Risk Management with Basis Risk
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Date
2018-06-19
Authors
Zhang, Jingong
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
Basis risk occurs naturally in a variety of financial and actuarial applications, and it introduces additional complexity to the risk management problems. Current literature on quantifying and managing basis risk is still quite limited, and one class of important questions that remains open is how to conduct effective risk mitigation when basis risk is involved and perfect hedging is either
impossible or too expensive. The theme of this thesis is to study risk management problems in the presence of basis risk under three settings: 1) hedging equity-linked financial derivatives; 2) hedging longevity risk; and 3) index insurance design.
First we consider the problem of hedging a vanilla European option using a liquidly traded asset which is not the underlying asset but correlates to the underlying and we investigate an optimal construction of hedging portfolio involving such an asset. The mean-variance criterion is adopted to evaluate the hedging performance, and a subgame Nash equilibrium is used to define the optimal solution. The problem is solved by resorting to a dynamic programming procedure and a change-of-measure technique. A closed-form optimal control process is obtained under a general diffusion model. The solution we obtain is highly tractable and to the best of our knowledge, this is the first time the analytical solution exists for dynamic hedging of general vanilla European options with basis risk under the mean-variance criterion. Examples on hedging European call options are presented to foster the feasibility and importance of our optimal hedging strategy in the presence of basis risk.
We then explore the problem of optimal dynamic longevity hedge. From a pension plan sponsor’s perspective, we study dynamic hedging strategies for longevity risk using standardized securities in a discrete-time setting. The hedging securities are linked to a population which may differ from the underlying population of the pension plan, and thus basis risk arises. Drawing
from the technique of dynamic programming, we develop a framework which allows us to obtain analytical optimal dynamic hedging strategies to achieve the minimum variance of hedging error. For the first time in the literature, analytical optimal solutions are obtained for such a hedging problem. The most striking advantage of the method lies in its flexibility. While q-forwards are considered in the specific implementation in the paper, our method is readily applicable to other securities such as longevity swaps. Further, our method is implementable for a variety of longevity models including Lee-Carter, Cairns-Blake-Dowd (CBD) and their variants. Extensive numerical experiments show that our hedging method significantly outperforms the standard “delta” hedging strategy which is commonly adopted in the literature.
Lastly we study the problem of optimal index insurance design under an expected utility maximization framework. For general utility functions, we formally prove the existence and uniqueness of optimal contract, and develop an effective numerical procedure to calculate the optimal solution. For exponential utility and quadratic utility functions, we obtain analytical expression of the optimal indemnity function. Our results show that the indemnity can be a highly non-linear and even non-monotonic function of the index variable in order to align with the actuarial loss variable so as to achieve the best reduction in basis risk. Due to the generality of model setup, our proposed method is readily applicable to a variety of insurance applications including
index-linked mortality securities, weather index agriculture insurance and index-based catastrophe insurance. Our method is illustrated by a numerical example where weather index insurance is designed for protection against the adverse rice yield using temperature and precipitation as the underlying indices. Numerical results show that our optimal index insurance significantly outperforms linear-type index insurance contracts in terms of reducing basis risk.
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Keywords
Basis Risk, Optimal Hedging, Longevity Risk, Index Insurance, Agricultural Insurance