Semi-Markov Driven Models: Limit Theorems and Financial Applications

Date
2015-06-12
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Abstract
This thesis deals with models driven by so-called semi-Markov processes, and studies some limit theorems and financial applications in this context. Given a system whose dynamics are governed by various regimes, a semi-Markov process is simply a process that ``keeps track" of the system regime at each time. It becomes fully Markovian if we ``add" to it the process keeping track of how long the system has been in its current regime. Chapter 1 consists of a global introduction to the thesis. We introduce the concepts of semi-Markov and Markov renewal processes, and give a brief overview of each chapter, together with the main results obtained. Chapter 2 introduces a semi-Markovian model of high frequency price dynamics: as suggested by empirical observations, it extends recent results to arbitrary distributions for limit order book events inter-arrival times, and both the nature of a new limit order book event and its corresponding inter-arrival time depend on the nature of the previous limit order book event. Chapter 3 establishes strong law of large numbers and central limit theorem results for time-inhomogeneous semi-Markov processes, for which the kernel is time-dependent. Chapter 4 develops a rigorous treatment of so-called inhomogeneous semi-Markov driven random evolutions, and extends already existing results related to the time-homogeneous case. Random evolutions allow to model a situation in which the dynamics of a system are governed by various regimes, and the system switches from one regime to another at random times. This phenomenon will be modeled by using semi-Markov processes. The notion of ``time-inhomogeneity" appears twice in our framework: random evolutions will be driven by inhomogeneous semi-Markov processes (using results from chapter 3), and constructed with propagators, which are time-inhomogeneous counterparts of semigroups. Chapter 5 presents a drift-adjusted version of the well-known Heston model - the delayed Heston model - which allows us to improve the implied volatility surface fitting. Pricing and hedging of variance and volatility swaps is also considered. Finally, chapter 6 concludes the thesis and presents some possible future research directions.
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Keywords
Economics--Finance, Mathematics
Citation
Vadori, N. N. (2015). Semi-Markov Driven Models: Limit Theorems and Financial Applications (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/27749