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Essays on supersolutions of BSDEs and equilibrium pricing in generalized capital asset pricing models
| Content Provider | Semantic Scholar |
|---|---|
| Author | Mainberger, Christoph |
| Copyright Year | 2014 |
| Abstract | In this thesis we study supersolutions of backward stochastic differential equations (BSDEs) and equilibrium pricing within two specific generalized capital asset pricing models (CAPMs). In the first part of the thesis we begin by assuming that the generators of the BSDEs under consideration are jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfy a specific normalization property. In our first main result we prove the existence and uniqueness of the minimal supersolution making use of a particular kind of semimartingale convergence and a suitably defined preorder on the set of supersolutions in combination with Zorn's lemma. In addition, we discuss possible relaxations of the assumptions imposed on the generator and extend our results from Brownian motion to arbitrary continuous local martingales. Next, we assume the generators to be convex and introduce constraints to our setting by restricting the admissible controls to continuous semimartingales Z of the form Z = z+ ∫ ∆du+ ∫ ΓdW and allowing for a dependence of the generator on the respective decomposition parts. We introduce a notion of constrained minimality and then prove existence of supersolutions that are minimal at finitely many fixed times within the class where controls coincide up to these times. Besides providing stability results for the non-linear operator E 0 (·) that maps a terminal condition to the value of the minimal supersolution at time zero, we give a dual representation of E 0 (·), including an explicit computation of the conjugate in the case of a quadratic generator, and derive conditions for the existence of solutions under constraints by means of the duality results. In the second part of the thesis we study equilibrium pricing in continuous time within affine and information-based CAPMs. Our model comprises finitely many economic agents and tradable securities. The agents seek to maximize exponential utilities and their endowments are spanned by the securities. In our first main result we show that an equilibrium exists and the agents' optimal trading strategies are constant and dependent on their respective risk aversion and endowment. In a next step, affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. Within both setups we derive semi-explicit pricing formulae which lend themselves to efficient numerical computations. In particular, no Monte Carlo methods are needed. Finally, we numerically analyze the impact of crucial parameters such as the agents' risk aversion or the intensity of jumps in the underlying's price on the implied volatility of simultaneously-traded European-style options within the affine framework, and investigate the dependence of credit-risky securities on the value of information about the financial standing of a company within the information-based framework. |
| File Format | PDF HTM / HTML |
| DOI | 10.18452/16916 |
| Alternate Webpage(s) | https://edoc.hu-berlin.de/bitstream/handle/18452/17568/mainberger.pdf?isAllowed=y&sequence=1 |
| Alternate Webpage(s) | http://edoc.hu-berlin.de/dissertationen/mainberger-christoph-2014-02-17/PDF/mainberger.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Thesis |
