A Two-Factor Lognormal Model of the Term Structure and the Valuation Of

A Two-factor Lognormal Mo del of the Term Structure and the Valuation of American-Style Options on Bonds 1 2 3 Sandra Peterson Richard C. Stapleton Marti G. Subrahmanyam February 25, 1999 1 Department of Accounting and Finance, The ManagementScho ol, Lancaster University, Lan- caster LA1 4YX, UK. Tel:44524{593637, Fax:44524{847321. 2 Department of Accounting and Finance, Strathclyde University, Glasgow, UK. Tel:44524{381 172, Fax:44524{846 874, e{mail:[email protected] 3 Leonard N. Stern Scho ol of Business, New York University, Management Education Center, 44 West 4th Street, Suite 9{190, New York, NY10012{1126, USA. Tel: 212998{0348, Fax: 212995{ 4233, e{mail:[email protected]. Abstract A Two-factor Lognormal Mo del of the Term Structure and the Valuation of American-Style Options on Bonds We build a no-arbitrage mo del of the term structure, using two sto chastic factors, the short- term interest rate and the premium of the forward rate over the short-term interest rate. The mo del extends the lognormal interest rate mo del of Black and Karasinski 1991 to two factors. It allows for mean reversion in the short rate and in the forward premium. The metho d is computationally ecient for several reasons. First, we use Lib or futures prices, enabling us to satisfy the no-arbitrage condition without resorting to iterative meth- o ds. Second, the multivariate-binomial metho dology of Ho, Stapleton and Subrahmanyam 1995 is extended so that a multip erio d tree of rates with the no-arbitrage prop erty can b e constructed using analytical metho ds. The metho d uses a recombining two-dimensional binomial lattice of interest rates that minimizes the number of states and term structures over time. Third, the problem of computing a large numb er of term structures is simpli ed by using a limited numb er of 'bucket rates' in each term structure scenario. In addition to these computational advantages, a key feature of the mo del is that it is consistent with the observed term structure of volatilities implied by the prices of interest rate caps and o ors. We illustrate the use of the mo del by pricing American-style and Bermudan-style options on b onds. Option prices for realistic examples using forty time p erio ds are shown to be computable in seconds. Atwo-factor lognormal mo del of the term structure 1 1 Intro duction One of the most imp ortant and dicult problems facing practitioners, in the eld of interest rate derivatives in recentyears, has b een to build inter-temp oral mo dels of the term structure of interest rates that are b oth analytically sound and computationally ecient. These mo dels are required b oth to help in the pricing and in the overall risk managementof a book of interest rate derivatives. Although many alternative mo dels have b een suggested in the literature and implemented in practice, there are serious disadvantages with most of them. For example, Gaussian mo dels of interest rates, whichhave the advantage of analytical tractability, have the drawbackof p ermitting negative interest rates, as well as failing to take into account the p ossibilityof skewness in the distribution of interest rates. Also, many of the term-structure mo dels used in practice are restricted to one sto chastic factor. Since the work of Ho and Lee 1986, it has b een widely recognized that term-structure mo dels must p ossess the no-arbitrage prop erty. In this context, a no-arbitrage mo del is one in which the forward price of a b ond is the exp ected value of the one-p erio d-ahead sp ot b ond price, under the risk-neutral measure. Building mo dels that p ossess this prop erty has b een a ma jor pre-o ccupation of b oth academics and practitioners in recent years. One mo del that achieves this ob jective in a one-factor context is the mo del prop osed by Black, Derman and Toy 1990BDT, and extended by Black and Karasinski 1991BK. In essence, the mo del whichwe build in this pap er is a two-factor extension of this typ e of mo del. In our mo del, interest rates are lognormal and are generated bytwo sto chastic factors. The general approachwe take is similar to that of Hull and White 1994HW, where the conditional mean of the short rate dep ends on the short rate and an additional sto chastic factor, which can be interpreted as the forward premium. In contrast to HW, and in line with BK, we build a mo del where the conditional variance of the short rate is a function of time. It follows that the mo del can be calibrated to the observed term structure of interest rate volatilities implied byinterest rate caps/ o ors. Essentially, the aim here is to build a term structure mo del which can b e applied to value American-style contingent claims on interest rates, which is consistent with the observed market prices of Europ ean-style contingent claims. Our approach to building a no-arbitrage term structure for pricing interest rate derivatives is consistent with the general framework prop osed by Heath, Jarrow and Morton 1992HJM. In the HJM formulation, assumptions are made ab out the volatility of the forward interest rates. Since the forward rates are related in a no-arbitrage mo del to the future sp ot interest rates, there is a close relationship between this approach and the one we are taking. In fact, the HJM forward-rate volatilities can be thought of as the outputs of our mo del. If the parameters of the HJM mo del are known, this represents a satisfactory alternative Atwo-factor lognormal mo del of the term structure 2 approach. However, the BDT-BK-HW approach has the advantage of requiring as inputs the volatilities of the short rate and of longer b ond yields which are more directly observable from market data on the pricing of caps, o ors and swaptions. Our mo del, which is an extension of BK, has similar advantages. We would like any mo del of the sto chastic term structure to have a number of desirable prop erties. Apart from satisfying the no-arbitrage prop erty,wewant the mo del inputs to b e consistent with the observed conditional volatilities of the variables and the mean-reversion of the short rate and the premium factor. We also require that the short term interest rates b e lognormal, so that they are b ounded from b elowby zero and skewed to the right. From a computational p ersp ective, we require the state space to b e non-explosive, i.e. re-combining, so that a reasonably large span of time-p erio ds can b e covered. The complexities caused by these mo del requirements are discussed in section 2. Weintro duce a numb er of new asp ects into our mo del that allow us to solve these requirements. Following Miltersen, Sandmann and Sondermann 1997 and Brace, Gatarek and Musiela 1997BGM we mo del the Lib or rate. We then extend and adapt the recombining binomial metho dology of Nelson and Ramaswamy 1990 and Ho, Stapleton and Subrah- manyam 1995HSS to mo del lognormal rates, rather than prices. The new computational techniques are discussed in detail in section 3. Section 4 presents the basic two-factor mo del, and discusses some of its principal characteristics. In section 5, we explain how the multi- p erio d tree of rates is built using a mo di cation of the HSS metho dology. In section 6, we present some numerical examples of the output of our mo del, apply the mo del to the pricing of American-style and Bermudan-style options and discuss the computational eciency of our metho dology. Section 7 presents our conclusions. 2 Requirements of the mo del There are several desirable features of any multi-factor mo del of the term structure of interest rates. Some of these features are requirements of theoretical consistency and others are necessary for tractability in implementation. Keeping in mind the latter requirements, it is imp ortant to recognize that the principal purp ose of building a mo del of the evolution of the term structure is to price interest rate options generally, and, in particular, those with path-dep endent payo s. The simplest examples of options that need to be valued using such a mo del are American-style and Bermudan-style options on an interest rate. First, since we wish to b e able to price any term-structure dep endent claim, it is imp ortant that the mo del output is a probability distribution of the term structure of interest rates Atwo-factor lognormal mo del of the term structure 3 at each p oint in time. A realistic mo del should be able to pro ject the term structure for ten or twenty years, at least on a quarterly basis. With the order of forty or eighty sub- p erio ds, the computational task is substantial and complex. If we do not compute each term structure p oint at each no de of the tree, then we need to b e able to interp olate, where necessary, to obtain required interest rates or b ond prices. As in the no-arbitrage mo dels of HL, HJM, HW, BDT, and BK, we rst build the risk-neutral or martingale distribution of the short-term interest rate, since other maturity rates and b ond prices can be computed from the short-term rate.

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