Term-structure modeling rate derivatives

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

We also postulated this simplified volatilty function in chapter (5), where we derived the mean-reverting short rate dynamics for a iV-factor term structure model. Nevertheless, the ODE can also be solved postulating a more general volatiUy function. 

The implications of this new model class are in contrast to most term structure models discussed in the literature, which assume that the bond markets are complete and fixed income derivatives are redundant securities. Collin-Dufresne and Goldstein [ 18] and Heiddari and Wu [36] show in an empirical work, using data of swap rates and caps/floors that there is evidence for one additional state variable that drives the volatility of the forward rates. Following from that empirical findings, they conclude that the bond market do not span all risks driving the term structure. This framework is rather similar to the affine models of equity derivatives, where the volatility of the underlying asset price dynamics is driven by a subordinated stochastic volatility process (see e.g. Heston [38], Stein and Stein [71] and Schobel and Zhu [69]). 

Chen R, Scott L (1995) Inters Rate Options in Multifactor Cox-lngersoll-Ross Models of the Term Structure. Journal of Derivatives 3 53-72. 

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. 

Klaus Sandmann and Dieter Sondermann, A Term Structure Model and the Pricing of Interest Rate Derivatives, Review of Futures Markets 12, no. 2 (1993), pp. 391 23. 

Equation (4.21) states that the dynamics of the forward-rate process, beginning with the initial rate/(0, J), are specified by the set of Brownian motion processes and the drift parameter. For practical applications, the evolution of the forward-rate term structure is usually derived in a binomial-type path-dependent process. Path-independent processes, however, have also been used, as has simulation modeling based on Monte Carlo techniques (see Jarrow (1996)). The HJM approach has become popular in the market, both for yield-curve modeling and for pricing derivative instruments, because it matches yield-curve maturities to different volatility levels realistically and is reasonably tractable when applied using the binomial-tree approach. 

Selecting the appropriate term-structure model is more of an art than a science, depending on the particular application involved and the users individual requirements. The Ho-Lee and BDT versions, for example, are arbitrage, or arbitrage-free, models, which means that they are designed to match the current term structure. With such models—assuming, of course, that they specify the evolution of the short rate correctly—the law of noarbitrage can be used to determine the price of interest rate derivatives. 

This chapter considers some of the techniques used to fit the model-derived term structure to the observed one. The Vasicek, Brennan-Schwartz, Cox-Ingersoll-Ross, and other models discussed in chapter 4 made various assumptions about the nature of the stochastic process that drives interest rates in defining the term structure. The zero-coupon curves derived by those models differ from those constructed from observed market rates or the spot rates implied by market yields. In general, market yield curves have more-variable shapes than those derived by term-structure models. The interest rate models described in chapter 4 must thus be calibrated to market yield curves. This is done in two ways either the model is calibrated to market instruments, such as money market products and interest rate swaps, which are used to construct a yield curve, or it is calibrated to a curve constructed from market-instrument rates. The latter approach may be implemented through a number of non-parametric methods. 

Beyond the cubic polynomial, there are two main approaches to fitting the term structure parametric and non-parametric curves. Parametric curves are based on term-structure models such as those discussed in chapter 4. As such, they need not be discussed here. Non-parametric curves, which are constructed employing spline-based methods, are not derived from any interest rate models. Instead, they are general approaches, described using sets of parameters. They are fitted using econometric principles rather than stochastic calculus, and are suitable for most purposes. 

A short-rate model can be used to derive a complete term structure. We can illustrate this by showing how the model can be used to price discount bonds of any maturity. The derivation is not shown here. Let P t, T) be the price of a risk-free zero-coupon bond at time t maturing at time T that has a maturity value of 1. This price is a random process, although we know that the price at time T will be 1. Assume that an investor holds this bond, which has been financed by borrowing funds of value C,. Therefore, at any time t the value of the short cash position must be C,= —P(t, T) otherwise, there would be an arbitrage position. The value of the short cash position is growing at a rate dictated by the short-term risk-free rate r, and this rate is given by... 

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 ... 

In this chapter we consider multi-factor and whole yield curve models. As we noted in the previous chapter, short-rate models have certain drawbacks, which, though not necessarily limiting their usefulness, do leave room for further development. The drawback is that as the single short-rate is used to derive the complete term structure, in practice, this can be unsuitable for the calculation of bond yields. When this happens, it becomes difficult to visualise the actual dynamics of the yield curve, and the model no longer fits observed changes in the curve. This means that the accuracy of the model cannot be observed. Another drawback is that in certain equilibrium model cases, the model cannot be fitted precisely to the observed yield curve, as they have constant parameters. In these cases, calibration of the model is on a goodness of fit or best fit approach. 

A no-arbitrage model that is implemented in a realistic approach matches precisely the term structure of interest rates that are implied by the current (or initial) observed market yields. It then derives a forward curve for the fumre that is dependent on the way it has modelled the dynamics of the interest-rate process. 

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. 

In Chapter 8, we described several models to measure the term structure of credit spread and we introduced the model proposed by Longstaff and Schwartz (1995) for pricing fixed-rate debt. The authors propose also a model to valuing floating-rate notes. The equation derived for pricing floating-rate bonds is given by (10.2) ... 

From oiu understanding of derivatives, we know that option pricing models such as Black-Scholes assume that asset price returns follow a lognormal distribution. The dynamics of interest rates and the term structure is the subject of... 

This chapter briefly describes the range of interest rate and bond options available on two of the major European derivative exchanges. It then moves on to establish a framework for option pricing by presenting some important theoretical models of the term structure of interest rates. The chapter then focuses on bond option pricing and discusses some of the main pricing theories, highlighting their assumptions and weaknesses. Some illustrative numerical examples are included in the text at appropriate junctures. 

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... 

The first two chapters of this section discuss bond pricing and yields, moving on to an explanation of such traditional interest rate risk measures as modified duration and convexity. Chapter 3 looks at spot and forward rates, the derivation of such rates from market yields, and the yield curve. Yield-curve analysis and the modeling of the term structure of interest rates are among the most heavily researched areas of financial economics. The treatment here has been kept as concise as possible. The References section at the end of the book directs interested readers to accessible and readable resources that provide more detail. 

Equilibrium interest rate models also exist. These make the same assumptions about the dynamics of the short rate as arbitrage models do, but they are not designed to match the current term structure. The prices of zero-coupon bonds derived using such models, therefore, do not match prices seen in the market. This means that the prices of bonds and interest rate derivatives are not given purely by the short-rate process. In brief, arbitrage models take as a given the current yield curve described by the... 

Because the derivatives isolate credit risk from other factors, such as client relationships and interest rate risk, they offer a formal mechanism for pricing credit issues only. A market in credit alone can thus evolve, allowing still more efficient pricing and the modeling of a term structure of credit rates. 

Thus OAS is a general stochastic model, with discount rates derived from the standard benchmark term structure of interest rates. This is an advantage over more traditional methods in which a single discount rate is used. The calculated spread is a spread over risk-free forward rates, accounting for both interest-rate uncertainty and the price of default risk. As with any methodol-ogy, OAS has both strengths and weaknesses however, it provides more realistic analysis than the traditional yield-to-maturity approach. Hence, it has been widely adopted by investots since its introduction in the late 1980s. 

PP model of micelle. This model generally gives a satisfactory fit of observed data in terms of residual errors (= kobs i - kcaicd where kobs i and i are, at the i-th independent reaction variables such as [D ], experimentally determined and calculated [in terms of micellar kinetic model] rate constants, respectively). The model also provides plausible values of kinetic parameters such as micellar binding constants of reactant molecules and rate constants for the reactions in the micellar pseudophase. The deviations of observed data points from reasonably good fit to a kinetic equation derived in terms of PP model for a specific bimo-lecular reaction under a specific reaction condition are generally understandable in view of the known limitations of the model. Such deviations provide indirect information regarding the fine, detailed structural features of micelles. 

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