Dissertation
A contribution to modeling tail dependence
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2020
DOI: 10.17077/etd.005530
Abstract
In our progressively interconnected world there is a plethora of examples of random phenomena that during normal times beguilingly show markedly less or insignificant interrelationship than during a crisis when they exhibit similarly large relative magnitude of departures from their respective norms. Examples include a set of disparate assets that move in tandem during market crashes such as the one during the 2007-2008 financial crisis, and lines of insurance that together suffer enormous losses during natural catastrophes such as Hurricane Katrina. Appropriate modeling of such phenomena is a challenge that risk managers face, which has only become more so with the opacity accompanying increasing layers of securitization. Nevertheless, not doing so can result in grossly underestimating risk, which in turn may cause imprudent investment decisions, or cause ill provisioning of capital, or even end up contributing towards making of a systemic crisis. This thesis is a contribution to the literature that seeks to facilitate better modeling of such random phenomena.
The interrelationship-in-extremes alluded to above is referred to as tail dependence, and a popular measure for it, when considering two random phenomena, is the tail dependence coefficient. For a set of d random phenomena, the bidimensional array of these coefficients across all pairs serves as a measure of multivariate tail dependence, and is called the Tail Dependence Matrix (TDM). While its construction is analogous to that of a correlation matrix, it is surprisingly computationally hard to verify if a given matrix is a TDM matrix, whereas a given d by d matrix can be determined to be a correlation matrix or otherwise in O(d^3) basic computational steps. In the first part of the thesis we lend further support to the belief that this problem, the TDM realization problem, is NP-hard. We do this via a Linear Programming (LP) formulation, and by showing that the combinatorial structure of the constraints is related to an intractable version of the max-cut problem in a weighted graph. Moreover, we significantly improve the performance of a recently proposed algorithm for the realization problem by substituting the solution of an NP-hard sub-problem with the solution of a polynomial time relaxation.
Despite the above efforts, the state-of-the-art algorithm for the realization problem has a ceiling on d below 100, with d>100 being of much practical import. Since especially in higher dimensions one resorts to lower dimensional parametrization of the TDMs, we posit that it is rather the complexity of the realization problem restricted to parametric classes of TDMs, and algorithms for it, that are more practically relevant. In the second part of this thesis, we show how the inherent symmetry and sparsity in a parametrization can be exploited to achieve a significant reduction in the LP formulation, which can lead to polynomial complexity of such realization problems - some parametrizations even resulting in the constraint set being free of d.
When considering a copula model for a set of random phenomena exhibiting tail dependence, we assert that an important factor must be the richness of the subset of TDMs that it can accommodate. In the final part of this thesis, we focus on the t-copula family that is a popular choice of modelers in the presence of tail dependence. We discuss some geometric properties of the subset of the TDMs that it supports, and importantly provide an efficient algorithm to determine its parameters to best capture the tail dependence specified by a target TDM. We also provide the source code of our implementation of this algorithm on the MATLAB software environment.
Details
- Title: Subtitle
- A contribution to modeling tail dependence
- Creators
- Siyang Tao
- Contributors
- Nariankadu D Shyamalkumar (Advisor)Thomas R Berry-Stöelzle (Committee Member)Ralph P Russo (Committee Member)Elias S W Shiu (Committee Member)Sanvesh Srivastava (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Statistics
- Date degree season
- Summer 2020
- DOI
- 10.17077/etd.005530
- Publisher
- University of Iowa
- Number of pages
- xii, 173 pages
- Copyright
- Copyright 2020 Siyang Tao
- Language
- English
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (pages 152-156).
- Public Abstract (ETD)
In our interconnected world there is a plethora of examples of random phenomena that during normal times beguilingly show markedly insignificant interrelationship than in a crisis. Examples include a set of disparate assets that move in tandem during market crashes such as the one in 2008, and lines of insurance that together suffer enormous losses during catastrophes such as Hurricane Katrina. Appropriate modeling of such phenomena is a challenge. Nevertheless, not doing so can result in grossly underestimating risk, which can even contribute towards making of a systemic crisis. This thesis contributes to better modeling of such random phenomena.
This interrelationship-in-extremes is known as tail dependence, and a popular measure for it, when considering two random phenomena, is the tail dependence co-efficient. For a set of phenomena, the bidimensional array of these coefficients across all pairs is called the Tail Dependence Matrix (TDM). The first part of this thesis contributes to the state-of-the-art of determining if a given matrix equals the TDM of any random phenomena, albeit synthetic - a problem that is surprisingly computationally hard. When a large number of phenomena are modeled, for the sake of parsimony, the indexed set of models considered is often low dimensional. The second part demonstrates that the former problem restricted to this setting can be computationally tractable even though it is infeasible in its full generality. The final part restricts attention to the popular t-copula indexed model, and provides an algorithm to determine its index to best model a target random phenomena.
- Academic Unit
- Statistics and Actuarial Science
- Record Identifier
- 9983988297402771
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