The Existence of Optimal Bang-Bang Controls for GMxB Contracts

A large collection of financial contracts offering guaranteed minimum benefits are often posed as control prob4 lems, in which at any point in the solution domain, a control is able to take any one of an uncountable number of 5 values from the admissible set. Often, such contracts specify that the holder exerts control at a finite number of 6 deterministic times. The existence of an optimal bang-bang control, an optimal control taking on only a finite 7 subset of values from the admissible set, is a common assumption in the literature. In this case, the numerical 8 complexity of searching for an optimal control is considerably reduced. However, no rigorous treatment as to 9 when an optimal bang-bang control exists is present in the literature. We provide the reader with a bang-bang 10 principle from which the existence of such a control can be established for contracts satisfying some simple 11 conditions. The bang-bang principle relies on the convexity and monotonicity of the solution and is developed 12 using basic results in convex analysis and parabolic partial differential equations. We show that a Guaranteed 13 Lifelong Withdrawal Benefits (GLWB) contract admits an optimal bang-bang control. In particular, we find that 14 the holder of a GLWB can maximize a writer’s losses by only ever performing nonwithdrawal, withdrawal at 15 exactly the contract rate, or full surrender. We demonstrate that the related Guaranteed Minimum Withdrawal 16 Benefits contract is not convexity preserving, and hence does not satisfy the bang-bang principle other than in 17 certain degenerate cases. 18