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First passage times of diffusion processes and their applications to finance

Li, Luting (2019) First passage times of diffusion processes and their applications to finance. PhD thesis, London School of Economics and Political Science.

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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 occasionally 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 various 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 allocation 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 different 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 strategies 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.

Item Type: Thesis (PhD)
Additional Information: © 2019 Luting Li
Library of Congress subject classification: H Social Sciences > HG Finance
Sets: Departments > Statistics
Supervisor: Xing, Hao and Dassios, Angelos

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