Two-Factor Interest Rate Models

This section briefly introduces a number of two-factor interest rate models. (The References section indicates sources for further research.) As their name suggests, these models specify the yield curve in terms of two factors, one of which is usually the short rate. A number of factors can be modeled when describing the dynamics of interest rates. Among them are 

Which factors the model incorporates depends in part on the purpose it is intended to serve—whether, for example, it is being used for pricing or hedging derivative instruments or for arbitrage trading. Other considerations also apply, such as the ease and readiness with which the parameters involved can be determined. 

---

Rebonato, R., Cooper, 1., 1996. The limitations of simple two-factor interest-rate models. J. Financ. Eng. 5 (1), 1-16. 

Francis A. Longstaff and Eduardo Schwartz, Interest Rate Volatility and the Term Structure A Two-Factor General Equilibrium Model, Journal of Finance 47 (1992), pp. 1259-1282 and Fletcher A. Longstaff and Eduardo Schwartz, A Two-Factor Interest Rate Model and Contingent Claim Valuation, Journal of Fixed Income 3 (1992), pp. 16-23. 

Longstaff FA, Schwartz ES (1992) Interest Rate Volatility an the Term Structure A Two-Factor General Equilibrium Model. Journal of Finance 47 1259-1282. 

Longstaff, F., Schwartz, E., 1992. Interest rate volatility and the term structure a two-factor general equUibrium model. J. Finance 47, 1259-1282. 

Althongh the sample option price is easy to read and interpret in respect of this screen, there is a mass of academic and practitioner research literatnre that provides a platform from which bond option prices in general can be calcnlated with integrity. The literature on modelling interest rate derivatives in this arena is freqnently divided into one-, two-factor, or mnltifactor, models. 

To understand the factors that are important in controlling the rates of ET reactions, it is best to refer to a specific theoretical model [35]. Choosing a model defines terms and allows us to analyze the results of experiments in precise ways. There are two main types of models classical and quantum mechanical. One way of smoothly moving from the use of a classical to a quantum mechanical model is provided by semiclassical (Landau-Zener) ET theory [36-38]. At high temperature, quantum mechanical models become equivalent in most respects to semiclassical ones. Thus the appropriate choice of model depends on the type of ET reaction that we are interested in studying. Ones in which the electron donor and acceptor have strong electronic interaction with each other prior to the ET event are well described by a classical model. In these systems an ET reaction always proceeds to products if the reactants reach the top of the reaction barrier (strong-... 

Using the prices of index-linked bonds, it is possible to estimate a term structure of real interest rates. The estimation of such a curve provides a real interest counterpart to the nominal term structure that was discussed in the previous chapters. More important it enables us to derive a real forward rate curve. This enables the real yield curve to be used as a somce of information on the market s view of expected future inflation. In the United Kingdom market, there are two factors that present problems for the estimation of the real term structure the first is the 8-month lag between the indexation uplift and the cash flow date, and the second is the fact that there are fewer index-linked bonds in issue, compared to the number of conventional bonds. The indexation lag means that in the absence of a measure of expected inflation, real bond yields are dependent to some extent on the assumed rate of future inflatiOTi. The second factor presents practical problems in curve estimation in December 1999 there were only 11 index-linked gilts in existence, and this is not sufficient for most models. Neither of these factors presents an insurmountable problem however, and it is stiU possible to estimate a real term structure. 

Heath, Jarrow, and Morton (HJM) derived one-factor and multifactor models for movements of the forward rates of interest. The models were complex enough to match the current observable term structure of forward rate and by equivalence the spot rates. Ritchken and Sankara-subramanian provide necessary and sufficient conditions for the HJM models with one source of error and two-state variables such that the ex post forward premium and the integrated variance factor are sufficient... 

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. 

Ren-Raw Chen and Louis Scott, Pricing Interest Rate Options in a Two-Factor Cox-Ingersoll-Ross Model of the Term Structure, The Review of Financial Studies 5, no. 4 (1992), pp. 613-636. 

Chen and Scott (1992) transformed the CIR model into a two-factor model specifying the interest rate as a function of two uncorrelated variables, both assumed to follow a stochastic process. The article demonstrated that this modification of the model has a number of advantages and useful applications. 

In the single-factor HJM model, forward rates of all maturities move in perfect correlation. For actual market applications— pricing an interest rate instrument that is dependent on the spread between two points on... 

An interesting question then arises as to why the dynamics of proton transfer for the benzophenone-i V, /V-dimethylaniline contact radical IP falls within the nonadiabatic regime while that for the napthol photoacids-carboxylic base pairs in water falls in the adiabatic regime given that both systems are intermolecular. For the benzophenone-A, A-dimethylaniline contact radical IP, the presumed structure of the complex is that of a 7t-stacked system that constrains the distance between the two heavy atoms involved in the proton transfer, C and O, to a distance of 3.3A (Scheme 2.10) [20]. Conversely, for the napthol photoacids-carboxylic base pairs no such constraints are imposed so that there can be close approach of the two heavy atoms. The distance associated with the crossover between nonadiabatic and adiabatic proton transfer has yet to be clearly defined and will be system specific. However, from model calculations, distances in excess of 2.5 A appear to lead to the realm of nonadiabatic proton transfer. Thus, a factor determining whether a bimolecular proton-transfer process falls within the adiabatic or nonadiabatic regimes lies in the rate expression Eq. (6) where 4>(R), the distribution function for molecular species with distance, and k(R), the rate constant as a function of distance, determine the mode of transfer. 

---