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This paper develops, under the framework of a small, open, and monetary economy, a stochastic model of inflation stabilization taking as a nominal anchor the exchange rate when credibility is imperfect. The agents have expectations driven by two processes: a diffusion-jump process for the devaluation rate where the size of a possible devaluation has an extreme value distribution, and a mean-reverting stochastic volatility process (a continuous version of the GARCH (1,1) model). This appropriately models that inflation is substantially more persistent than the devaluation rate; as showed the stylized facts about extreme devaluations registered in Mexico in 1994 and in Argentina in 2001. It is assumed that there is no a derivatives market to hedge against future devaluations, that is, the financial markets are incomplete. Under this framework, interior and corner solutions are examined when a stabilization plan with imperfect credibility is implemented. It is also studied an experiment in which the mean expected inflation takes a greater value from some time in the future and stays there forever, taking into account the probability that this monetary policy occurs. The case of a stochastic horizon of stabilization with the exponential distribution is studied. Moreover, the real option to postpone consumption is valued when a stabilization plan is to be abandoned. Also, exogenous shocks on consumption and economic welfare are assessed. Finally, the proposed model is used to carry out simulations that reproduce the booms of private consumption before the anti-inflationary plans were abandoned in Mexico in 1990-1994 and in Argentine in 2001-2003, when extreme devaluations took place. [ABSTRACT FROM AUTHOR]