Pricing forward rates

We overcome this inconsistency, by deriving a unified framework that directly leads to consistent cap/floor and swaption prices. Thus, in general we start from a HJM-like framework. This framework includes the traditional HIM model as well as an extended approach, where the forward rates are driven by multiple Random Fields. Furthermore, even in the case of a multifactor unspanned stochastic volatility (USV) model we are able to compute the bond option prices very accurately. First, we make an exponential affine guess for the solution of an expectation, which is comparable to the solu-... 

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). 

In recent works Collin-Dufresne and Goldstein [18], Heiddari and Wu [36], Jarrow, Li, and Zhao [45] and Li, Zhao [54] have extended the HJM approach to a framework, where either the volatility of forward rates, or the volatility of bond prices is driven by a subordinated stochastic process. One major implication of these new type of models is an additional source of uncertainty driving the volatility. This implies the existence of an additional market price of risk. Intuitively, this market price of risk cannot be hedged only by bonds. As a result of this, we have a new class of models causing incomplete bond markets ... 

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]). 

Shirakawa H (1991) Interest Rate Option Pricing with Poisson-Ganssian Forward Rate Cnrve Processes. Mathematical Finance 1 77-94. 

There is still a consistency problem if we want to price interest rate derivatives on zero bonds, like caplets or floorlets, and on swaps, like swaptions, at the same time within one model. The popular market models concentrate either on the valuation of caps and floors or on swaptions, respectively. Musiela and Rutkowski (2005) put it this way We conclude that lognormal market models of forward LIBORs and forward swap rates are inherently inconsistent with each other. A challenging practical question of the choice of a benchmark model for simultaneous pricing and hedging of LIBOR and swap derivatives thus arises. ... 

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. 

Bond Prices as a Function of Spot and Forward Rates... 

In this section, we describe the relationship between the price of a zero-coupon bond and spot and forward rates. We assume a risk-free zero-coupon bond of nominal value 1, priced at time t and maturing at time T. We also assume a money market bank account of initial value P t, T) invested at time t. The money market account is denoted M. The price of the bond at time t is denoted P t, T) and if today is time 0 (so that t > 0), then the bmid price today is unknown and a random factor (similar to a future interest rate). The bond price can be related to the spot rate or forward rate that is in force at time t. 

Following the relationship between spot and forward rates, it is also possible to describe the bond price in terms of forward rates. We show the result here only. First we know that... 

Expression (3.20) states that the bond price is a function of the range of forward rates that apply for all/(r, s) that is, the forward rates for aU time periods s from t to T (where tforward rate/(f, s) that results for each s arises as a result of a random or stochastic process that is assumed to start today at time 0. Therefore, the bond price Pit, T) also results from a random process, in this case all the random processes for all the forward rates/(t, i). 

Equation (3.22) describes the bond price as a function of the spot rate only, as opposed to the multiple processes that apply for aU the forward rates from t to T. As the bond has a nominal value of 1, the value given by Equation (3.22) is the discount factor for that term the range of zero-coupon bond prices would give us the discount function. 

For derived forward rates, the bond price functirm P t, T) is continuously differentiable with respect to t. Therefore, the model produces the following for the instantaneous forward rates ... 

The forward rate is a function of the short-rate and is normally distributed. Figure 3.4 shows the forward rate curves that correspond to the price curves in Figure 3.3, under the same parameters. 

The price fiinction above can be continuously differentiated as a function of t. The forward rate is given by Equation (3.54) ... 

The different models can lend themselves to a particular calibration method. In the Ho-Lee model, rally parallel yield curve shifts are captured and the current yield curve is a direct input therefore, a constant volatility parameter is used. This implies that all the forward rate implied volatilities are identical. In practice, this is not necessarily realistic, as long-dated bond prices often experience lower volatility than short-dated bond prices. The model also assumes... 

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. 

The models are based on the fact that the price of a bond, which exhibits a pull-to-par effect, and the forward rate, are both Ito processes. For the bond... 

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 default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

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. 

The relationship between forward rates and the price of a zero-coupon bond at time t is given by Equation (4.34) ... 

In selecting the model, a practitioner will select the market variables that are incorporated in the model these can be directly observed such as zero-coupon rates or forward rates, or swap rates, or they can be indeterminate such as the mean of the short rate. The practitioner will then decide the dynamics of these market or state variables, so, for example, the short rate may be assumed to be mean reverting. Finally, the model must be calibrated to market prices so, the model parameter values input must be those that produce market prices as accurately as possible. There are a number of ways that parameters can be estimated the most common techniques of calibrating time series data such as interest rate data are general method of moments and the maximum likelihood method. For information on these estimation methods, refer to the bibliography. 

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. 

Central banks and market practitioners use interest rates prevailing in the government bond market to extract certain information, the most important of which is implied forward rates. These are an estimate of the market s expectations about the future directirMi of short-term interest rates. They are important because they signify the market s expectafirMis about the future path of interest rates however, they are also used in derivative pricing and to create synthetic bond prices from the extent of credit spreads of corporate bonds. 

Forward rates may be calculated using the discount function or spot interest rates. If spot interest rates are known, then the bond price equation can be set as ... 

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. 

Expression (7.1) states that the price of a zero-coupon bond is equal to the discount factor from time t to its maturity date or the average of the discount factors under all interest-rate scenarios, weighted by their probabilities. It can be shown that the T-maturity forward rate at time t is given by... 

From Figure 7.3, we see fliat to price a very Iraig-dated bond off the yield of the 30-year government bond would lead to errors. The unbiased expectations hypothesis suggests that 100-year bond yields are essentially identical to 30-year yields however, this is in fact incorrect. The theoretical 100-year yield in fact will be approximately 20-25 basis points lower. This reflects the convexity bias in longer dated yields. In our illustration, we used a hypothetical scenario where only three possible interest-rate states were permitted. Dybvig and Marshall showed that in a more realistic environment, with forward rates calculated using a Monte Carlo simulation, similar observations would result. Therefore, the observations have a practical relevance. 

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