《金融投资学》教学资源（英文文献）Fixed Income Pricing.pdf_大学文库

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, including 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 explicitly, 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 (conditional 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 conceptual approaches to fixed-income pricing, highlighting some of the tradeoffs that have emerged in the literature between the complexity of the probability 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 payoffs 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. Initially, 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 particular 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 deterministic 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

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 . (Equation (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

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 securities, 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 cumulative 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