A Fresh View on the Ho-Lee Model of the Term Structure from a Stochastic Discounting Perspective

Abstract

In the present paper we reformulated a slightly generalised version of the Ho-Lee-model of the term structure of interest rates by using the method of stochastic discounting. The main results can be summarized as follows: The time discrete Ho-Lee-model of the interest rate structure can be developed by constructing a suitable stochastic discount factor from which the whole structure of the model can be deduced; interest rates become binomial random walks with a drift which is mainly determined by the initial term structure. Besides using the empirical probability distribution, the model allows for a specific market's risk aversion to be imposed. This parameter is reponsible for how futures contracts are valued by the market. In a limiting process which is similar to the work of Heath, Jarrow and Morton (1990), it is shown that the interest rate becomes a normally distributed process the drift of which is determined by the initial term structure and the market's risk aversion and by the jump probability of the underlying binomial process. The latter property is in contrast with the Heath, Jarrow, Morton (1990) result where these probabilities play no specific role. Whether or not they do play a role depends on the market's risk aversion being zero or not. Furthermore, it turns out that the stochastic discount factor is log-normal, but, surprisingly, it is driven by two independent normally distributed factors. One of these factors is completely correlated with the short rate. An expression which facilitates the valuation of futures contracts is also provided; clearly, it severely depends on the market's risk aversion and becomes unity if there is no risk aversion in the market; in this case futures prices are, actually, expected future prices. The importance of knowing the stochastic discount factor lies in its capability to consistently value any arbitrary asset in the market, derivative or not.