Investing for General Utility: Results for the Black-Scholes-Hull-White Model

  • Thijs Kamma Thijs Kamma

This study is concerned with the analyses and derivations of a stochastic optimal terminal wealth problem in the context of the DB-DC trend regarding its transition within the pension industry. A comparison in terms of this problem is conducted by means of a thorough analytical as well as an empirical analysis of the effects due to a Black-Scholes economy specification and an equivalent economy with inclusion of stochastic interest rates following a Hull-White diffusion process (BSHW), such that the con-
sequences of these included stochastic interest rates become apparent. The problem describes a setting in which an agent maximizes her expected utility under some budget constraint. This essential optimal control problem deviates from the well-known Merton’s terminal wealth and consumption problems in the sense that a Cox-Huang methodology is utilized rather than the widely-applied HJB approach. Inherent in this problem and model specification, the phenomenon of the stochastic discount factor plays a central role, for which the analytical representations are derived given the former Black-Scholes and Black-Scholes-Hull-White economies. The problem is cast into a simplified non-robust framework for a complete market stetting, being nested under the class of infinite-dimensional optimization problems, as addressed in a general setting as well. Along with the economies, we consider a set of three utility specifications, i.e. power, exponential, and kinked utility, implicit in the agent’s risk prolfie, in order to compare the results among i.a. ARA and RRA utilities. In accordance with the latterly addressed optimal wealth, the correspondent hedging strategies are derived in an equivalent sense, whereafter these are simulated under recently calibrated parameters, altogether, providing an illustration of the effects of interest. The results indicate an extremely apparent non-negligible difference in respect of the agent’s optimal wealth due to inclusion of a stochastic nature for the interest rates. Within the borders of this framework, given the desired feature of stochasticity with regard to interest rates, we conclude that these mark an essential characteristic that should be included in most of the life-cycle literature, for purposes that serve its relevance for applied settings; specifically the individualized DC setting at hand.

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