Abstract
The state price deflators introduced in the previous chapter have several weaknesses such as they do not allow for sufficient modeling flexibility in practice and as they do not provide convincing fits to real data. In the present chapter we introduce the Heath–Jarrow–Morton (HJM) framework which is much more flexible and allows for potentially infinite dimensional term structure curves. Crucial in the HJM framework is the no-arbitrage condition which leads to the so-called HJM term that is analyzed in this chapter. We give explicit examples for the HJM framework in terms of the Cairns forward rate model and of the Teichmann–Wüthrich yield curve prediction model.
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Appendix: Proofs of Chap. 4
Appendix: Proofs of Chap. 4
In this section we prove the statements of Chap. 4. We start with the proof of Theorem 4.1. In a first step we re-express the ZCB price process in terms of the spot rate process .
Lemma 4.13
In the discrete time HJM framework (4.1), we have under the equivalent martingale measure \(\mathbb{P}^{\ast}\) for the ZCB price process
for 0≤t≤m.
Proof
The proof is analogous to the continuous time case (see, e.g., Filipović [67], Lemma 7.1.1). The goal essentially is to rearrange the terms such that forward rates are expressed in terms of spot rates. We calculate the logarithm of the ZCB price for 0<t≤m
Next we use the definition of v and we rewrite the last term (using (4.3))
This proves the lemma. □
Proof of Theorem 4.1
For the bank account numeraire discounted price process Lemma 4.13 provides
Consistency property (4.4) w.r.t. \(\mathbb {P}^{\ast}\) implies that this discounted price process needs to be a \((\mathbb {P}^{\ast},\mathbb{F} )\)-martingale, that is, we require
In view of the above expression for \(B_{t}^{-1}P(t,m)\) this gives under the finiteness assumption in (4.6) the requirement
This means that \(\sum_{u=t+1}^{m}\alpha(t,u)=h(t,m)\) for 0<t<m. For 0<t=m−1 we therefore obtain
and for 0<t<m−1
Note that h(t,t)=0, proving the claim. □
Next we prove Theorem 4.5. We start with a preliminary lemma.
Lemma 4.14
We have
Proof
Rearranging the terms provides
This proves the claim. □
Proof of Theorem 4.5
First we rewrite the spot rate. We have because of (4.13) and Lemma 4.14 that
Since the price process of the ZCB needs to be consistent w.r.t. \(\mathbb{P}^{\ast}\), see (4.4), we obtain from Corollary 2.19
The last expected value can be calculated explicitly, providing for an appropriate constant k 3
So there remains the calculations of the constants k 1,k 2,k 3. We have, see Theorem 4.2,
where in the last equality we define g(u,s)=α(u,s+1) to be the term under the summation. From this we can calculate k 2 similar to Lemma 4.14 and obtain
Moreover, we have
This implies that
For k 3, using the independence and multivariate Gaussian distribution of \(\boldsymbol{\varepsilon }_{u}^{\ast}\) (with independent components), we obtain
This implies
This proves the theorem. □
Proof of Theorem 4.8
\(\mathbb{P}^{\ast}\)-consistency implies
Solving this requirement and using the Gaussian properties of \(\boldsymbol{\varepsilon}^{\ast}_{t}\) under \(\mathbb{P}^{\ast}\) proves the claim. □
Proof of Theorem 4.12
In the first step we apply the tower property for conditional expectation which decouples the problem into several steps. We have . Thus, we need to calculate the inner conditional expectation of the d×d matrix S (K)(y). We define the auxiliary matrix
This implies that we can rewrite \(C_{(K)}= K^{-1/2}~\widetilde{C}_{(K)}\). Moreover, we rewrite the matrix \(\widetilde{C}_{(K)}\) as follows
where \(\widetilde{C}_{(K-1)}\in\mathbb{R}^{d\times(K-1)}\) is -measurable. This gives the following decomposition
This implies for the conditional expectation of S (K)(y)
To calculate the conditional expectation in the last term, we start with the conditional covariance. From Corollary 4.10 we obtain
This implies
Iterating this provides the result. □
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Wüthrich, M.V., Merz, M. (2013). Stochastic Forward Rate and Yield Curve Modeling. In: Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31392-9_4
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