Stochastic Forward Rate and Yield Curve Modeling

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.

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

$$ \log P(t,m) =\log P(0,m)+\sum_{s=1}^{t} \Biggl[r_{s-1}+\sum_{u=s+1}^{m}- \alpha(s,u) -\mathbf{v}(s,m)~ \boldsymbol{\varepsilon}^\ast_{s} \Biggr], $$

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

$$ B_t^{-1}{P(t,m)} =P(0,m)~\exp \Biggl\{\sum _{s=1}^{t} \Biggl[\sum_{u=s+1}^{m}- \alpha(s,u) -\mathbf{v}(s,m) \boldsymbol{\varepsilon}^\ast_{s} \Biggr] \Biggr\}. $$

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

$$ \alpha(m-1,m)=h(m-1,m), $$

and for 0<t<m−1

$$ \alpha(t,m)= \sum_{u=t+1}^{m} \alpha(t,u) - \sum_{u=t+1}^{m-1} \alpha(t,u)=h(t,m)-h(t,m-1). $$

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

$$ \sum_{s=t}^{m-1} \sum _{u=1}^s \boldsymbol{\sigma}(u,s+1) \boldsymbol{ \varepsilon}_u^\ast = \sum_{u=1}^t \bigl[\mathbf{v}(u,m)-\mathbf{v}(u,t) \bigr] \boldsymbol{\varepsilon}_u^\ast+ \sum_{u=t+1}^{m-1} \mathbf{v}(u,m) \boldsymbol{ \varepsilon}_u^\ast. $$

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

$$ P(t,m) =\frac{P(0,m)}{P(0,t)} ~ \exp \Biggl\{-k_3- \sum _{u=1}^t \bigl[\mathbf{v}(u,m)-\mathbf{v}(u,t) \bigr] \boldsymbol{\varepsilon}_u^\ast \Biggr\}. $$

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

$$ k_2= \frac{1}{2}~ \sum_{u=1}^{t} \bigl[ \bigl \Vert \mathbf{v}(u,m)\bigr \Vert ^2 -\bigl \Vert \mathbf{v}(u,t)\bigr \Vert ^2 \bigr] + \frac{1}{2}~\sum _{u=t+1}^{m-1} \bigl \Vert \mathbf{v}(u,m)\bigr \Vert ^2. $$

For k ₃, using the independence and multivariate Gaussian distribution of $\boldsymbol{\varepsilon }_{u}^{\ast}$ (with independent components), we obtain

This implies

$$ k_3=k_2-\frac{1}{2}~ \sum _{u=t+1}^{m-1} \bigl \Vert \mathbf{v}(u,m)\bigr \Vert ^2 =\frac{1}{2}~ \sum_{u=1}^{t} \bigl[ \bigl \Vert \mathbf{v}(u,m)\bigr \Vert ^2 -\bigl \Vert \mathbf{v}(u,t)\bigr \Vert ^2 \bigr]. $$

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

$$ \widetilde{C}_{(K)} = \bigl( \bigl[ \varsigma(\mathbf{R}_{\delta k,-})^{-1}~ \boldsymbol{\varUpsilon}_{\delta k} \bigr]_j \bigr)_{j=1,\ldots, d; ~k=1, \ldots, K}\in\mathbb{R}^{d\times K}. $$

This implies that we can rewrite $C_{(K)}= K^{-1/2}~\widetilde{C}_{(K)}$. Moreover, we rewrite the matrix $\widetilde{C}_{(K)}$ as follows

$$ \widetilde{C}_{(K)}= \bigl[\widetilde{C}_{(K-1)}, \varsigma( \mathbf{R}_{\delta K,-})^{-1}~ \boldsymbol{\varUpsilon}_{\delta K} \bigr], $$

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. □