(see,e.g.,Lando [1998]) P(t;{q(Y,T)})=E EQ e-frta)d g ds Vi This pricing equation is a special case of(6)with hs =Asqs,gr=0,and an “effective riskless rate”ofrs+入s. In "structural"pricing models of defaultable securities,the default time is typically modeled as the first passage time of firm value below some default boundary.With a constant default boundary and exogenous firm value pro- cess (e.g.,Merton [1974],Black and Cox [1976],and Longstaff and Schwartz [1995]),the pricing of the default risk amounts to the computation of the first- passage probability under the forward measure.With an endogenously de- termined default boundary (e.g.,Leland [1994]and Leland and Toft [1996]), the probability of the first passage time and the value of the risky debt must be jointly determined.5 3 DTSMs for Default-free Bonds In this section we overview the pricing of default-free bonds within DTSMs. We begin with an overview of one-factor models (N=1)and then turn to the case of multi-factor models. 3.1 One-factor DTSMs Some of the more widely studied one-factors models are: Nonlinear CEV Model r follows the one-dimensional Feller [1951] process dr(t)=(KOr(t)20-1-kr(t))dt+ar(t)"dw(t) (10) In this model,the admissible range for n is [0,1),and the zero boundary is entrance (cannot be reached from the interior of the state space)if 5Similar to an American option,the price of the risky debt can be characterized as the solution to a PDE with a "free boundary",with the boundary conditions given by the so-called“vaue-matching”and the“smooth-pasting”conditions. 9(see, e.g., Lando [1998]) P(t; {q(Yτ , τ )}) = EQ Z T t e− R s t ru du qs d Pr(τ ≤ s;t|Yu : t ≤ u ≤ s)     Yt  = EQ Z T t e− R s t (ru+λu) du λs qs ds     Yt  . This pricing equation is a special case of (6) with hs = λsqs, gT = 0, and an “effective riskless rate” of rs + λs. In “structural” pricing models of defaultable securities, the default time is typically modeled as the first passage time of firm value below some default boundary. With a constant default boundary and exogenous firm value pro￾cess (e.g., Merton [1974], Black and Cox [1976], and Longstaff and Schwartz [1995]), the pricing of the default risk amounts to the computation of the first￾passage probability under the forward measure. With an endogenously de￾termined default boundary (e.g., Leland [1994] and Leland and Toft [1996]), the probability of the first passage time and the value of the risky debt must be jointly determined.5 3 DTSMs for Default-free Bonds In this section we overview the pricing of default-free bonds within DTSMs. We begin with an overview of one-factor models (N = 1) and then turn to the case of multi-factor models. 3.1 One-factor DTSMs Some of the more widely studied one-factors models are: • Nonlinear CEV Model r follows the one-dimensional Feller [1951] process dr(t)=(κθr(t) 2η−1 − κr(t)) dt + σr(t) ηdWQ(t). (10) In this model, the admissible range for η is [0, 1), and the zero boundary is entrance (cannot be reached from the interior of the state space) if 5Similar to an American option, the price of the risky debt can be characterized as the solution to a PDE with a “free boundary”, with the boundary conditions given by the so-called “value-matching” and the “smooth-pasting” conditions. 9