a modified version of the third paper [75]. We consider the risk capital allo- cation issue under the forthcoming regulatory framework, namely the Fundamental Review of Trading Book. Apart from studying coherent properties of the new risk measure, we propose two alternative capital allocation schemes within the range of Internal Modelling Approach. Our analysis shows that, different choices in allocation methods can lead significantly dif- iv ferent allocated capitals, therefore, impacting on bank's performance measure and capital optimisation. Our current working paper about systematic trading is demonstrated in Chapter 8. We propose two mathematical frameworks for, respectively, defining executable trading strate- gies and identifying the strategy-associated trading signals. Based on our definitions, we show how the first passage time can be employed in systematic trading. As a summary of applications to previous chapters, we use simulation analysis to illustrate the trading idea and the implementation of risk capital allocation. In the end, real data backtest on China stock market indicates that the first passage time could be an effective tool in recognising trading opportunities. v Contents 1 Introduction 1 2 Preliminary Definitions and Results 5 2.1 Stochastic Differential Equation . .5 2.2 First Passage Time . 10 2.3 Inverse Laplace Transform Algorithm . 14 2.4 Portfolio Selection, Risk Management and Capital Allocation . 17 2.5 Table of Nomenclature . 23 2.5.1 Abbreviation . 23 2.5.2 Set and Space . 24 2.5.3 Probability and Stochastic Process . 25 2.5.4 Operation and Operator . 26 2.5.5 Function . 27 3 Explicit Asymptotics on the First Passage Times of Diffusion Processes 29 3.1 Introduction, Motivation, and Literature Review . 29 3.2 Perturbed Dirichlet Problem . 32 3.3 Truncation Error and Convergence . 34 3.4 Recursion under Frequency Domain . 37 Appendix 3.A Alternative Error Estimation . 39 (N) Appendix 3.B Probabilistic Representation of pτ (t)................ 40 4 First Passage Time of Ornstein-Uhlenbeck Process 43 4.1 N-th Order Perturbed FPTD .