Short-term interest rates

Term structure models are essentially models of the interest-rate process. The problem being posed is, what behaviour is exhibited by interest rates, and by the short-term interest rate in particular An excellent description of the three most common processes that are used to describe the dynamics of the short-rate is given in Phoa (1998), who describes ... 

Chan, K., et al., 1992. An empirical comparison of alternative models of the short-term interest rate. J. Financ. 47, 1209-1227. 

Central banks and market practitioners use interest rates prevailing in the government bond market to extract certain information, the most important of which is implied forward rates. These are an estimate of the market s expectations about the future directirMi of short-term interest rates. They are important because they signify the market s expectafirMis about the future path of interest rates however, they are also used in derivative pricing and to create synthetic bond prices from the extent of credit spreads of corporate bonds. 

The Bank of England uses a variation of the Svensson yield curve model, a one-dimensional paranetric yield curve model. This is similar to the Nelson and Siegel model and defines the forward rate curve/(/n) as a function of a set of unknown parameters, which are related to the short-term interest rate and the slope of the yield curve. The model is summarised in Appendix B. Anderson and Sleath (1999) assess parametric models, including the Svensson model, against spline-based methods such as those described by Waggoner (1997), and we summarise their results later in this chapter. 

The application of risk-neutral valuation, which we discussed in chapter 45, requires that we know the sequence of short-term rates for each scenario, which is provided in some interest-rate models. For this reason, many yield curve models are essentially models of the stochastic evolution of the short-term rate. They assume that changes in the short-term interest-rate is aMarkov process. (It is outside the scope of this book to review the mathematics of such processes, but references are provided in subsequent chapters.) This describes an evolution of short-term rates in which the evolution of the rate is a function only of its current level, and not the path by which it arrived there. The practical significance of this is that the valuation of interest-rate products can be reduced by the solution of a single partial differential equation. 

In a later section we will examine why investors and others might want to use these exchange-traded bond options. Before this, however, we will turn our attention towards the other major exchange-traded interest rate product, short-term interest rate options. 

A cap is an interest rate derivative offering protection against unexpected fluctuations in short-term interest rates, but over an extended period of time. An example will make this clearer. 

Range accrual notes can be based on almost any qnoted market rate. In this example, the return on a dollar-denominated instrnment is linked to European short-term interest rates, but it could also be linked to exchange rates, stock or commodity prices. In most cases they are structured using strips of digital options. 

Kalok Chan, G. Andrew Karolyi, Francis A. Longstaff, and Anthony B. Sanders, An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, Journal of Finance 47 (1992), pp. 1209-1227. 

Euronext-LIFFE s 3-month EURIBOR futures option, traded on the Chicago Mercantile Exchange, is an actively traded short-term interest rate option that enjoys high trading volume. If these options are exercised, the buyer and the seller of the option take positions in an underlying 3-month EURIBOR futures contract. The futures contract is cash-settled and the final price at delivery is equal to 100 minus the 3-month LIBOR. 

Mean Reversion Rate EsIilltfjM Convexity bias estimation requires an estimate of the mean reversion rate (a) and the standard deviation (a) of the change in short-term interest rates. There are several alternative methodologies for estimating a and o. The first methodology uses historical data to estimate the parameters. 

We assume that the short-term interest rates follow the following Vasicek discount bond prices stochastic process ... 

Estimating o flows from the mean reversion estimation process. It estimates the conditional standard deviation of short-term interest rates using the GARCH(1, 1) model ... 

The approach is similar to the historical simulation method, except that it creates the hypothetical changes in prices by random draws from a stochastic process. It consists of simulating various outcomes of a state variable (or more than one in case of multifactor models), whose distribution has to be assumed, and pricing the portfolio with each of the results. A state variable is the factor underlying the price of the asset that we want to estimate. It could be specified as a macroeconomic variable, the short-term interest rate or the stock price, depending on the economic problem. 

In order to illustrate the application of Monte Carlo simulation, we present two methods in detail below. The first considers price movements, and the second, which also handles pull-to-par, is a short-term interest rate model. 

The Monte Carlo experiment consists of first simulating movements in short-term interest rates, then using the simulated term structure to price the securities at the target rate. 

In the academic literature, the risk-neutral price of a zero-coupon bond is expressed in terms of the evolution of the short-term interest rate, r t)—the rate earned on a money market account or on a short-dated risk-free security such as the T-bill—which is assumed to be continuously compounded. These assumptions make the mathematical treatment simpler. Consider a zero-coupon bond that makes one payment, of 1, on its maturity date T. Its value at time ris given by equation (3.14), which is the redemption value of 1 divided by the value of the money market account, given by (3.12). 

The Binomial Tree of Short-Term Interest Rates... 

A rise in volatility generates a range of possible future paths around the expected path. The actual expected path that corresponds to a zero-coupon bond price incorporating zero OAS is a function of the dispersion of the rai e of alternative paths around it. This dispersion is the result of the dynamics of the interest-rate process, so this process must be specified for the current term structure. We can illustrate this with a simple binomial model example. Consider again the spot rate structure in Table 12.1. Assume that there are only two possible future interest rate scenarios, outcome 1 and outcome 2, both of equal probability. The dynamics of the short-term interest rate are described by a constant drift rate a, together with a volatility rate a. These two parameters describe the evolution of the short-term interest rate. If outcome 1 occurs, the one-period interest rate one period from now will be... 

Traders who believe the cost of carry will decrease can sell the spread to exploit this view. Those with longer time horizons might trade the spread between the short-term interest rate contract and the long bond future. 

It is surprising that bond-fund buyers are not fully aware of the inverse relationship between interest rates and bond prices, considering that short-term interest rates are at their lowest levels in four decades. The interest rates on some types of bonds are likely to increase whenever the economy eventually rebounds from the current slump. 

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