endogenously determined (in the sense that it must be determined jointly with the price of the security under consideration). The optimal exercise policy of an American option can be characterized as an endogenous stopping time.Valuation of American options in general, and valuation of fixed-income securities containing features of an American option in particular,is challenging,because closed-form solutions are rarely available and numerical computations (finite-difference,binomial-lattice,or Monte Carlo simulation)are typically very expensive (especially when there are multiple risk factors).As a result,approximation schemes are often used (see,e.g.,Longstaff and Schwartz [2001]),and considerable attention has been given to establishing upper and lower bounds on American option prices (e.g.,Haugh and Kogan [2001]and Anderson and Broadie [2001]). In the light of these complexities in pricing,some have questioned whether the optimal exercise strategies implicit in the parsimonious models typically used in practice are correctly valuing the American option feature of many products (e.g.,Andersen and Andreasen [2001]and Longstaff,Santa-Clara, and Schwartz [2001]).Of course,characterizing the optimal exercise policy itself can be challenging,particularly in the case of mortgage backed securi- ties,because factors other than interest rates may influence the prepayment behavior (e.g.,Stanton [1995). In "reduced-form"pricing models for defaultable securities(e.g.,Jarrow, Lando,and Turnbull [1997],Lando [1998],Madan and Unal [1998,and Duffie and Singleton [1999),the default time is typically modeled as the exogenous arrival time of an autonomous counting process.The claim to the recovery value of a defaultable security with maturity T is the present value of the payoff gr=g(Y,T)(recovery upon default)at the default arrival time T whenever T≤T: P(ti(a(Y,))=Bef rdg,l(rsn (9) This expression simplifies if r is the arrival time of a doubly stochastic Poisson process with state-dependent intensity A=A(Yi,t).At date t,the cumula- tive distribution of arrival of a stopping time before date s,conditional on {Yu:t≤u≤s}isPr(r≤s;tYu:t≤u≤s)=l-ex..It follows that 8endogenously determined (in the sense that it must be determined jointly with the price of the security under consideration). The optimal exercise policy of an American option can be characterized as an endogenous stopping time. Valuation of American options in general, and valuation of fixed-income securities containing features of an American option in particular, is challenging, because closed-form solutions are rarely available and numerical computations (finite-difference, binomial-lattice, or Monte Carlo simulation) are typically very expensive (especially when there are multiple risk factors). As a result, approximation schemes are often used (see, e.g., Longstaff and Schwartz [2001]), and considerable attention has been given to establishing upper and lower bounds on American option prices (e.g., Haugh and Kogan [2001] and Anderson and Broadie [2001]). In the light of these complexities in pricing, some have questioned whether the optimal exercise strategies implicit in the parsimonious models typically used in practice are correctly valuing the American option feature of many products (e.g., Andersen and Andreasen [2001] and Longstaff, Santa-Clara, and Schwartz [2001]). Of course, characterizing the optimal exercise policy itself can be challenging, particularly in the case of mortgage backed securi￾ties, because factors other than interest rates may influence the prepayment behavior (e.g., Stanton [1995]). In “reduced-form” pricing models for defaultable securities (e.g., Jarrow, Lando, and Turnbull [1997], Lando [1998], Madan and Unal [1998], and Duffie and Singleton [1999]), the default time is typically modeled as the exogenous arrival time of an autonomous counting process. The claim to the recovery value of a defaultable security with maturity T is the present value of the payoff qτ = q(Yτ , τ ) (recovery upon default) at the default arrival time τ whenever τ ≤ T: P(t; {q(Yτ , τ )}) = EQ h e− R τ t ru duqτ 1{τ≤T}    Yt i . (9) This expression simplifies if τ is the arrival time of a doubly stochastic Poisson process with state-dependent intensity λt = λ(Yt, t). At date t, the cumula￾tive distribution of arrival of a stopping time before date s, conditional on {Yu : t ≤ u ≤ s} is Pr(τ ≤ s;t|Yu : t ≤ u ≤ s)=1−e− R s t λudu. It follows that 8