Robust Optimizations For Equity-Linked Products

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The main objective of this proposal is to develop robust hedging strategies for equity-linked products (ELPs). ELPs form a class of insurance products that offer limited participation in the performance of an equity index (Equity-indexed Annuity) or a mutual fund (Variable Annuity) while providing a predetermined guaranteed amount. ELPs are considered as long-term financial derivatives that include death, surrender, withdrawal, and/or accumulation guarantees. In analyzing the risk underlying these guarantees, Augustyniak & Boudreault (2012) study several econometric models and conduct out-of-sample analyses of these models, using the financial crisis (and the associated observed equity-linked returns) as the in-sample period. They observe that tail risk measures significantly vary across the various models. This stresses the importance of carefully selecting the model when hedging investment guarantees. Given that the Canadian Institute of Actuaries recommends the use of stochastic models for reserving future losses on ELPs, it is important to have a measure that can serve as a yardstick to both insurers and regulators in comparing various models, hence the consideration for robust approaches to evaluate ELPs. I propose to derive hedging strategies under worst-case scenarios and robust control approaches. Both concepts are robust adaptations of risk-control strategies introduced by Gaillardetz & Hachem (2019). Osei Mireku* (2019) investigates the robust counterpart hedging strategy when the CVaR is used in the constraint. He presents approximate solutions by sampling different uncertainty sets of probability mass functions. I intend to investigate other approaches (e.g. numerical methods) since simulation results are not consistent. Zhu & Fukushima (2009) apply the concept of worst-case local CVaR in portfolio management. Gaillardetz & Hachem (2019) show that hedging strategies obtained by minimizing the local CVaR are outperformed by strategies based on the coherent dynamic risk measure, which minimizes the local CVaR while penalizing the future losses. I propose to apply the results from Zhu & Fukushima (2009) and derive the worst-case coherent dynamic risk measure. I propose to investigate robust hedging strategies, where the risk measures are replaced by non-stochastic measurements in the risk-control strategies. These generalize the expensive super-replicating strategy in which no positive loss is allowed. The non-stochastic measurements may concede some positive losses, but restrain them by imposing constraints. Based on the results of Ben-Tal et al. (2009), computationally tractable equivalent reformulations can be used to relax the discrete assumption in the underlying financial process. The relaxation assumes a bounded continuous financial process, which is usually constrained using some upper and lower bounds. In this case, the solutions can be represented by the convex combination of the extremes.