《金融投资学》教学资源（英文文献）Fixed Income Pricing

Fixed Income Pricing Qiang Dai and Kenneth Singleton This draft:July 1,2002 1Dai is with the Stern School of Business,New York University,New York, NY,dai@stern.nyu.edu.Singleton is with the Graduate School of Business,Stan- ford University,Stanford,CA 94305 and NBER,ken@future.stanford.edu.We are grateful to Len Umantsev and Mariusz Rabus for research assistance;to Len Umantsev for comments and suggestions;and for financial support from the Fi- nancial Research Initiative,The Stanford Program in Finance,and the Gifford Fong Associates Fund,at the Graduate School of Business,Stanford University

Fixed Income Pricing Qiang Dai and Kenneth Singleton1 This draft : July 1, 2002 1Dai is with the Stern School of Business, New York University, New York, NY, dai@stern.nyu.edu. Singleton is with the Graduate School of Business, Stan￾ford University, Stanford, CA 94305 and NBER, ken@future.stanford.edu. We are grateful to Len Umantsev and Mariusz Rabus for research assistance; to Len Umantsev for comments and suggestions; and for financial support from the Fi￾nancial Research Initiative, The Stanford Program in Finance, and the Gifford Fong Associates Fund, at the Graduate School of Business, Stanford University

Contents 1 Introduction 2 2 Fixed-income Pricing in a Diffusion Setting 3 21 The Term Structure......·········· 4 2.2 FIS with Deterministic Payoffs.. 5 2.3 FIS with State-dependent Payoffs 5 2.4 FIS with Stopping Times..············ 7 3 DTSMs for Default-free Bonds 9 3.1One-factor DTSMs...·...·....·····.· 9 3.2 Multi--factor DTSMs·....············· 12 4 DTSMs with Jump Diffusions 16 5 DTSMs with Regime Shifts 17 6 DTSMs with Rating Migrations 20 6.1 Fractional Recovery of Market Value....·.,.·····. 21 6.2 Fractional Recovery of Par,Payable at Maturity........ 24 6.3 Fractional Recovery of Par,Payable at Default ........ 25 6.4 Pricing Defaultable Coupon Bonds.......... 25 6.5 Pricing Eurodollar Swaps..........···. 26 7 Pricing of Fixed-Income Derivatives 27 7.1 Derivatives Pricing using DTSMs 27 7.2 Derivatives Pricing using Forward Rate Models 29 7.3 Defaultable Forward Rate Models with Rating Migrations 31 7.4 The LIBOR Market Model..·················· 34 7.5 The Swaption Market Model...·.........···... 38 1

Contents 1 Introduction 2 2 Fixed-income Pricing in a Diffusion Setting 3 2.1 The Term Structure . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 FIS with Deterministic Payoffs . . . . . . . . . . . . . . . . . . 5 2.3 FIS with State-dependent Payoffs . . . . . . . . . . . . . . . . 5 2.4 FIS with Stopping Times . . . . . . . . . . . . . . . . . . . . . 7 3 DTSMs for Default-free Bonds 9 3.1 One-factor DTSMs ........................ 9 3.2 Multi-factor DTSMs . . . . . . . . . . . . . . . . . . . . . . . 12 4 DTSMs with Jump Diffusions 16 5 DTSMs with Regime Shifts 17 6 DTSMs with Rating Migrations 20 6.1 Fractional Recovery of Market Value . . . . . . . . . . . . . . 21 6.2 Fractional Recovery of Par, Payable at Maturity . . . . . . . . 24 6.3 Fractional Recovery of Par, Payable at Default . . . . . . . . . 25 6.4 Pricing Defaultable Coupon Bonds . . . . . . . . . . . . . . . 25 6.5 Pricing Eurodollar Swaps . . . . . . . . . . . . . . . . . . . . . 26 7 Pricing of Fixed-Income Derivatives 27 7.1 Derivatives Pricing using DTSMs . . . . . . . . . . . . . . . . 27 7.2 Derivatives Pricing using Forward Rate Models . . . . . . . . 29 7.3 Defaultable Forward Rate Models with Rating Migrations . . 31 7.4 The LIBOR Market Model . . . . . . . . . . . . . . . . . . . . 34 7.5 The Swaption Market Model . . . . . . . . . . . . . . . . . . . 38 1

1 Introduction This chapter surveys the literature on fixed-income pricing models,includ- ing dynamic term structure models (DTSMs)and interest rate sensitive, derivative pricing models.This literature is vast with both the academic and practitioner communities having proposed a wide variety of models and model-selection criteria.Central to all pricing models,implicitly or explic- itly,are:(i)the identity of the state vector:whether it is latent or observable and,in the latter case,which observable series;(ii)the law of motion (con- ditional distribution)of the state vector under the pricing measure;and (iii) the functional dependence of the short-term interest rate on this state vector. A primary objective,then,of research on fixed-income pricing has been the selection of these ingredients to capture relevant features of history,given the objectives of the modeler,while maintaining tractability,given available data and computational algorithms.Accordingly,we overview alternative concep- tual approaches to fixed-income pricing,highlighting some of the tradeoffs that have emerged in the literature between the complexity of the proba- bility model for the state,data availability,the pricing objective,and the tractability of the resulting model. A pricing model may be "monolithic"in the sense that it prices both bonds (as functions of a set of underlying state variables or "risk factors" -i.e.,is a "term structure model")and fixed-income derivatives (with pay- offs expressed in terms of the prices or yields on these underlying bonds). Alternatively,a model may be designed to price fixed-income derivatives, taking as given the current shape of the underlying yield curve.The former modeling strategy is certainly more comprehensive than the latter.However, researchers have often found that the latter approach offers more flexibility in calibration and tractability in computation when pricing certain derivatives. Initially,taking the monolithic approach,we overview a variety of models for pricing default-free bonds and associated derivatives written on these (or portfolios of these)bonds.Basic issues in pricing fixed-income securities (FIS)for the case where the state vector follows a diffusion are discussed in Section 2."Yield-based"DTSMs are reviewed in Section 3.Extensions of these pricing models to allow for jumps or regime shifts are explored in Sections 4 and 5,respectively. Then,in Section 6,we turn to the case of defaultable securities.Here we start by considering a quite general framework in which there are multiple credit classes (possibly indexed by rating)and deriving pricing relations for 2

1 Introduction This chapter surveys the literature on fixed-income pricing models, includ￾ing dynamic term structure models (DTSMs) and interest rate sensitive, derivative pricing models. This literature is vast with both the academic and practitioner communities having proposed a wide variety of models and model-selection criteria. Central to all pricing models, implicitly or explic￾itly, are: (i) the identity of the state vector: whether it is latent or observable and, in the latter case, which observable series; (ii) the law of motion (con￾ditional distribution) of the state vector under the pricing measure; and (iii) the functional dependence of the short-term interest rate on this state vector. A primary objective, then, of research on fixed-income pricing has been the selection of these ingredients to capture relevant features of history, given the objectives of the modeler, while maintaining tractability, given available data and computational algorithms. Accordingly, we overview alternative concep￾tual approaches to fixed-income pricing, highlighting some of the tradeoffs that have emerged in the literature between the complexity of the proba￾bility model for the state, data availability, the pricing objective, and the tractability of the resulting model. A pricing model may be “monolithic” in the sense that it prices both bonds (as functions of a set of underlying state variables or “risk factors” – i.e., is a “term structure model”) and fixed-income derivatives (with pay￾offs expressed in terms of the prices or yields on these underlying bonds). Alternatively, a model may be designed to price fixed-income derivatives, taking as given the current shape of the underlying yield curve. The former modeling strategy is certainly more comprehensive than the latter. However, researchers have often found that the latter approach offers more flexibility in calibration and tractability in computation when pricing certain derivatives. Initially, taking the monolithic approach, we overview a variety of models for pricing default-free bonds and associated derivatives written on these (or portfolios of these) bonds. Basic issues in pricing fixed-income securities (FIS) for the case where the state vector follows a diffusion are discussed in Section 2. “Yield-based” DTSMs are reviewed in Section 3. Extensions of these pricing models to allow for jumps or regime shifts are explored in Sections 4 and 5, respectively. Then, in Section 6, we turn to the case of defaultable securities. Here we start by considering a quite general framework in which there are multiple credit classes (possibly indexed by rating) and deriving pricing relations for 2

the case where issuers may transition between classes according to a Markov process.Several of the most widely studied models for pricing defaultable bonds are compared by specializing to the case of a single credit class. The pricing of fixed-income derivatives is overviewed in Section 7.Ini- tially,we continue our discussion of DTSMs and overview recent research on the pricing of derivatives using yield-based term structure models.Then we shift our focus from monolithic models to models for pricing derivatives in which the current yield curve,and possibly the associated yield volatilities, are taken as inputs into the pricing problem.These include models based on forward rates (both for default-free and defaultable securities),and the LIBOR and Swaption Market models. To keep our overview of the literature manageable we focus,for the most part,on term structure models and fairly standard derivatives on zero-coupon and coupon bonds (both default-free and defaultable),plain-vanilla swaps, caps,and swaptions.In particular,we do not delve deeply into many of the complex structured products that are increasingly being traded.Of particu- lar note,we have chosen to side-step the important issue of pricing securities in which correlated defaults play a central role in valuation.Additionally, we focus almost exclusively on pricing and the associated "pricing measures." Our companion paper Dai and Singleton [2002]explores in depth the speci- fications of the market prices of risk that connect the pricing with the actual measures,as well as the empirical goodness-of-fit of models2 under alternative specifications of the market prices of risks. 2 Fixed-income Pricing in a Diffusion Setting A standard framework for pricing FIS has the riskless rate rt being a deter- ministic function of an N x 1 vector of risk factors Yt, Tt=r(Yi,t), (1) 1Musiela and Rutkowski [1997b]discuss the pricing of a wide variety of fixed-income products,and Duffie and Singleton [2001]discuss pricing of structured products in which correlated default is a central consideration. 2 See also Chapman and Pearson [2001]for another surveys of the empirical term structure literature. 3

the case where issuers may transition between classes according to a Markov process. Several of the most widely studied models for pricing defaultable bonds are compared by specializing to the case of a single credit class. The pricing of fixed-income derivatives is overviewed in Section 7. Ini￾tially, we continue our discussion of DTSMs and overview recent research on the pricing of derivatives using yield-based term structure models. Then we shift our focus from monolithic models to models for pricing derivatives in which the current yield curve, and possibly the associated yield volatilities, are taken as inputs into the pricing problem. These include models based on forward rates (both for default-free and defaultable securities), and the LIBOR and Swaption Market models. To keep our overview of the literature manageable we focus, for the most part, on term structure models and fairly standard derivatives on zero-coupon and coupon bonds (both default-free and defaultable), plain-vanilla swaps, caps, and swaptions. In particular, we do not delve deeply into many of the complex structured products that are increasingly being traded. Of particu￾lar note, we have chosen to side-step the important issue of pricing securities in which correlated defaults play a central role in valuation.1 Additionally, we focus almost exclusively on pricing and the associated “pricing measures.” Our companion paper Dai and Singleton [2002] explores in depth the speci- fications of the market prices of risk that connect the pricing with the actual measures, as well as the empirical goodness-of-fit of models2 under alternative specifications of the market prices of risks. 2 Fixed-income Pricing in a Diffusion Setting A standard framework for pricing FIS has the riskless rate rt being a deter￾ministic function of an N × 1 vector of risk factors Yt, rt = r(Yt, t), (1) 1Musiela and Rutkowski [1997b] discuss the pricing of a wide variety of fixed-income products, and Duffie and Singleton [2001] discuss pricing of structured products in which correlated default is a central consideration. 2 See also Chapman and Pearson [2001] for another surveys of the empirical term structure literature. 3

and the risk-neutral dynamics of Y:following a diffusion process,3 dYi=u(Yi,t)dt+a(Yi,t)dw (2) Here,We is a Kx 1 vector of standard and independent Brownian motions under the risk-neutral measure Q,u(Y,t)is a N x 1 vector of determinis- tic functions of Y and possibly time t,and o(Y,t)is a N x K matrix of deterministic functions of Y and possibly t. 2.1 The Term Structure Central to the pricing of FIS is the term structure of zero-coupon bond prices. The time-t price of a zero-coupon bond with maturity T and face value of $1 is given by D(t,T)e (3) where F is the information set at time t,and EF denotes the con- ditional expectation under the risk-neutral measure Q.Since a diffusion process is Markov,we can take F to be the information set generated by Yi. Thus,the discount function [D(t,T):T>t}is completely determined by the risk-neutral distribution of the riskless rate and Yi.4 As an application of the Feynman-Kac theorem,the price of a zero-coupon bond can alternatively be characterized as a solution to a partial differential equation(PDE).Heuristically,this PDE is obtained by applying Ito's lemma to the pricing function D(t,T),for some fixed T>t: dD(t,T)=u(Yi,t;T)dt+a(Yi,t;T)'dw, (Y,t;T)= 成+4 D传，T),KT)=taD卫 where A is the infinitesimal generator for the diffusion Y: A=4品+ ay+Trace (Y,)σ(Y,元 2 YOY 3See Duffie [1996]for sufficient technical conditions for a solution to(2)to exist. 4Here we assume that sufficient regularity conditions conditions(that may depend on the functional form ofr(Y,t))have been imposed to ensure that the conditional expectation in (3)is well-defined and finite

and the risk-neutral dynamics of Yt following a diffusion process,3 dYt = µ(Yt, t) dt + σ(Yt, t) dWQ t . (2) Here, WQ t is a K × 1 vector of standard and independent Brownian motions under the risk-neutral measure Q, µ(Y, t) is a N × 1 vector of determinis￾tic functions of Y and possibly time t, and σ(Y, t) is a N × K matrix of deterministic functions of Y and possibly t. 2.1 The Term Structure Central to the pricing of FIS is the term structure of zero-coupon bond prices. The time-t price of a zero-coupon bond with maturity T and face value of $1 is given by D(t, T) = EQ h e− R T t rs ds    Ft i , (3) where Ft is the information set at time t, and EQ[ · |Ft] denotes the con￾ditional expectation under the risk-neutral measure Q. Since a diffusion process is Markov, we can take Ft to be the information set generated by Yt. Thus, the discount function {D(t, T) : T ≥ t} is completely determined by the risk-neutral distribution of the riskless rate and Yt. 4 As an application of the Feynman-Kac theorem, the price of a zero-coupon bond can alternatively be characterized as a solution to a partial differential equation (PDE). Heuristically, this PDE is obtained by applying Ito’s lemma to the pricing function D(t, T), for some fixed T ≥ t: dD(t, T) = µ(Yt, t; T) dt + σ(Yt, t; T) 0 dWQ t , µ(Y, t; T) =  ∂ ∂t + A  D(t, T), σ(Y, t; T) = σ(Y, t) 0 ∂D(t, T) ∂Y , where A is the infinitesimal generator for the diffusion Yt: A = µ(Y, t) 0 ∂ ∂Y + 1 2 Trace  σ(Y, t) σ(Y, t) 0 ∂2 ∂Y ∂Y 0  . 3See Duffie [1996] for sufficient technical conditions for a solution to (2) to exist. 4Here we assume that sufficient regularity conditions conditions (that may depend on the functional form of r(Y, t)) have been imposed to ensure that the conditional expectation in (3) is well-defined and finite. 4

No-arbitrage requires that,under Q,the instantaneous expected return on the bond be equal to the riskless rate rt.Imposing this requirement gives [品+ D(t,T)-r(Y,t)D(t,T)=0, (4) with the boundary condition D(T,T)=$1 for all Yr. 2.2 FIS with Deterministic Payoffs The price of a security with a set of deterministic cash fows [Ci:j= 1,2,...,n}at some given relative payoff dates Ti(j=1,2,...,n)is given by Pe{C,:j=1,2,,m)=∑C,D,t+r) 1 In particular,the price of a coupon-bond with face value F,semi-annual coupon rate of c,and maturity T=Jx.5 years (where J is an integer)is J Pt{c,T)=∑F×5xDt,t+5)+FxD6,T. 1=1 It follows that the par yield-i.e.,the semi-annually compounded yield on a par bond (with P=F)-is given by PY(t,T)= [1-D(t,T)] ∑1Dt,t+.5列 (5) 2.3 FIS with State-dependent Payoffs The price of a FIS with coupon flow payment hs,t<s <T,and terminal payoff gr is P(t;{hs:t≤s≤T;gr}) -B[eia%叫+[eRg如小 (6) When ru=r(Yu,u),hu h(Yu,u),and gu=g(Yu,u)are deterministic functions of the state vector Yu,this price is obtained as a solution to the 5

No-arbitrage requires that, under Q, the instantaneous expected return on the bond be equal to the riskless rate rt. Imposing this requirement gives  ∂ ∂t + A  D(t, T) − r(Y, t) D(t, T)=0, (4) with the boundary condition D(T,T) = $1 for all YT . 2.2 FIS with Deterministic Payoffs The price of a security with a set of deterministic cash flows {Cj : j = 1, 2,... ,n} at some given relative payoff dates τj (j = 1, 2,... ,n) is given by P(t; {Cj , τj : j = 1, 2,... ,n}) = Xn j=1 CjD(t, t + τ j ). In particular, the price of a coupon-bond with face value F, semi-annual coupon rate of c, and maturity T = J × .5 years (where J is an integer) is P(t; {c, T}) = X J j=1 F × c 2 × D(t, t + .5j) + F × D(t, T). It follows that the par yield – i.e., the semi-annually compounded yield on a par bond (with Pt = F) – is given by PY(t, T) = 2 [1 − D(t, T)] PJ j=1 D(t, t + .5j) . (5) 2.3 FIS with State-dependent Payoffs The price of a FIS with coupon flow payment hs, t ≤ s ≤ T, and terminal payoff gT is P (t; {hs : t ≤ s ≤ T; gT }) =EQ Z T t e− R s t ru du hs ds     Ft  + EQ h e− R T t ru du gT    Ft i . (6) When ru = r(Yu, u), hu = h(Yu, u), and gu = g(Yu, u) are deterministic functions of the state vector Yu, this price is obtained as a solution to the 5

PDE +A P(t)-r(Y,t)P(t)+h(Y,t)=0, (7) under the boundary condition P(T;[hr;gr})=g(Yr,T),for all Yr.(Equa- tion (4)is obtained as the special case of (7)with hu=0 and gr=$1.) A mathematically equivalent way of characterizing the price P(t;ths t<s<T;gr})is in terms of the Green's function.Let 6(x)denote the Dirac function,with the property that 6(x)=0 at x0,6(0)=oo,and fdr6(x-y)f(x)=f(y)for any continuous and bounded function f().The price,G(Yi,t;Y,T),of a security with a payoff (Yr-Y)at T,and nothing otherwise,is referred to as the Green's function.By definition,the Green's function is given by G(t,YiT,Y)=E er(Yr-Y). It is easy to see that G solves the PDE (7)with h(Y,t)=0 under the boundary condition G(Yr,T;Y,T)=6(Yr-Y).If G is known,then any FIS with payment flow h(Yi,t)and terminal payoff g(Yr,T)is given by P(t:{hs:t≤s≤T;gr}) -ds dr G,tys)hny到+& (8) dYG(Yi,t;Y,T)g(Y,T). Essentially,the Green's function represents the set of Arrow-Debreu prices for the case of a continuous state space.When the Green's function is known, equation (8)is often convenient for the numerical computation of the prices of a wide variety of FIS (see Steenkiste and Foresi [1999]for applications of the Green's function for affine term structure models). In the absence of default risk,some fixed-income derivative securities with state-dependent payoffs can be priced using the discount function alone, because they can be perfectly hedged or replicated by a(static)portfolio of spot instruments.These include: Forward Contracts:a forward contract with settlement date T and forward price F on a zero-coupon bond with par $100 and maturity date T+r can be replicated by a portfolio of spot instruments consisting of long a zero-coupon bond with maturity T+r and par $100 and short a 6

PDE  ∂ ∂t + A  P(t) − r(Y, t) P(t) + h(Y, t)=0, (7) under the boundary condition P(T; {hT ; gT }) = g(YT , T), for all YT . (Equa￾tion (4) is obtained as the special case of (7) with hu ≡ 0 and gT = $1.) A mathematically equivalent way of characterizing the price P(t; {hs : t ≤ s ≤ T; gT }) is in terms of the Green’s function. Let δ(x) denote the Dirac function, with the property that δ(x) = 0 at x 6= 0, δ(0) = ∞, and R dx δ(x−y) f(x) = f(y) for any continuous and bounded function f(·). The price, G(Yt, t; Y, T), of a security with a payoff δ(YT − Y ) at T, and nothing otherwise, is referred to as the Green’s function. By definition, the Green’s function is given by G(t, Yt; T,Y ) = EQ h e− R T t ru du δ(YT − Y ) i . It is easy to see that G solves the PDE (7) with h(Y, t) ≡ 0 under the boundary condition G(YT , T; Y, T) = δ(YT − Y ). If G is known, then any FIS with payment flow h(Yt, t) and terminal payoff g(YT , T) is given by P (t; {hs : t ≤ s ≤ T; gT }) = Z T t ds Z dY G(Yt, t; Y, s) h(Y, s) + Z dY G(Yt, t; Y, T) g(Y, T). (8) Essentially, the Green’s function represents the set of Arrow-Debreu prices for the case of a continuous state space. When the Green’s function is known, equation (8) is often convenient for the numerical computation of the prices of a wide variety of FIS (see Steenkiste and Foresi [1999] for applications of the Green’s function for affine term structure models). In the absence of default risk, some fixed-income derivative securities with state-dependent payoffs can be priced using the discount function alone, because they can be perfectly hedged or replicated by a (static) portfolio of spot instruments. These include: • Forward Contracts: a forward contract with settlement date T and forward price F on a zero-coupon bond with par $100 and maturity date T + τ can be replicated by a portfolio of spot instruments consisting of long a zero-coupon bond with maturity T +τ and par $100 and short a 6

zero-coupon bond with maturity T and par F.Thus,the market value of the forward contract is $100x D(t,T+T)-Fx D(t,T).Consequently the forward price is given by F=$100xD D(t,T) A Floating Payment:a floating payment indexed to a riskless rate with tenor r,with coupon rate reset at T and payment made at T+r, can be replicated by a portfolio of spot instruments consisting of long a zero-coupon bond with maturity T and par $100 and short a zero- coupon bond with maturity T+r and par $100.Thus,the price of the floating payment is $100 x [D(t,T)-D(t,T+)].This implies immediately that a floating rate note with payment in arrears is always priced at par on any reset date. A Plain Vanilla Interest Rate Swap:a plain-vanilla interest rate swap with the tenor of the floating index matching the payment fre- quency can be perfectly replicated by a portfolio of spot instruments consisting of long a floating rate note with the same floating index, payment frequency,and maturity and short a coupon bond with the same maturity and payment frequency,and with coupon rate equal to the swap rate.It follows that,at the inception of the swap,the swap rate is equal to the par rate: s(t,T)= 1-D(t,T) ∑0i,Dt,T where t =To<T<...<TN =T,6j Tj+1-Tj is the length of the accrual payment period indexed by j,0<j<N-1,based on an appropriate day-count convention,N is the number of payments,and T is the maturity of the swap. In the presence of default risk,the above pricing results may not hold ex- cept under specific conditions(see,e.g.,Section 6.5 for pricing of Eurodollar swaps). 2.4 FIS with Stopping Times For some fixed-income securities,including American options and defaultable securities,the cash flow payoff dates are also random.A random payoff date is typically modeled as a stopping time,that may be exogenously given or 7

zero-coupon bond with maturity T and par F. Thus, the market value of the forward contract is $100×D(t, T +τ )−F ×D(t, T). Consequently the forward price is given by F = $100 × D(t,T +τ) D(t,T) . • A Floating Payment: a floating payment indexed to a riskless rate with tenor τ , with coupon rate reset at T and payment made at T + τ , can be replicated by a portfolio of spot instruments consisting of long a zero-coupon bond with maturity T and par $100 and short a zero￾coupon bond with maturity T + τ and par $100. Thus, the price of the floating payment is $100 × [D(t, T) − D(t, T + τ )]. This implies immediately that a floating rate note with payment in arrears is always priced at par on any reset date. • A Plain Vanilla Interest Rate Swap: a plain-vanilla interest rate swap with the tenor of the floating index matching the payment fre￾quency can be perfectly replicated by a portfolio of spot instruments consisting of long a floating rate note with the same floating index, payment frequency, and maturity and short a coupon bond with the same maturity and payment frequency, and with coupon rate equal to the swap rate. It follows that, at the inception of the swap, the swap rate is equal to the par rate: s(t, T) = 1 − D(t, T) PN−1 j=0 δjD(t, Tj ) , where t ≡ T0 < T1 < ... < TN ≡ T, δj = Tj+1 − Tj is the length of the accrual payment period indexed by j, 0 ≤ j ≤ N − 1, based on an appropriate day-count convention, N is the number of payments, and T is the maturity of the swap. In the presence of default risk, the above pricing results may not hold ex￾cept under specific conditions (see, e.g., Section 6.5 for pricing of Eurodollar swaps). 2.4 FIS with Stopping Times For some fixed-income securities, including American options and defaultable securities, the cash flow payoff dates are also random. A random payoff date is typically modeled as a stopping time, that may be exogenously given or 7

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 8

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 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

(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