Forward rate structure

APPENDIX ILLUSTRATION OF FORWARD RATE STRUCTURE WHEN SPOT RATE STRUCTURE IS INCREASING... 

Estimated rate constants for the various electron-transfer steps, together with approximate reduction potentials, are displayed in Figure 6.39. For each step, the forward rate is orders of magnitude faster than the reverse reaction. The rapid rates suggest that attempts to obtain x-ray structures of intermediates (especially the early ones ) will not be successful. However, molecular dynamics methods are being explored in computer simulations of the structures of various intermediates. Within a few years we may begin to understand why the initial steps are so fast. 

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). 

The first generation of term structure models started with a finite factor modeling of the process dynamics with constant coefficients (e.g. Vasicek [73], Brennan and Schwartz [10], Cox, Ingersoll, and Ross [22]). Due to the fact that this type of models are inconsistent with the current term structure, the second generation of models exhibits time dependent coefficients (e.g. Hull and White [41]). A completely different approaeh starts from the direct modeling of the forward rate dynamies, by using the initial term strueture as an input (e.g. Ho, and Lee [39], Heath, Jarrow, and Morton [35]). 

From empirical investigations we know that the correlation should converge to unity as the difference in its maturities approaches zero. One the other hand, the correlation should vanish as the difference in the maturities goes to infinity. Another empirical implication is the relative smoothness of the observed forward rate curved Hence, we are able to separate the class of RF models according to the existence or absence of this smoothness property. Obviously, the non-differentiable class leads to non-smoothed forward rate curves, whereas the T-differentiable Random Fields enforces smoothed yield curves. Even if we restrict the number of admissible RF models to the non-differentiable Field dZ t,T) and the r-differentiable counterpart dU we obtain a new degree of freedom to improve the possible fluctuations of the entire term structure. 

This non-differential process has been used by Kennedy [51] and Goldstein [33] to model the dynamics of the term structure of forward rates. 

Now, assuming (hat the term structure of forward rates is driven by the T -differential RF dU(t, T) impUes that the correlation function is determined by... 

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]). 

Glasserman P, Kou S (2003) The Term Structure of Simple Forward Rates with Jump Risk. Mathematical Finance 13 383-410. 

This is convenient because this means that the price at time t of a zero-coupon bond maturing at T is given by Equation (3.7), and forward rates can be calculated from the current term structure or vice versa. 

The relationship described by Equation (3.12) states that the spot rate is given by the arithmetic average of the forward rates/(t, s) where tdiffer from the relationship in a discrete time environment We know that the spot rate in such a framework is the geometric average of the forward rates, and this is the key difference in introducing the continuous time structure. Equation (3.12) can be rearranged to... 

What Equation (3.14) implies is that if the spot rate increases, then by definition the forward rate (or marginal rate as has been suggested that it may be called ) will be greater. From Equation (3.14), we deduce that the forward rate will be equal to the spot rate plus a value that is the product of the rate of increase of the spot rate and the time period (T-1). In fact, the conclusions simply confirm that the forward rate for any period will lie above the spot rate if the spot rate term structure is increasing, and will lie below the spot rate if it is decreasing. In a constant spot rate environment, the forward rate will be equal to the spot rate. 

What is the importance of this result for our understanding of the term structure of interest rates First, we see (again, but this time in continuous time) that spot rates, forward rates and the discount function are all closely related, and... 

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 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. 

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. 

In seeking to develop a model for the entire term structure, the requirement is to model the behaviour of the entire forward yield curve, that is, the behaviour of the forward short-rate/(f, T) for all forward dates T. Therefore, we require the random process f(T) for all forward dates T. Given this, it can be shown that the yield R on a T-maturity zero-coupon bond at time t is the average of the forward rates at that time on all the forward dates s between t and T, given by Equation (4.1) ... 

In the HIM model, the processes for the bond price and the spot rate are not independent of each other. As an arbitrage-free pricing model, it differs in crucial respects from the equilibrium models presented in the previous chapter. The core of the HIM model is that given a current forward rate curve, and a function capturing the dynamics of the forward rate process, it models the entire term structure. 

Ritchken, P., Sankarasubramanian, L., 1995. Volatility structures of forward rates and the dynamics of the term structure. Math. Financ. 5, 55-72. 

Remember of course that the forward rate is derived from the current spot rate term stracture, and therefore although it is an expectation based on all currently known information, it is not a prediction of the term structure in the future. Nevertheless the fra-ward rate is important because it enables market makers to price and hedge financial instruments, most especially contracts with a forward starting date. 

Implied forward rates indicate the expected short-term (one-period) future interest rate for a specific point along the term structure they reflect the spread on the marginal rate of return that the market requires if it is investing in debt instruments of longer and longer maturities. 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

---