Optimal early withdrawal and valuation of finite-horizon fund protection options

ABSTRACT When embedded in equity-linked annuities, the dynamic fund protection option automatically increases the number of primary fund units so that their total value does not fall below the (possibly stochastic) guaranteed level. As the option payoff is determined by the running maximum of a price ratio, the “standard” fund protection option can be excessively expensive. In this paper, we modify the design of the embedded option by relaxing the guarantee to take effect only when the upgraded fund value has fallen a certain level (the “spread”) below the reference fund value. Through a change of numeraire and appealing to the Markov and scaling property of the state variables, we are led to consider the fund protection option at inception, for which a decomposition formula for the American value can be derived. For the standard option, the early withdrawal premium is given by a single integral whose integrand is an explicit function of the early withdrawal boundary, whereas for the option with spread (K, say), the early withdrawal premium is given by a double integral whose integrand is an explicit function of a continuum of early withdrawal boundaries corresponding to options with spread not exceeding K. In both cases, a piecewise linear approximation of the early withdrawal boundary substantially simplifies the evaluation of the option value. Detailed algorithms are provided for the computation of both the prices and an entire family of early withdrawal boundaries for American fund protection options. For completeness, we also determine the prices and stationary boundaries of perpetual options with spread. Our numerical results show that the introduction of a spread in the option payoff is effective in making fund protection options more economically attractive.