Volatility and dividend risk in perpetual American options

American options are financial instruments that can be exercised at any time before expiration. In this paper we study the problem of pricing this kind of derivatives within a framework in which some of the properties —volatility and dividend policy— of the underlaying stock can change at a random instant of time, but in such a way that we can forecast the final conditions. These peculiar assumptions can still model some actual market conditions: some hypothetical but relevant facts may have sharp predictable consequences on a firm. We will show the effects of this potential risk on perpetual American derivatives, a topic connected with a wide class of recurrent problems in physics: holders of American options must look for the fair price and the optimal exercise strategy at once, a typical question of free absorbing boundaries. We present explicit solutions to the most common contract specifications and derive analytical expressions concerning the mean and higher moments of the exercise time. PACS numbers: 89.65.Gh, 02.50.Ey, 05.40.Jc