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No- Approach to Pricing Spread Derivatives

CHI CHIU CHU AND YUE KUEN KWOK

CHI CHIU CHU No arbitrage in the case of pricing should the actual spot spread breach some is a Ph.D. student in derivatives refers to determination of the time-depen- strike level. Alternatively, the payoff may mathematics at Hong dent drift terms in the mean reversion pro- depend on the difference between the spot Kong University of Sci- ence and Technology in cesses of the instantaneous spot rate and spot spread spreads of two reference entities. Credit spread Hong Kong, China. by fitting the current term structures of -free options may be used either to earn income [email protected] and defaultable . The riskless rate and the from the premium, to bet one’s view credit spread of a reference entity are taken to be cor- on credit spread change, or to target the pur- YUE KUEN KWOK related stochastic state variables in this pricing model. chase of a reference entity on a forward basis is an associate professor of at a favored . mathematics at Hong When the spot rate and spot spread both follow the By following a reduced-form approach Kong University of Sci- ence and Technology. Hull and White model, one can derive an analytic that models the occasion of default as a point [email protected] representation for the time-dependent drift terms and process, Jarrow, Lando, and Turnbull [1997] analytic price formulas for credit spread options. Algo- construct a Markov chain model for valuation rithms for the numerical valuation of credit spread of risky and credit derivatives that incor- derivatives are developed, and the pricing behaviors porates the credit ratings of a firm as an indi- of credit spread options are examined. cator of the likelihood of default. The Markov model provides the evolution of an arbitrage- free term structure of the credit spread. redit spread derivatives are credit Problems arise in numerical implemen- rate-sensitive financial instruments tation of the model because bonds with high designed to against or cap- credit ratings may experience no default within italize on changes in the credit the sample period, so the risk premium adjust- Cspread of a reference entity such as a risky ments become ill-defined as the estimated bond. The credit spread of a risky bond is the default probabilities are zero. Kijima and difference between the current of the Komoribayashi [1998] propose a technique of risky bond and a benchmark rate of compa- risk premium adjustment to overcome this dif- rable maturity. The spread represents the risk ficulty with high-rated bonds. premium the market demands for holding the In related work, Duffie and Singleton risky bond. [1999] and Schönbucher [1998] use intensity Unlike default swaps, credit spread models to analyze the term structure of derivatives do not depend upon any specific defaultable bonds and compute the price of credit event occurring. The buyer of a credit credit derivatives. Schönbucher [1999] con- spread option pays an up-front premium, and structs a two-factor tree model for fitting both in return the writer agrees to pay an amount defaultable and default-free term structures of

SPRING 2003 THE JOURNAL OF DERIVATIVES 51 bond prices, following the Hull and White framework formulas for credit spread options. Mougeot [2000] with time-dependent drift terms. He uses the Hull and extends the two-factor Longstaff-Schwartz model to a White [1990] model for the dynamics of the risk-free three-factor model for pricing options on the spread of and default intensity. Garcia, van Gindersen, the values of two reference . and Garcia [2001] use a two-factor tree algorithm that is We extend the Longstaff and Schwartz model by composed of a Hull and White tree for the risk-free assuming time-dependent drift functions in the mean interest rate dynamics and a Black and Karasinski [1991] reversion diffusion processes of the riskless interest rate tree for the intensity dynamics. and credit spread. The spot riskless interest rate is taken Das and Sundaram [2000] propose a discrete-time to follow the Hull-White model, while the spot credit Heath-Jarrow-Morton [1992] model for valuation of spread can be taken to follow either the Hull-White model credit derivatives. They derive risk-neutral drifts for the [1990] or the Black-Karasinski model [1991]. When both processes of the risk-free forward rate and forward spread. state variables follow the Hull-White model, we can Combining a recovery of market value condition, they derive analytic expressions for the time-dependent drift obtain a recursive representation of drifts that are path- functions that fit the current term structures of default- dependent. free and defaultable bond prices. Also, we can obtain ana- This formulation has the advantage that it can easily lytic pricing formulas for credit spread options. handle the American feature in the pricing of Unfortunately, the time-dependent drift function the credit derivatives, and the corresponding algorithm also in the spot spread process is not analytically tractable when includes information about default probabilities inferred the logarithm of the spot spread is assumed to follow a from . Its major disadvantage is that its com- mean-reverting diffusion (this is the assumption in the putational complexity grows exponentially with the Black-Karasinski model). By extending the Hull-White number of time steps, since the corresponding binomial fitting methodologies [1994], we extend tree is non-recombining. the fitting algorithms to allow numerical valuation of A structural approach derives the term structure of credit spread options. the credit spread of a credit entity by examining the credit quality of the issuer. In structural models, the credit spread I. CONTINUOUS MODELS is not one of the state variables in the model, but rather is a derived quantity. Default occurs when the issuer firm We begin with the pricing of credit spread deriva- value falls below some threshold value. The structural tives under the Hull-White [1990] model. This is an exten- approach has been used by Das [1995] to price sion of the one-factor Hull-White model to the two-factor derivatives. version, where both the instantaneous spot riskless interest Longstaff and Schwartz [1995] develop a valuation rate and the credit spread follow a mean-reverting model model for pricing credit spread derivatives by assuming that with time-dependent drift terms. Like the one-factor ver- the riskless interest rate and the spread follow correlated sion, the two-factor model exhibits nice analytical mean reversion diffusion processes. From empirical analysis tractability and ease of numerical implementation. of observed credit spreads, they argue that the logarithm of the credit spread follows a mean reversion process. They Pricing Framework Under assume the parameters in the characterization of the pro- Two-Factor Hull-White Model cesses to be constant, and derive closed-form price formulas for the credit spread options. A disadvantage of the process We specify the processes of the instantaneous spot is that their model does not incorporate information about riskless interest rate r and the spot spread s to follow the the current term structures of bond prices. extended Vasicek [1977] model, where the drift terms are Chacko and Das [2002] have since then derived assumed to be time-dependent functions. Under a risk- price formulas for credit spread options based on the neutral valuation framework, we assume that the stochastic uncorrelated Cox-Ingersoll-Ross [1985] processes for the processes followed by r and s are given by spot rate and spot spread, and again the parameters in the stochastic processes are assumed to be constant. Tahani dr =[θ(t) − ar]dt + σrdWr, (1-A) [2000] models the process of the logarithm of credit spread using the GARCH model and obtains closed-form price

52 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003 then ds =[φ(t) − bs]dt + σsdWs, (1-B)

σ σ (t, T )=− r [1 − e−a(T−t)]. where θ(t) and φ(t) are time-dependent functions, and a P a (6) and b are positive constant parameters that give the rates σ of mean reversion of r and s, respectively. Further, r and By following a similar procedure, it is possible to σ φ s are the constant parameters for the spot rate determine the other time-dependent function (t) in the r and spot spread s, respectively, and dWr and dWs are stan- mean reversion process of s by fitting the current term dard Brownian motions under the risk-neutral measure structure of defaultable bond prices. This is achieved by ρ P. Let denote the correlation coefficient between dWr changing the to the forward-adjusted ρ T and dWs; that is, dWrdWs = dt. measure Q with delivery at future time T. Let D(t, T) Determination of the Time-Dependent Drift Terms. denote the price at time t of a defaultable bond maturing Hull and White [1990] demonstrate how to determine the at time T. By the , we have: time-dependent function θ(t) in the drift term of the    default-free spot rate by fitting the current term structure T of default-free bond prices. Their procedure can be sum- D(t, T)=EP exp − [r(u)+s(u)] du marized as follows. Let P(t, T) denote the price at time t t    of a discount bond maturing at time T, which is given by: T T = P(t, T)EQ exp − s(u) du (7) t T P P(t, T )=E [exp(− r(u) du)], (2) t Accordingly, the process followed by the spot spread s under the forward measure QT is given by where EP denotes the expectation under the risk-neutral measure P. By taking the risk-neutral expectation in Equa- T tion (2), where r is defined in Equation (1-A), one obtains: ds =[φ(t) − bs + ρσsσP (t, T)]dt + σsdW , (8)  r(t) −a(T−t) P(t, T )=exp − [1 − e ] − T T a where dW is a standard Brownian motion under Q . Under the process defined in Equation (8), the quantity  T 1 − e−a(T−u) T σ2 ∫T θ(u) du + r [1 − e−a(T −u)]2 du ts(u) du follows the normal distribution with mean: 2 t a t 2a   T   (3) T s(t) EQ s(u) du = 1 − e−b(T−t) + b Differentiating Equation (3) with respect to T twice, t we obtain: T 1 − e−b(T−u) [φ(u)+ρσsσP (u, T )] du t b σ2 ∂2 ∂ θ(T)= r [1 − e−2a(T−t)] − ln P(t, T) − a ln P(t, T). (9-A) 2a ∂T2 ∂T and variance (4)    This equation gives the dependence of the drift term T T σ2 θ(t) using the information of the current term structure of var s(u) du = s [1 − e−b(T −u)]2 du. b2 the default-free bond price P(t, T) over the period [t, T]. t t Suppose we write the process followed by P(t, T) as: (9-B)

dP (t, T ) Since = rdt + σ (t, T)dW , P(t, T ) P r (5)

SPRING 2003 THE JOURNAL OF DERIVATIVES 53    T T dξ =[φ(t)+ρσsσP (t, T)]s(t)(−b(T − t))dt EQ exp − s(u) du = T t + σss(t)(−b(T − t))dW . (14)     T  T QT 1 ξ exp −E s(u) du + var s(u) du Note that T is conditionally normally distributed t 2 t 2 with mean µξ and variance σξ, where:

(10-A)  ρσ σ 1 − e−(a+b)(T −t) 1 − e−b(T−t) µ = s(t)e−b(T−t) + s r − we obtain ξ a a + b b

 T     −b(T−u) D(t, T) s(t) T 1 − e−b(T −u) + φ(u)e du =exp − [1 − e−b(T−t)] − [φ(u)+ρσ σ (u, T )] du t P(t, T ) b s P b  t σ2 ∂ D(t, T )   = s [1 − e−b(T−t)][1 − e−2bte−b(T−t)] − ln T σ2 2 + s [1 − e−b(T −u)]2 du . 2b ∂T P(t, T ) 2 t 2b  ρσ σ 1 − e−(a+b)(T−t) 1 − e−b(T−t) e−at[e−a(T−t) − e−b(T −t)] (10-B) + s r − − a a + b b b  Again, by differentiating Equation (10-B) with e−(a+b)te−b(T −t)[1 − e−a(T−t)] − , respect to T twice and observing Equation (6), we obtain: b (15-A)     σ2 ∂ D(t, T ) ∂2 D(t, T ) φ(T)= s [1 − e−2b(T−t)] − b ln − ln 2b ∂T P(t, T) ∂T2 P(t, T) 2   2 σs −2b(T−t) 1 − e−a(T −t) 1 − e−b(T−t) σξ = [1 − e ]. (15-B) + ρσ σ + e−a(T−t) . 2b s r a b (11) Note that µξ depends not only on the time to expi- Credit Spread . Let psp(r, s, t) denote the ration T – t but also on the current time t. Once the mean ξ price of the credit spread put option, whose terminal and variance of T are available, the price formula for the payoff is given by credit spread put is given by:

psp(r, s, T )=max(s − K, 0). (12) psp(r, s, t)=P(t, T)  σ 2 2 µ − K ξ −(K−µξ ) /2σ ξ √ e ξ +(µξ − K)N Here K is the strike spread at which the option holder has 2π σξ the right to put the risky bond to the option writer. It (16) is called a credit spread put option although the terminal payoff resembles a call payoff. When the spot credit spread is higher than the strike spread, this signifies that the price Combined Hull-White of the risky bond drops below the and Black-Karasinski Model of the bond corresponding to the strike spread. Since the terminal payoff depends only on s, we may use the default Longstaff and Schwartz [1995)] propose that the free bond price as the numeraire and obtain: interest rate and the logarithm of the credit spread follow mean-reverting Vasicek processes. We extend their model p (r, s, t)=P(t, T)EQT [max(s − K, 0)]. by assuming time-dependent functions in the drift terms; sp (13) that is, we choose the Hull-White model for the interest rate process and the Black-Karasinski model for the credit Suppose we define ξ(t) = s(t)exp(–b[T – t]). Then spread process. Assume that y = ln s, and the stochastic ξ follows the : processes followed by r and y are governed by:

dr =[θ(t) − ar]dt + σrdWr, (17-A)

54 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003 dy =[ψ(t) − cy]dt + σ dW , σ2 y y (17-B) σ2 = y [1 − e−2c(T −t)]. ζ 2c (20-B) σ where c is the rate of mean reversion of y, y is the con- ^ρ stant volatility parameter for y, and dWrdWy = dt. Unfor- Heath-Jarrow-Morton Model tunately, the Black-Karasinski model lacks the level of analytic tractability exhibited by the Hull-White model. The two-factor extended Vasicek model is a special Under the Black-Karasinski assumption of the credit spread case of the Heath-Jarrow-Morton (HJM) model corre- process, we find that Equation (7) has to be modified as: sponding to a specific choice of the volatility structures. First, we present the general HJM formulation of the     T pricing model with the two state variables r and s. D(t, T) T = EQ exp − exp(y(u)) du . Let f(t, T ) denote the riskless instantaneous forward P(t, T) t rate. Following the HJM framework, the dynamic of f(t, T ) (18) is assumed to be

∫T ∫T Unlike ts(u) du, however, the quantity texp[y(u)]du (t, T )=α (t, T)dt + σ (t, T)dW , is no longer normally distributed. Hence, there is not a simple df f f 1 (21) relation between [D(t, T )]/[P(t, T )] and the time-depen- ψ α σ dent drift function (t), like that shown in Equation (11). where f is the drift, and f is the volatility. Also, dW1 is If θ(t) and ψ(t) are known functions, we may derive a standard Brownian motion under the risk-neutral mea- α the price formula for the credit spread put option. In sure P. Under the no-arbitrage condition, the drift f σ terms of y, the terminal payoff of the credit spread put is and volatility f must observe: given by max(ey – K, 0). The put price formula takes the usual Black-Scholes form:  T αf (t, T)=σf (t, T) σf (t, u) du. (22)  t µ +σ2/2 psp(s, r, t)=P(t, T ) e ζ ζ N(d) − KN(d − σζ) From the dynamics of the risky bond prices, we (19-A) can define the risky instantaneous forward rate fd(t, T ) in where the same manner as the risk-free instantaneous forward rate. Next, the instantaneous forward spread h(t, T) is 2 defined to be f (t, T ) – f(t, T ); accordingly, the spot spread − lnK + µζ + σ d d = ζ . s(t) = f (t, t) – f(t, t). Assume that h(t, T ) follows the σ (19-B) d ζ stochastic process:

2 Similar to µξ and σξ as given in Equations (15-A) 2 dh(t, T)=αh(t, T)dt + σh1 (t, T)dW1 + σh2 (t, T )dW2, and (15-B), the mean µξ and variance σξ are given by: (23)

−c(T−t) ρσ yσr µζ = y(t)e + α σ σ a where h is the drift, h1 and h2 are volatilities, and dW1   and dW are independent standard Brownian motions 1 − e−(a+c)(T −t) 1 − e−c(T−t) 2 − + under the risk-neutral measure P. Again, the no-arbi- a + c c α σ σ trage condition dictates that h, h1, and h2, satisfy (see  T Schönbucher [1998]): ψ(u)e−c(T −u) du, (20-A) t

SPRING 2003 THE JOURNAL OF DERIVATIVES 55   2 T T II. CONSTRUCTION OF NUMERICAL α (t, T )= σ (t, T) σ (t, u) du + σ (t, T) σ (t, u) du h hi hi h1 f + i=1 t t ALGORITHMS  T σ (t, T ) σ (t, u) du. f h1 (24) Now we can extend the Hull and White [1994] t methodology to fitting the current term structures of default-free and defaultable bond prices. By recalling that s(t) = h(t, t), one can show that the process followed by s is governed by: Two-Factor Algorithms

   T 2 Let x = x(r) and z = z(s) be some specified func- ∂h(t, T) 2 ds(T)= + σ (u, T )+2σh (u, T)σf (u, T ) du tions of the spot interest rate and spot credit spread, respec- ∂T hi i + t i=1 tively. Assume that both x and z follow mean reversion    T T 2 ∂σ (u, T ) ∂σ (u, T ) processes: σ (u, v) hi + σ (u, v) f hi ∂T h1 ∂T + t u i=1    ∂σ (u, T ) 2 T ∂σ (u, T ) σ (u, v) h1 dv du + hi dW (u) dT f ∂T ∂T i + dx =[θ (t) − ax]dt + σ dW , i=1 t x x (29-A) 2

σh (T,T)dWi(T). dz =[φ(t) − z ]dt + σ dW , i (25) b z z (29-B) i=1

Equation (25) reveals that the drift term of s is in gen- where dW dW = ρ~dt. We would like to construct a two- eral path-dependent. If instead we choose the volatility x z factor tree that fits the current term structures of prices of structures of the riskless instantaneous forward rate and the default-free and defaultable bonds through numerical pro- instantaneous forward spread to be: cedures. For example, the combined Hull-White and Black- Karasinski model corresponds to x(r) = r and z(s) = ln s. −a(T−t) σf (t, T)=σre , (26-A) We first briefly review some of the key steps in the Hull-White fitting procedure for the one-factor tree, and σ (t, T)=ρσ e−b(T−t), h1 s (26-B) follow by generalization to the two-factor tree. In the  one-factor Hull-White tree, we first build the trinomial σ (t, T)= 1 − ρ2σ e−b(T−t), h2 s (26-C) tree for the process x* with x = 0 that corresponds to ~ 0 θ(t) = 0, where: then the process of s becomes Markovian. By comparing ∗ ∗ ∗ the resulting expression for ds and that given in Equation dx = −ax dt + σxdWx,x(0) = 0. (30-A) (1-B), we obtain: Note that [x*(t + ∆t) – x*(t)] is normally distributed σ2  ∂h(t, T ) φ(T)= s 1 − e−2b(T−t) + bh(t, T)+ with mean M and variance V as given by 2b ∂T + x x  ~∆ 1 − e−a(T −t) 1 − e−b(T −t) M = –[1 – e–a t]x*(t) ρσ σ + e−a(T−t) x s r a b and (27)

2 This is consistent with the expression given in Equa- σx −2a∆t Vx = [1 − e ]. (30-B) tion (11) by noting that: 2a

 ∂ D(t, T ) h(t, T)=− ln . ∂T P(t, T) (28)

56 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003 We first construct a trinomial tree that simulates the where α~ is the mean reversion rate of x [see Equation process x*. The node probabilities are obtained by equating (29-A)]. the mean and variance of the discrete tree process and the In the combined two-factor tree that simulates the continuous process as given by Equation (30-B). correlated evolution of r and s, there are nine branches ∆ ∆ A key step in the Hull-White fitting procedure is emanating from the (m, j, k) node at tm = m t, xj = j x, α ∆ determination of the time-varying drift m at time tm so and zk = k z. We use the Hull-White approach of con- that the lattice tree can price default-free bonds at dif- structing probability matrices that give the node proba- ferent maturities as observed in the market at the current bilities of moving from a node at tm to the different nine β time. Let Ym be the current yield of a default-free bond nodes at tm+1. We let m denote the time-varying drift for maturing at time tm; Qm, j be the state price of node (m, the discrete credit spread process. Similarly, we try to β j), which is the of a that pays off $ determine m numerically by fitting the current term 1 if node (m, j) is reached and zero otherwise; and (m, j) structure of defaultable bond prices. α ∆ indicates the node at time tm at which x = m + j x. Ini- Let Qm,j,k denote the state price of the (m, j, k) node α ~ tially, we have Q0, 0 = 1, 0 = x(Y1). Also, we define rm, j and Ym be the current yield of a defaultable bond maturing -1 α ∆ -1 β ~ = x ( m + j x), where x denotes the inverse function at time tm. At initiation, 0 = z(Y1 – Y1); and the succes- β ≥ of x(r). sive m, m 1, are obtained recursively by the relationship: α If we have obtained m–1 and Qm–1,j for all values of (j) (k) j, the state price at time t is given by: nm nm m −1 −Ym+1(m+1)∆t −(rm,j+z (βm+k∆z))∆t e = Qm,j,ke , (j) (k) j=−nm k=−nm j −rm−1,j∗ ∆t Qm,j = Qm−1,j ∗ pj∗ e , (31) j∗ (35)

(k) (j) β where n m is defined similarly as nm. When z(s) = s, m i where pj* is the node probability of moving from node xj* can be solved explicitly by the formula: at time tm–1 to node xj at time tm, and the summation is * *   taken over all values of j for which the nodes x are con- n(j) n(k) j m m −(rm,j+k∆s)∆t ln (j) (k) Qm,j,ke nected to the node x . j=−nm k=−nm  j βm = +(m +1)Ym+1. From the relation in Equation (2), we deduce that: ∆t (36)

(j) nm ∼ −1 −Ym+1(m+1)∆t −x (αm+j∆x)∆t β φ e = Qm,je , Once m is known, the value of (tm) can be esti- (j) mated by: j=−nm (32)  βm − βm−1  (j) φm = + bβm, (37) where n m is the number of nodes on each side of the cen- ∆t α tral node at time tm. The time-varying drift m can then be determined numerically in a recursive manner. α ~ When x = r, m can be solved explicitly by: where b is the mean reversion rate of z[see Equation (29-B)]. When the are known at the nodes, the    (j) nm −j∆r∆t value of any European-style credit at time t0 ln (j) Qm,je α = j=−nm +(m +1)Y . whose value depends on r and s can be obtained by: m ∆t m+1 (33) p (r ,s ,t )= Q p (r ,s ,t ), Moreover, the value of the time-dependent drift sp 0 0 0 M,j,k sp M,j M,k M ~ j k θ(t ) can be estimated by: m (38)

 αm − αm−1 where psp(rM,j, sM,k, tM) is the terminal payoff at maturity θm = + aαm, (34) ∆t date tM, and the summation is taken over all terminal nodes.

SPRING 2003 THE JOURNAL OF DERIVATIVES 57 It is quite computationally demanding to calculate σ2 γ2 = r [e−2a∆t(1 − e−2am∆t) − the state prices since this two-factor algorithm requires a m 2a(1 − e−a∆t)2 search process at each node. When x(r) = r and z(s) = s, α 2e−a∆t(1 − e−am∆t)(1 + e−a∆t) one can derive explicit formulas for the time drifts m and β without calculating the state prices. Further, when the m −2a∆t terminal payoff of a is independent of the + m(1 − e )]. (42) interest rate, we can develop a one-factor fitting algorithm via the use of valuation in the forward-adjusted risk-neu- Since the price at time t0 of a default-free bond tral measure. This reduced one-factor fitting algorithm can maturing at time tm+1 is given by: be considered an extension of the Grant-Vora [2001]    approach to derive an explicit analytic expression for the m P    time drift function of the one-factor Hull-White model. P(t0,tm+1)=E exp − rj∆t (43-A) j=0 One-Factor Tree for Joint Extended Vasicek Processes so that The no-arbitrage fitting procedure developed by Grant and Vora [2001] is indeed the discrete analogue of the ana- lnP(t ,t ) m−1 γ2 θ EP[r ]=− 0 m+1 − EP[r ]+ m , lytic procedure for deriving the time-dependent drift (t). m ∆t j 2 The discrete analogue of the extended Vasicek process for j=0 r as defined in Equation (1-A) can be represented by: (43-B) we then have ∆rm = rm+1 − rm = Mrrm + δm + VrZm, (39) 2 2 P 1 P(t0,tm+1) γm − γm−1 E [rm]=− ln + , (44) ∆t P(t0,tm) 2 where the mean Mr and variance Vr are defined much the same as in Equation (30-B), Zm is a standard normal vari- δ α able, and m is an adjustment parameter for the no-arbi- which gives the time-varying drift m. α P trage fitting procedure. Note that m is equal to E [rm]. As in the continuous model, we can reduce the two- Deductively, we obtain: factor algorithm to a one-factor version by choosing the default-free bond price as the numeraire. First, we would m−1 β like to find the time-varying drift m of the spot spread m m−1−j rm = r0(1 + Mr) + (1 + Mr) δj + process under the risk-neutral measure P and the forward j=0 measure QT. Note from Equations (1-B) and (8) that the m−1 two measures are related by m−1−j Vr(1 + Mr) Zj, (40) j=0 QT P E [s(tm)] = E [s(tm)] + X0,m,M , (45) where the normal variables Zj, j = 0, …, m – 1, are inde- pendently and identically distributed. From Equation (39), one can deduce that: where the last term

δ = EP[r ] − (1 + M )EP[r ]. tm m m+1 r m (41) −b(tm −u) Xj,m,M = ρσsσP (u, tM )e du (46) tj m γ 2 ∆ Σ ∆ Let m t be the variance of rj t. It can be shown represents the drift adjustment required for effecting the that j=0 change of measure.

58 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003 Using the bond volatility σ (t, T) defined in Equation P p (r ,s ,t )=P(t ,t ) U p (s ,t ), (6), and assuming constant values for ρ and σ , we have sp 0 0 0 0 M M,k sp M,k M (52) s k

−a(t −t ) −at −bt +(a+b)t −b(t −t ) ρσsσr e M m − e M m j 1 − e m j Xj,m,M = − a a + b b where psp(sM,k, tM) is the terminal payoff on maturity date tM, and UM,k is the cumulative probability under the for- (47) ward measure reaching the (M, k) node from the starting β ∆ β The discrete analogue of the extended Vasicek pro- (0, 0) node. Note that U0,0 = 1, sm,k = m + k s, and m = ξ χ cess followed by s can be represented by m + 0,m,m. ∆sm = sm+1 − sm = Mssm + ξm + Xm,m+1,M + VsZm, III. NUMERICAL RESULTS (48) We implement the two-factor and one-factor fitting ξ where Ms, Vs, and m are defined in an analogous manner algorithms using current term structures of default-free and as in Equation (39). Similarly, the adjustment parameter defaultable yield curves so as to check the accuracy and the ξ m is determined by fitting the term structure of default- effectiveness of the numerical algorithms against the ana- able bond prices. lytic price formulas. In our numerical experiments, we gen- Solution of the recursive relation (48) gives erate the current yield of the default-free Treasury bond by:

m−1 Y (T)=0.08 − 0.05e−1.8T . m m−1−j (53) sm = s0(1 + Ms) + (1 + Ms) (ξj + Xj,j+1,M ) + j=0 m−1 We use the Briys and de Varenne [1997] intertem- m−1−j Vs(1 + Ms) Zj. (49) poral default model to generate the yield for a high-rated j=0 bond and a low-rated bond. The spread curves are obtained by subtracting the yield of the default-free Trea- The discrete version of Equation (7) is seen to be sury bonds from the yield curves of the defaultable bonds. Note that the yield curves of the default-free Treasury    m−1 bond and the high-rated bond increase monotonically Qtm D(t0,tm)=P(t0,tm)E exp − sj∆t with time to maturity, while the yield curve of the low- j=0 rated bond exhibits the typical hump shape. (50) These yield curves (shown in Exhibit 1) are used as input in all subsequent calculations. In Exhibits 2-A and 2-B, we plot the time-depen- P φ from which we deduce the formula for E [sm] (this is equal dent drift function (t) in the mean reversion of the spot β to m, the time-varying drift for the credit spread process): spread process, obtained by fitting the yield curves of the high-rated bond and the low-rated bond, respectively. The m−1 drift φ(t) is shown to be an increasing function of the spot P 1 D(t0,tm+1) P E [sm]= − ln − E [sj] – spread volatility σ for both high-rated and low-rated bonds. ∆t P(t0,tm+1) s j=0 Since the spread approaches zero as time approaches m−1 2 maturity, the drift goes to zero as t tends to zero. The Xj,j+1,m m−j γm [(1 + Ms) − 1] + , drift functions for both high-rated and low-rated bonds Ms 2 j=0 reach some peak value, then decline at a higher value of (51) t and settle at some asymptotic value at infinite t. γ~2 γ 2 where m, is defined similarly as m, except that rm, a, and The drift is greater for the low-rated bond since the σ σ r are replaced by sm, b, and s, respectively. corresponding spot spread has a higher value. Also, the drift Under the forward measure, the time t0 price of a shows less sensitivity to the spot spread volatility for the credit spread derivative is given by: low-rated bond.

SPRING 2003 THE JOURNAL OF DERIVATIVES 59 E XHIBIT 1 Yield Curve Plots

0.12

low-rated yield

0.1

0.08

low-rated

0.06 high-rated yield Yield / Spread

Treasury yield 0.04

0.02 high-rated yield spread

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time to Maturity

E XHIBIT 2-A Time-Dependent Drift Function for High-Rated Bond

0.1

σ = 0.3 s 0.08

0.06 σ = 0.2 s

0.04 (t) φ σ = 0.1 s

0.02

0

0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

60 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003 E XHIBIT 2-B Time-Dependent Drift Function for Low-Rated Bond 0.6

0.5

0.4

0.3 (t) φ 0.2

σ = 0.2 s σ = 0.1 0.1 s σ = 0.3 s

0

0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

E XHIBIT 3 Comparison of One-Factor and Two-Factor Tree Calculations Average Relative Errors in Average Relative Errors in Number of Time Steps Two-Factor Tree Calculations One-Factor Tree Calculations

83.18% 2.92% 16 1.175% 1.63% 32 1.10% 1.03%

To test the convergence of the two-factor and one- O(M 3), where M is the number of time steps. This prop- factor fitting algorithms, we apply the algorithms to price erty of a polynomial order of complexity is highly desir- a credit spread put with payoff as given in Equation (12). able, as numerical implementation does not take an We use different combinations of parameter values in the unreasonable amount of computation time. pricing model, and compare the numerical results with We also investigate the pricing behavior of the credit the results obtained from the analytic price formula in spread put option with respect to time to and Equation (16) with varying numbers of time steps. spot spread volatility. In Exhibit 4-A, we plot the spread Exhibit 3 shows the convergence behaviors of the put option function against time to expiration with varying two fitting algorithms in terms of average relative errors spot spread volatility where the underlying is a high-rated (in percentages). The relative errors have relatively low bond. The spread put price increases with longer time to values that decline in a roughly linear manner with expiration and higher spot spread volatility. increasing numbers of time steps. When the underlying is a low-rated bond, however, The one-factor algorithm provides better accuracy. the time-dependent behavior of the spread put price In terms of computational complexity, the one-factor exhibits a humped shape (see Exhibit 4-B). This may be algorithm is O(M 2), while the two-factor algorithm is due to a declining credit spread at very maturities for

SPRING 2003 THE JOURNAL OF DERIVATIVES 61 E XHIBIT 4-A Price of Credit Spread Put Against Time to Expiration—High-Rated Bond 0.12 σ = 0.3 s

0.1

0.08 σ = 0.2 s

0.06 Option Price

0.04 σ = 0.1 s

0.02

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time to Expiration

E XHIBIT 4-B Price of Credit Spread Put Against Time to Expiration—Low-Rated Bond

0.14

σ = 0.3 s 0.12

0.1

σ = 0.2 0.08 s

Option Price 0.06

σ = 0.1 s 0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time to Expiration

62 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003 low-rated bonds. Spread put prices are always seen to be price of the credit spread put is an increasing function of increasing functions of volatility, regardless of the credit the volatility of the spread process. This result holds for both quality of the underlying bond. high-rated and low-rated underlying risky bonds. When spread volatility is relatively low and time to expiration is IV. CONCLUSION relatively long, the credit spread put price function may become a decreasing function of time to expiration for The credit spread is a common state variable in lower-rated underlying bonds. This property contrasts pricing models for credit spread derivatives. By assuming with the behavior of most other option products, that the instantaneous spot riskless interest rate and the but it is consistent with the decline of the spread of a long- credit spread follow correlated stochastic processes, we term lower-rated bond under low spread volatility. have developed a no-arbitrage approach for pricing credit Our work extends pricing models for credit deriva- spread derivatives. tives in several respects. Under the assumption of extended Our pricing model assumes the riskless interest rate Vasicek processes for the underlying state variables, one follows a mean-reverting process with time-dependent can obtain analytic price formulas for credit spread deriva- drift, and the credit spread (or its logarithm) follows a sim- tives and closed-form representations of the time-depen- ilar process. The time-dependent drift functions in the dent drift terms using a no-arbitrage approach that fits mean reversion processes are determined by fitting current the current term structures of bond prices. Our one- term structures of default-free and defaultable bond prices. factor and two-factor fitting algorithms for pricing credit When both the spot rate and the spot spread follow spread derivatives exhibit only a polynomial order of com- an extended Vasicek process, the time-dependent drift putational complexity. And we show how to choose functions can be obtained in closed form. Also, we take an volatility structures so that the Heath-Jarrow-Morton expectation under the risk-adjusted forward measure to model reduces to the Hull and White model. derive analytic price formulas for credit spread derivatives. When the logarithm of the spread follows the REFERENCES extended Vasicek process, the Black-Karasinki model does not exhibit the same level of analytic tractability, but one Black, F., and P. Karasinski. “Bond and Option Pricing when can always rely on numerical fitting algorithms to deter- the Rates are Lognormal.’’ Financial Analysts Journal, mine the drift term and compute the price of the credit July/August 1991, pp. 52-59. spread option. We also show that the two-factor extended Vasicek model is a special case of the Heath-Jarrow- Briys, E., and F. de Varenne. “Valuing Risky Fixed Rate Debt: An Extension.’’ Journal of Financial and Quantitative Analysis, 32 Morton model corresponding to some specific choice of (1997), pp. 239-248. the volatility structures. By following the Hull-White tree-fitting algorithm, Chacko, G., and S. Das. “Pricing Interest Rate Derivatives: A we have developed two-factor and one-factor fitting algo- General Approach.’’ Review of Financial Studies, 15 (2002), pp. rithms for pricing credit spread derivatives. Since our tri- 195-241. nomial tree construction has recombining properties, a polynomial order of computational complexity is observed Cox, J.C., J.E. Ingersoll, Jr., and S.A. Ross. “A Theory of the in our numerical algorithm. The polynomial order prop- Term Structure of Interest Rates.” Econometrica, 53 (1985), pp. erty is an overriding advantage over the Heath-Jarrow- 385-407. Morton (HJM) framework, since the non-recombining tree in the HJM model leads to an exponential order of Das, S.R. “Credit Risk Derivatives.’’ The Journal of Derivatives, complexity. 2, 3 (Spring 1995), pp. 7-23. The validity of our analytic price formulas and the Das, S.R., and R.K. Sundaram. “A Discrete-Time Approach versatility of the numerical schemes are verified through to Arbitrage-Free Pricing of Credit Derivatives.’’ numerical comparison of the computed results and the Science, 46 (2000), pp. 46-62. results obtained by analytic formulas. The algorithms are numerically accurate even with relatively few time steps. Duffie, D., and K. Singleton. “Modeling Term Structures of Assuming that both the spot rate and the spot spread Defaultable Bonds.’’ Review of Financial Studies, 12 (1999), pp. follow extended Vasicek processes, we observe that the 687-720.

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64 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003