the optimal exercise boundary is implicitly defined by the integral equation that determines the price of the option. Only under certain circumstances closed expressions for American option prices do exist: e.g. in the case in which the properties of the derivative lead to a constant early exercise price, as in Rubinstein and Reiner [@RR91], or when the option is perpetual, as in Kim [@IJK90], and also in Elliott and Chan [@EC04], where the stock is driven by fractional Brownian motion. In the most general scenario analytical or numerical approximate methods must be used instead —see, for instance, Barone-Adesi and Whaley [@BAW87], Broadie and Detemple [@BD96], Ju [@NJ98], Broadie [*et al*]{} [@BDGT00], and references therein as well.

In this article we will generalize a market model first introduced by Herzel [@H98] as a simplified version of Naik’s work. Naik [@N93] developed a model in which the volatility can take only two known values, and the market switches back and forth between them in a random way. Herzel let the volatility jump at most once: a suitable way for encoding a market that may undergo a severe change in volatility only if some forthcoming event takes place. Herzel formally solved the problem 