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 . (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
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 zerocoupon 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 frequency 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 except 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 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
(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 process (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 firstpassage probability under the forward measure. With an endogenously determined 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