This article develops a family of option pricing models when the underlying stock price dynamic is modeled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (1989) , in which volatility in°uences returns. In addition, our models allow for feedback e®ects from returns to volatilities. Our family also includes GARCH option models as a special

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... limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an e±cient algorithm for the computation of American option prices for the general case. This article develops a family of option pricing models obtained when the underlying stock dynamic is modeled by a Markov regime switching process. In such models, the stochastic process remains in one regime for a random amount of time before switching over into a new regime. In our case, the regimes are characterized by di®erent volatility levels. Rather than permitting volatilities to follow a continuous time, continuous state process, as in most stochastic volatility models, our primary focus is on cases where volatilities can take on a¯nite set of values, and can only switch regimes at¯nite times. In this regard, our models can be viewed as special cases of the large family of stochastic volatility models, in which the number of distributions for the logarithmic return are constrained to a¯nite collection. While this may, at rst glance, appear unnecessarily restrictive, our family of models includes as special limiting cases, many well-known models, including the family of GARCH option pricing models discussed by Duan (1995) . We also can obtain more general limiting models in which variance updating schemes depend not only on levels of variance and on asset innovations, but also on a second factor that is uncorrelated with asset returns. As a result variance levels are not completely determined by the path of prices. This second factor allows for further°exibility in capturing the properties of stock return processes. Our family of models also include the regime switching models of Hamilton (1989) as a special case. His models allow volatility regimes to impact returns, but they do not allow returns to impact future volatilities. Our models do permit this feedback e®ect. Thus our models allow us to capture the correlation between asset and volatility innovations, or equivalently, the asymmetric volatility response to good and bad news in asset returns. Our family of models¯lls the gap of models between the Black Scholes model, which in our framework can be viewed as a single volatility regime model with no feedback e®ects, and the extended GARCH models, which have in¯nitely many volatility regimes with feedback e®ects. We demonstrate that it is possible to establish models with a relatively small number of volatility regimes that produce option prices indistinguishable from models with a continuum of volatility states. We present an algorithm that permits American derivatives to be priced for our most general bi-directional regime switching process. If one is only interested in the rich family of GARCH option models, then our algorithm provides an alternative to the numerical procedure of Ritchken and Trevor (1999) and Duan and Simonato (1999) . The addition of the second orthogonal factor causes little complication for the algorithm, and provides meaningful extensions to processes beyond the GARCH family. Our study is certainly not the¯rst to consider regime switching mechanisms nor option pricing under regime switching. The early regime switching models were primarily designed to capture changes in the underlying economic mechanism that generated the data. Examples include Hamilton (1989) and Gray (1996) , Bekaert and Hodrick (1993) and Durland and McCurdy (1994) . Recently, attention has been placed on volatility regime switching models, solely for the purpose of better understanding option price behavior. Bollen, Gray and Whaley (1999) , for example, show that a very simple regime switching model with independent shifts in the mean and variance dominate a range of GARCH models in the foreign exchange market. Bollen (1998) presents a lattice based algorithm that permits American options