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Mean-ρ portfolio selection
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Khan, Nazem (2022) Mean-ρ portfolio selection. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3817641
Abstract
The three pillars of Mathematical Finance are optimal investment, pricing, and risk management. In this thesis, we intertwine all three in the context of a one-period economy by replacing the variance in mean-variance portfolio selection by a risk measure ρ. This entanglement stems from ρ-arbitrage, which is a generalisation of ordinary arbitrage where – unlike in the classical theory of Markowitz – no efficient portfolios exist.
We first assume an Expected Shortfall (ES) risk constraint and prove that the market does not admit ES-arbitrage at confidence level α if and only if there exists an equivalent martingale measure Q ≈ P such that dQ dP ∞ < 1 α.
We then quantify risk by a general positively homogeneous risk measure. After providing a primal characterisation of ρ-arbitrage we prove that it cannot be excluded in this setting unless ρ is as conservative as the worst-case risk measure. In the case where ρ is a coherent risk measure that admits a dual representation, we further give a necessary and sufficient dual characterisation of ρ-arbitrage. This is intimately linked to the interplay between the set of equivalent martingale measures for the discounted risky assets and the set of absolutely continuous measures in the dual representation of ρ.
We end our exploration by considering star-shaped risk measures. We introduce the new axiom of strong sensitivity to large losses and show it is key to ensure the absence of ρ-arbitrage. This leads to a new class of risk measures that are suitable for portfolio selection. Specialising to the case that ρ is convex and admits a dual representation allows us to derive equivalent dual characterisations of ρ-arbitrage as well as the property that ρ is suitable for portfolio selection. Finally, we introduce the new risk measure of Loss Sensitive Expected Shortfall, which is similar to and not more complicated to compute than Expected Shortfall, but suitable for portfolio selection – which Expected Shortfall is not.
Item Type: | Thesis (PhD) | ||||
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Subjects: | H Social Sciences > HG Finance Q Science > QA Mathematics |
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Library of Congress Subject Headings (LCSH): | Business mathematics, Finance -- Mathematical models, Financial risk, Arbitrage, Arbitrage, Portfolio management | ||||
Official Date: | March 2022 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Department of Statistics | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Henderson, Vicky ; Herdegen, Martin | ||||
Format of File: | |||||
Extent: | xi, 150 leaves : charts | ||||
Language: | eng |
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