Term-Structure Model

Vol. 565 W. Lemke, Term Structure Modeling and Estimation in a State Space Framework. IX, 224 pages. 2006. Vol. 566 M. Genser, A Structural Framework for the Pricing of Corporate Securities. XIX, 176 pages. 2006. 

Thus, by applying the lEE algorithm we are able to compute the price of an option on a eoupon bearing bond, even if the term structure model is determined by a multi-faetor FIJM-model with USV (see chapter (7)). 

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. 

Multiple-Random Fields term structure models... 

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]). 

As proxy for the T-differential type of term structure models we postulate the following deterministic correlation function... 

In the following, we compute the price of bond options assuming these two types of Random Fields as correlated sources of uncertainty, while dZ(t T) leads to anon-differential and dU (t, T) to a T-differential type of term structure model. Note that the computation of the particular option price differs only in the proposed type of correlation function. 

This has been shown e.g. by Santa-Clara and Sornette [67] assuming a single-field term structure model. 

Unlike the traditional type of HIM term structure models, where the drift is completely determined by the volatility function we now have to specify an additional correlation function. Hence, the drift p t,T) is completely determined by the volatility function (t,T), together with the correlation structure U). 

Similar to section (5.3.4), where we postulated a traditional multi-factor HJM-model, there exists no closed-form solution for the price of a coupon bond option assuming a multiple field term structure model. In the following, we show that the moments of the random variable V(7o, 7i ) can be computed in closed-form, even if the underlying random variable is driven by N admissible Random Fields. 

The option price based on a non-differentiable RF term structure model can be easily derived by plugging the new correlation function... 

As aforementioned, it can be shown that the expectation Yi zm ) is determined by an exponential affine form fulfilling the martingale condition. Now, postulating a multiple-Field term structure model the bond price dynamics are given by... 

Note that the lEE approach again can be easily extended to multi-factor or multiple-Field term structure models. This property is directly linked to the fact that the moments of the underlying random variable can be computed in closed-form, regardless of the dimensionality of the underlying model structure. 

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]). 

Again, we start with a one-factor term structure model with USV and show that the exponential affine guess... 

Like Collin-Dufresne and Goldstein [18], we started from a HJM-like term structure model, where the stochastic volatility is driven by an addi-... 

Eberlein E, Kluge W (2006) Exact Pricing Formulae for Caps and Swaptions in a Levy Term Structure Model. Journal of Computational Finance 9 1-27. 

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

The traditional one-, two- and multi-factor equilibrium models, known as ajfine term structure models (see James and Webber, 2000 or Duffie, 1996, p. 136). These include Gaussian affine models such as Vasicek, Hull-White and Steeley, where the model describes a process with constant volatility and models that have a square-root volatility such as Cox-Ingersoll-Ross (CIR) ... 

A function SH — SH is affine if there are constants a and b such that for all values of x, H x) = a+bx. This describes certain term structure models drift and diffusion functions. 

Term structure models are essentially models of the interest-rate process. The problem being posed is, what behaviour is exhibited by interest rates, and by the short-term interest rate in particular An excellent description of the three most common processes that are used to describe the dynamics of the short-rate is given in Phoa (1998), who describes ... 

Therefore, once we have a full description of the random behaviour of the short-rate r, we can calculate the price and yield of any zero-coupon bond at any time, by calculating this expected value. The implication is clear specifying the process r t) determines the behaviour of the entire term structure so, if we wish to build a term structure model, we need only (under these assumptions) specify the process for r t). 

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