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Review of Continuous-Time Term-Structure Models

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Review of Continuous-Time Term-Structure Models Review of Continuous-Time Term-Structure Mo dels Part I: equilibrium classical mo dels Jesp er Lund Department of Finance The Aarhus Scho ol of Business DK-8210 Aarhus V, Denmark E-mail: [email protected] Homepage: www.hha.dk/~jel/ First draft: April, 1997 Current draft: March 2, 1998 I am grateful to Peter Honor e for helpful comments. 1 Intro duction This pap er contains a survey of continuous-time term-structure mo dels. The general idea is intro ducing the reader student to the sub ject, so that he or she hop efully will b e able to read journal articles on the sub ject. The technical and mathematical nature of this pap er re ects that ob jective. It is now customary to divide term-structure mo dels into two groups, called arbitrage-free and equilibrium mo dels, resp ectively. The former group contains mo d- els which t the initial yield exactly, and prominent examples are the Heath, Jarrow and Morton 1992 mo del and the Hull and White 1990 extended Vasicek mo del. On the other hand, the equilibrium mo dels do not p er construction t the yield curve exactly. Sometimes, the mo dels in equilibrium framework are called \classical" mo dels since this approach dates back earlier than the more recent arbitrage-free 1 mo dels. To a large extent, the choice between arbitrage-free and equilibrium mo dels is dictated by the purp ose of the analysis. If we are interested in identifying b onds that are mispriced relative to other b onds, we can only use equilibrium mo dels. On the other hand, when pricing xed-income derivatives it is generally preferable to use an arbitrage-free mo del, see chapter 9 in Tuckman 1995 for an excellent discussion. For this reason, our survey pap er deals with b oth mo dels. In Part I, we discuss equilibrium mo dels classical mo dels with either a single or multiple factors, whereas Part II forthcoming describ es two typ es of arbitrage-free mo dels: equilibrium-typ e mo dels with time-dep endent parameters calibrated mo dels and mo dels in Heath, Jarrow and Morton class. The outline of part I is as follows: section 2 contains a brief summary of the de nition of the term structure using continuous comp ounding. Sections 3 presents a general analysis of one-factor mo dels, whereas section 4 describ es three examples. Finally, section 5 extends the mo deling framework to multiple factors, including a discussion of the exp onential-ane class of mo dels. Most multi-factor mo dels with known analytical solutions for the term structure b elong to this class. 1 Unfortunately, the literature is not consistent with resp ect to these de nitions. In some pap ers, esp ecially pre-1990 pap ers, a mo del is only called an equilibrium mo del if derived from the utility function of the representative agent. The Vasicek 1977 mo del is sometimes referred to as an arbitrage-free or partial equilibrium mo del b ecause an \absence of arbitrage" argument is used in the mo del derivation, cf. also section 3 in the present pap er. In summary, there is b ound to b e some confusion, and the reader should b e careful when encountering de nitions along these lines in, esp ecially, the \older" literature. 1 2 De nitions There are three di erentways to represent the term structure of interest rates: 2 P t; T is the price, at time t, of a zero-coup on b ond maturing at time T the maturity date . The time to maturity of this b ond is = T t. It is imp ortant to note the distinction b etween the maturity date and the time to maturity | they are only identical when t =0. In general, we assume that P t; T exists for all T >t. Rt; T The yield-to-maturity with continuous comp ounding at time t, for a zero- coup on b ond maturing at time T . f t; T The instantaneous forward rate at time t, for a zero-coup on b ond matur- ing at time T . The yield-to-maturity R t; T and forward rate f t; T are de ned as follows: log P t; T R t; T = 1 T t @P t; T =@ T @ log P t; T f t; T = = : 2 P t; T @T Note that log denotes the natural base e logarithm. The inverse relationship ex- presses the b ond price, P t; T , in terms of either R t; T or ft; T : R t; T T t P t; T = e 3 R T f t; sds t P t; T = e : 4 The rst formula, 3, follows simply by rearranging the de nition of R t; T in 1. To derive 4, rst note that Z Z T T @ log P t; s log P t; T log P t; t = ds = f t; sds; 5 @s t t and since P t; t=1, we get 4. Furthermore, by equating the terms in the exp onents in 3 and 4, we get the following relationship b etween yield-to-maturity and forward rates: Z T 1 ft; sds; 6 R t; T = T t t which may be interpreted as the average forward rate over the remaining time to maturityof the b ond. 2 Unless we state otherwise in the text, all b onds are assumed to be zero-coup on b onds which havea single paymentof one \unit of account" at time T . Bonds with more than one remaining payment, for example bullets, are called coupon bonds. 2 3 A general one-factor mo del In this section we carefully explain the common mathematical structure of term- structure mo dels with a single factor, a framework that encompasses the Merton 1973, Vasicek 1977 and Cox, Ingersoll and Ross CIR 1985 mo dels. The reader is assumed to be familiar with sto chastic di erential equations SDEs, including Ito's lemma, at a level comparable to Hull 1997, Luenb erger 1997, or a similar non-mathematically oriented text. We make the following assumptions: A{1 The b ond market is frictionless: no distorting taxes, no transactions costs, no short-sale restrictions, and all b onds are in nitely divisible. A{2 Investors always prefer more wealth to less, i.e., the marginal utilityofwealth is p ositive at all levels of wealth. In e ect, this assumption rules out that arbitrage opp ortunities can exist. A{3 All b ond prices, i.e. P t; T for all T > t, dep end only on a single state variable: the short rate r in addition to t and T . By implication, changes t 3 in the yield curve at di erent maturities are p erfectly correlated. A{4 The short rate instantaneous interest rate follows the general SDE: dr = r dt + r dW ; 7 t t t t where r and r are the drift and volatility functions, resp ectively, and W a Brownian motion Wiener pro cess. t It is imp ortant to realize that we do not assume that the relationship b etween P t; T and the short rate, r , is known. On the contrary, the entire purp ose of the following t is deriving that function endogenously from the ab ove assumptions, esp ecially from the assumption ab out absence of arbitrage. In the rst step, we determine the sto chastic pro cess SDE for the b ond price P t; T . Note that T is xed, and t denotes calendar time. By Ito's lemma we get: dP t; T = t; T P t; T dt + t; T P t; T dW ; 8 P P t where 2 @P @P 1 @ P 2 t; T P t; T = r+ + r 9 P 2 @r @t 2 @r @P t; T P t; T = r: 10 P @r In equation 8, t; T is the exp ected instantaneous return of the b ond with ma- P turity date T , and t; T is the volatility standard deviation of the b ond return. P 3 Of course, this is highly restrictive, but later we relax the assumption with the so-called multi- factor mo dels. Rightnowwewanttokeep things simple! 3 The exp ected return and the volatility dep end on the short rate, r , but to simplify t the notation this dep endence is suppressed here. The problem is that equilibrium exp ected returns t; T for di erent T 's are P unknown, so a general expression for the b ond price P t; T cannot be determined at this stage. The intermediate goal in the following is developing some form of equilibrium mo del for the exp ected returns, t; T , for all T . Concretely, we use P the principle of no-arbitrage to reduce this problem to sp ecifying a single market price risk preference parameter. Supp ose we construct a p ortfolio consisting of w b onds with maturity date T 1 1 4 and w b onds with maturity date T . We require T 6= T , but apart from that T 2 2 1 2 1 and T can b e arbitrary. The value of the resulting p ortfolio, at time t, is denoted by 2 = w P t; T +w Pt; T ; 11 t 1 1 2 2 and the value, , satis es the SDE: t d =[w t; T P t; T + w t; T P t; T ] dt + t 1 P 1 1 2 P 2 2 [w t; T P t; T + w t; T P t; T ] dW : 12 1 P 1 1 2 P 2 2 t Since there are two b onds and only one source of risk, it must b e p ossible to eliminate the risk by cho osing w and w such that 1 2 w t; T P t; T +w t; T P t; T =0: 13 1 P 1 1 2 P 2 2 In general, this requires continuous adjustment of the p ortfolio which can be done costlessly since wehave assumed away transactions costs.
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