One-factor models

Eberlein and Kluge [29] find a closed-form solution for swaptions using a L6vy term stnicture model. A solution for bond options assuming a one-factor model has been derived by Jamishidian [42]. 

So, now we have determined that a short-rate model is related to the dynamics of bond yields and therefore may be used to derive a complete term structure. We also said that in the same way the model can be used to value bonds of any maturity. The original models were one-factor models, which describe the process for the short-rate r in terms of one source of uncertainty. This is used to capture the short-rate in the following form ... 

From the previous section, we see that under a model that assumes the short-rate to follow a normal distribution, there can arise instances of negative forward rates. The Cox et al. (1985) is a one-factor model and as originally presented removed the possibility of negative rates. Under the CIR model, the dynamics of the short-rate are described by Equation (3.38) ... 

Black, F., Derman, E., Toy, W., 1990. A one-factor model of interest rates and its application to Treasury bond options. Financ. Anal. J. 46, 33-39. 

It is generally accepted that one-factor models can be used for most bond applications where multi-factor models are more appropriate may be in the following situations ... 

The optimum approach would appear to be a combination of a one-factor model and a multi-factor model to suit individual requirements. However, this may not be practical it might not be ideal to have different parts of a bank using different models (although this does happen, desks across the larger investment banks sometimes use different models) and valuing instruments using different models. The key factors to focus on are accessibility, accuracy, appropriateness and speed of computation. 

When calcnlating option prices in a one-factor model, a frequently made assnmption is that the process is driven by the short rate often with a mean reversion featnre linked to the short rate. There are several popnlar models which fall into this category, for example, the Vasicek model, and the Cox, Ingersoll, and Ross model both of which will be discussed in more detail later. Calculating option prices in a two-factor model involves both the short- and long-term rates linked by a mean reversion process. 

CIR wrote arguably the first of several papers developing one-factor models of the term structure of interest rates. Other papers which were... 

Wolfgang M. Schmidt, On a General Class of One-Factor Models for the Term Structure of Interest Rates, Finance and Stochastics 1 (1997), pp. 3-24. 

Rabinovitch advocated the idea that the bond follows a log-normal process (similar to equity prices). Chen pointed out that this assumption is grossly misleading since the bond price is a contingent claim on the same interest rate. As a result the bond option pricing model cannot be a two-factor model as proposed by Rabinovitch rather it collapses to a one-factor model, in which case the formulae are the same with those proved respectively by Chaplin and by Jamshidian. 

Consider a bond paying a periodic cash payment p at times Ti,T2,...,T , and the principal at maturity T = T j. A coupon bond can be mapped into a portfolio of discount bonds with corresponding maturities (under one source of uncertainty, that is one factor model). The value of a coupon bearing bond at time t [Pg.594]

Because we started with a one-factor model for the short interest rates we can use the decomposition property emphasized by Jamshidian and calculate the required coupon-bond European call price as the sum of all the elements in the last column in Exhibit 18.7, which include the coupon rate factor p. Thus, the value of this option is 0.344. 

It is a one-factor model of interest rates, which is driven by movements in the short-term rates drf. In this model, movements in longer-term interest rates are perfectly correlated with movements in the shortterm rate throngh dz. 

The short rate follows a stochastic process, or probability distribution. So, although the rate itself can assume a range of possible future values, the process by which it changes from value to value can be modeled. A one-factor model of interest rates specifies the stochastic process that describes the movement of the short rate. 

Impose no-arbitrage conditions, based on the principle of hedging a position in one bond with a position in another bond (for a one-factor model a two-factor model requires two bonds as a hedge) of a different maturity, to derive the partial differential equation of the zero-coupon bond price. 

Although published officially in 1985, the Cox-Ingersoll-Ross model was described in academic circles in 1977, or perhaps even earlier, which would make it the first interest rate model. Like Vasiceks it is a one-factor model that defines interest rate movements in terms of the dynamics of the short rate. It differs, however, in incorporating an additional feature, which relates the variation of the short rate to the level of interest rates. This feature precludes negative interest rates. It also reflects the fact that interest rate volatility rises when rates are high and correspondingly decreases when rates are low. The Cox-lngersoll-Ross model is expressed by equation (4.11). 

Black, E, E. Derman, and W. Toy. 1990. A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options. Einancial Analysts Journal, Spring, 33-39. 

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