Armstrong, Sebastian Peter (2018) Coherent risk measures, reserving, and transaction costs. PhD thesis, University of Warwick.

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Abstract

This thesis deals with reserving for risk in a dynamic multi-asset market. Chapter 1 contains an exposition of the basic concepts of reserving for risks under convex and coherent risk measures.

In Chapter 2, we provide a dual characterisation of the weak∗-closure of a finite sum of cones in L∞ adapted to a discrete time filtration Ft: the tth cone in the sum contains bounded random variables that are Ft-measurable. Hence we obtain a generalisation of Delbaen's m-stability condition [Delbaen, 2006a] for the problem of reserving in a collection of numéraires V, called V-m-stability, provided these cones arise from acceptance sets of a dynamic coherent measure of risk [Artzner et al., 1997, Artzner et al., 1999]. We also prove that V-m-stability is equivalent to time-consistency when reserving in portfolios of V, which is of particular interest to insurers.

In Chapter 3, we examine the problem of dynamic reserving for risk in multiple currencies under a general coherent risk measure. The reserver requires to hedge risk in a time-consistent manner by trading in baskets of currencies. We show that reserving portfolios in multiple currencies V are time-consistent when (and only when) a generalisation of Delbaen's m-stability condition [Delbaen, 2006a], termed optional V-m-stability, holds. We prove a version of the Fundamental Theorem of Asset Pricing in this context. We show that this problem is equivalent to dynamic trading across baskets of currencies (rather than just pairwise trades) in a market with proportional transaction costs and with a frictionless final period.

Chapter 4 deals with the related problem of trading to acceptability, where a claim X is acceptable if and only if the expected gain under each measure in a collection exceeds an associated floor.

Item Type:	Thesis (PhD)
Subjects:	H Social Sciences > HD Industries. Land use. Labor Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Mathematical statistics, Risk management -- Mathematical models
Official Date:	May 2018
Dates:	Date Event May 2018 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Jacka, Saul D.
Sponsors:	Engineering and Physical Sciences Research Council
Extent:	vi, 96 leaves
Language:	eng

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