Extensions of the Ho and Lee interest-rate model to the multinomial case☆
Introduction
No-arbitrage bond models are characterised by the assumptions that the initial term structure is given and the intertemporal changes in the term structure follow a stochastic process with one or more factors.
In spite of the fact that two or three factor models provide the more realistic description of term structure movements, most of the recent literature focuses on one-factor models which provide large tractability.
Heath et al., 1990, Heath et al., 1992 proposed a general approach for no-arbitrage one-factor models in which term structure movements are calculated specifying the volatility of all forward rates at all times and the initial values of the forward rates are chosen to be consistent with the initial term structure. Their models are usually non-Markovian and require large computational efforts. In addition accurate valuation of derivatives is not provided.
An alternative approach involves a Markov process for the short-term interest rate. The first no-arbitrage model following this latter approach was proposed by Ho and Lee (1986) in the form of a binomial tree of discount bond prices. In this model all short-term interest rates are assumed to be normally distributed and have the same variance in time. The drift of the interest-rate process is time dependent with no mean reversion.
The model belongs to the class of affine single-factor term structure models discussed by Duffie (1996), Hull and White (1990), Vasicek and Fong (1982), Black et al. (1990), Cox et al. (1985) and Flesaker (1993).
Hull and White (1990) extended the Ho and Lee model introducing two volatility parameters: the overall value of volatility and a reversion rate parameter. Hull and White, 1994, Hull and White, 1996 provide a procedure to produce a trinomial tree by a forward induction either for the Ho–Lee and the Hull–White models respectively.
A version of the Ho–Lee model for multinomial tree was proposed by Abaffy et al. (1994). This generalisation also extends the results proposed by Bliss and Ronn (1989) based on a trinomial process.
Looking at other choices for short-term interest-rate models, Black–Derman–Toy suggest a lognormal process whose mains feature is that the short rate cannot become negative. An efficient procedure for implementing this latter model involving a binomial tree was provided by Black and Karasinski (1991). An efficient procedure, suggested by Bjerksund and Stensland (1996), is based on the iterative update of an approximated tree obtained by approximation formulas.
This paper deals with further theoretical insight and some empirical results concerning the multinomial model which seems to be of particular interest for the Italian market where the pricing of interest-rate sensitive securities is not well fitted by the binomial process, as shown in Bertocchi and Gnudi (1995).
Section 2 introduces the model and outlines the main theorems. 3 Determination of functions, 4 Local expectations hypothesis and the term premium present the form of the perturbation functions as solution of first-order difference equations and provides a closed formula to evaluate a contingent claim on the tree generated by the model. In Section 5, a theorem concerning the local expectations hypothesis is presented. A test of the theory for the Italian market appears in Section 6. Conclusions appears in Section 7.
Section snippets
A multinomial model for pricing interest-rate derivative securities
We develop a multinomial model for short-term interest rates using a tree allowing k+1 paths from each node, with k⩾1.
We assume, as in the Ho–Lee model, that:
- (1)
Zero-coupon bonds with maturities of T=1,…,m are traded at discrete points in time t=0,…,n.
- (2)
The zero-coupon bond market is frictionless.
- (3)
At point t in time, there are several possible discount functions, Pit(T), t=0,…,n, i=0,…,k, T=1,…,m.
- (4)
The transition behaviour of a discount function is characterised by perturbation functions hti(T) that
Determination of functions hjt
For simplicity we suppose in this paragraph, that functions hjt do not change with time, therefore, we omit the subscript t. The functions hj must satisfy, from Theorem 2.2, Eqs. (21) and, from Theorem 2.1, Eq. (2). Using relation (30), we must solve the following equations:Set γ=h1(1) and δ=h0(1) andand rearranging Eq. (32) yieldsEq. (35) is a first-order difference equation. Its general
Local expectations hypothesis and the term premium
We now generalise the necessary and sufficient condition for the local expectation hypothesis (LEH) given in Ho and Lee (1986).
Before formulating the theorem, we recall the definition of LEH following Cox et al. (1981). The LEH means that the expected holding period returns are equal for all possible default-free bonds, where the holding period is the next shortest (one-period) interval. Thus, when the LEH does not hold, we expect that longer bonds have higher expected returns and we define a T
A closed formula for computing discount factors
The model is an extension of the Ho–Lee model and it is possible to derive closed formulas for evaluating interest-rate derivative instruments along a tree built up using the functions hj. In this section we address how to evaluate a contingent claim on the tree generated by our model. We follow the approach of Montessoro et al. (1994), generalising it to the multinomial case. Let the price of an interest-rate contingent claim at time t and state i be C(t,i). Theorem 5.1 Consider any interest-rate
Empirical appropriateness of a multinomial model to the Italian market
To test the proposed model we choose a period in time where the yields move in a large range and a simple derivative instrument based on bond, as the CTO, was easy available on the market.
In the empirical tests we used BTP (Italian Treasury coupon bonds) weekly prices available on the wireless market over the period 1990–1994 to fit the initial term structure.
To calibrate the model propose in Section 5 to the market, we use puttable bonds (CTOs), that were available during the chosen period. We
Conclusions
This paper presents the theoretical framework of a multinomial model for the evolution of the term structure of interest rates. The appropriateness of a multinomial model for the Italian market in a well-defined period is shown. The calibration of the model is strictly related to the availability on the market of quoted derivative securities based on bonds. A possible further analysis is related to generalise the proposed model to include the possibility that the number of states changes along
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The work was supported by MURST 40% 96–97, CNR n.96.01313.CT10 and 97.01205.CT10, MIUR 60% 2002 grants.