Pricing Perpetual Fund Protection with Withdrawal Option

Abstract

Consider an American option that provides the amount if it is exercised at time t, t ≥0. For simplicity of language, we interpret S1(t) and S2(t) as the prices of two stocks. The option payoff is guaranteed not to fall below the price of stock 1 and is indexed by the price of stock 2 in the sense that, if F(t) > S1(t), the instantaneous growth rate of F(t) is that of S2(t). We call this option the dynamic fund protection option. For the two stock prices, the bivariate Black-Scholes model with constant dividend-yield rates is assumed. In the case of a perpetual option, closed-form expressions for the optimal exercise strategy and the price of the option are given. Furthermore, this price is compared with the price of the perpetual maximum option, and it is shown that the optimal exercise of the maximum option occurs before that of the dynamic fund protection option. Two general concepts in the theory of option pricing are illustrated: the smooth pasting condition and the construction of the replicating portfolio. The general result can be applied to two special cases. One is where the guaranteed level S1(t) is a deterministic exponential or constant function. The other is where S2(t) is an exponential or constant function; in this case, known results concerning the pricing of Russian options are retrieved. Finally, we consider a generalization of the perpetual lookback put option that has payoff [F(t) − κS1(t)], if it is exercised at time t. This option can be priced with the same technique.