of pricing European options if the market price of volatility risk was constant. The problem was revisited in [@MM04], where different risk premiums were considered and some explicit solutions were found. Here we will tackle the problem of pricing perpetual American options within a framework where volatility but also dividend rate may perform a single transition at some unknown instant in the future.

The paper is structured as follows: In we present the market model and its general properties. In we introduce the concept of financial derivative, and explore the links between finance and physics in the context of American derivatives. is devoted to the subject of pricing perpetual American options: we stress the financial interest of these ideal derivatives and emphasize the potentials of the analytical expressions found. In we show some illustrative examples and discuss their implications. Conclusions are drawn in , and the paper ends with two appendices. In \[SA\] we revisit the problem by following a different approach, and \[SB\] deals with the smooth pasting condition for vanilla options.

The market model {#S2}
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Let us begin with the general description of our set-up. We will consider a financial market where the non-deterministic stock $S$ is 