Interest-rate models

We obtain a closed-form solution for the special case of a coupon bond option containing only one payment date (see section 5.3.1). Furthermore, there exists a closed-form solution assuming one-factor interest rate models (see Jamishidian [42]). 

Starting from the dynamics of the short rates, extensive work has been done in implementing jumps in interest rates models (see e.g. Ahn and Thompson (1988) andChako and Das [15]). However only a few authors implemented jumps in a HJM-framework (see. e.g. Shirakawa [70] and Glasser-man and Kou [32]). Further work could be done in implementing jumps in the aforementioned framework combined with USV and correlated sources of uncertainty. Another area of research could result from combining a HJM-Uke multiple RF-framework with the class of USV models given by... 

Johannes M (2004) The Statistical and Economic Role of Jumps in Continous-Time Interest Rate Models. Journal of Finance 59 227-260. 

Many interest rate models assume that the movement of interest rates over time follows a Weiner process. 

James, J., Webber, N., 2000. Interest Rate Modelling. Wiley, Chichester, Chapters 3-5,7-9,15-16. Jamshidian, F., 1991. Bond and option valuation in the Gaussian interest rate model. Res. Finance 9, 131-170. 

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

An interest-rate model provides a description of the dynamic process by which rates change over time, in terms of a statistical construct, as well as a means by which interest-rate derivatives such as options can be priced. It is often the practical implementation of the model that dictates which type is used, rather than mathematical neatness or more realistic assumptions. An excellent categorisation is given in James and Webber (2000), who list models as being one of the following types ... 

Although formally published in 1985, the Cox-Ingeisoll-Ross model was being circulated in academic circles from the mid-1970s onwards, which would make it one of the earliest interest-rate models. 

Implementing an interest-rate model requires the input of the term structure yields and volatility parameters, which are used in the prcx ess of calibrating... 

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

Jamshidian, F., 1996. Bond, futures and option valuation in the quadratic interest rate model. Appl. Math. Finance 3, 93-115. 

In response to these issues, interest-rate models have been developed that model the entire yield curve. In a whole yield curve, the dynamics of the entire term structure are modelled. The Ho-Lee model is a simple type of whole curve... 

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. 

A landmark development in the longstanding research into yield ciuve modelling was presented by David Heath, Robert Jarrow and Andrew Morton in their 1989 paper, which formally appeared in volume 60 of Econometrica (1992). The paper considered interest-rate modelling as a stochastic process, but applied to the entire term structure rather than only the short-rate. The importance of the HJM presentation is this in a market that permits no arbitrage, where interest... 

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