Spot rates volatility term structure

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

These models are two more general families of models incorporating Vasicek model and CIR model, respectively. The first one is used more often as it can be calibrated to the observable term structure of interest rates and the volatility term structure of spot or forward rates. However, its implied volatility structures may be unrealistic. Hence, it may be wise to use a constant coefficient P(t) = P and a constant volatility parameter a(t) = a and then calibrate the model using only the term structure of market interest rates. It is still theoretically possible that the short rate r may go negative. The risk-neutral probability for the occurrence of such an event is... 

The Ho and Lee model is straightforward to implement and is regarded by practitioners as convenient because it uses the information available from the current term structure so that it produces a model that precisely fits the current term structure. It also requires only two parameters. However, it assigns the same volatility to all spot and forward rates, so the volatility structure is restrictive for some market participants. In addition, the model does not incorporate mean reversion. 

In an arbitrage-free model, the initial term structure described by spot rates today is an input to the model. In fact such models could be described not as models per se, but essentially a description of an arbitrary process that governs changes in the yield curve, and projects a forward curve that results from the mean and volatility of the current short-term rate. An equilibrium term structure model is rather more a true model of the term structure process in an equilibrium model the current term structure is an output from the model. An equilibrium model employs a statistical approach, assuming that market prices are observed with some statistical error, so that the term structure must be estimated, rather than taken as given. 

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... 

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