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Semimartingales and Contemporary Issues in Quantitative Finance

Abstract : In this thesis, we study various contemporary issues in quantitative finance. The first chapter is dedicated to the stability of the semimartingale property under filtration expansion. We study first progressive filtration expansions with random times. We show how semimartingale decompositions in the expanded filtration can be obtained using a natural link between progressive and initial expansions. The link is, on an intuitive level, that the two coincide after the random time. We make this idea precise and use it to establish known and new results in the case of expansion with a single random time. The methods are then extended to the multiple time case, without any restrictions on the ordering of the individual times. We then look to the expanded filtrations from the point of view of filtration shrinkage. We turn then to studying progressive filtration expansions with processes. Using results from the weak convergence of sigma fields theory, we first establish a semimartingale convergence theorem, which we apply in a filtration expansion with a process setting and provide sufficient conditions for a semimartingale of the base filtration to remain a semimartingale in the expanded filtration. A first set of results is based on a Jacod's type criterion for the increments of the process we want to expand with. An application to the expansion of a Brownian filtration with a time reversed diffusion is given through a detailed study and some known examples in the litterature are recovered and generalized. Finally, we focus on filtration expansion with continuous processes and derive two new results. The first one is based on a Jacod's type criterion for the successive hitting times of some levels and the second one is based on honest times assumptions for these hitting times. We provide examples and see how those can be used as first steps toward harmful dynamic insider trading models. In the expanded filtration the finite variation term of the price process can become singular and arbitrage opportunities (in the sense of FLVR) can therefore arise in these models. In the second chapter, we reconcile structural models and reduced form models in credit risk from the perspective of the information induced credit contagion effect. That is, given multiple firms, we are interested on the behaviour of the default intensity of one firm at the default times of the other firms. We first study this effect within different specifications of structural models and different levels of information. Since almost all examples are non tractable and computationally very involved, we then work with the simplifying assumption that conditional densities of the default times exist. The classical reduced-form and filtration expansion framework is therefore extended to the case of multiple, non-ordered defaults times having conditional densities. Intensities and pricing formulas are derived, revealing how information-driven default contagion arises in these models. We then analyze the impact of ordering the default times before expanding the filtration. While not important for pricing, the effect is significant in the context of risk management, and becomes even more pronounced for highly correlated and asymmetrically distributed defaults. We provide a general scheme for constructing and simulating the default times, given that a model for the conditional densities has been chosen. Finally, we study particular conditional density models and the information induced credit contagion effect within them. In the third chapter, we provide a methodology for a real time detection of bubbles. After the 2007 credit crisis, financial bubbles have once again emerged as a topic of current concern. An open problem is to determine in real time whether or not a given asset's price process exhibits a bubble. Due to recent progress in the characterization of asset price bubbles using the arbitrage-free martingale pricing technology, we are able to propose a new methodology for answering this question based on the asset's price volatility. We limit ourselves to the special case of a risky asset's price being modeled by a Brownian driven stochastic differential equation. Such models are ubiquitous both in theory and in practice. Our methods use non parametric volatility estimation techniques combined with the extrapolation method of reproducing kernel Hilbert spaces. We illustrate these techniques using several stocks from the alleged internet dot-com episode of 1998 - 2001, where price bubbles were widely thought to have existed. Our results support these beliefs. During May 2011, there was speculation in the financial press concerning the existence of a price bubble in the aftermath of the recent IPO of LinkedIn. We analyzed stock price tick data from the short lifetime of this stock through May 24, 2011, and we found that LinkedIn has a price bubble. The last chapter is about discretely sampled variance swaps, which are volatility derivatives that trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this chapter is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled variance swap values are finite, given the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled variance swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both variance swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values.
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Submitted on : Thursday, October 27, 2011 - 10:23:57 AM
Last modification on : Wednesday, March 27, 2019 - 4:08:30 PM
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Younes Kchia. Semimartingales and Contemporary Issues in Quantitative Finance. Computational Finance [q-fin.CP]. Ecole Polytechnique X, 2011. English. ⟨pastel-00635436⟩

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