Long-term factorization in Heath–Jarrow–Morton models

Abstract

The long-term factorization decomposes the stochastic discount factor (SDF) into discounting at the rate of return on the long bond and a martingale that defines a long-term forward measure. We establish sufficient conditions for existence of the long-term factorization in HJM models. A condition on the forward rate volatility ensures existence of the long bond volatility. This yields existence of the long bond and convergence of $T$-forward measures to the long forward measure. It contrasts with the familiar risk-neutral factorization that decomposes the SDF into discounting at the short rate and a martingale defining the risk-neutral measure.

Appendix: Proof of Theorem 2.4

The proof consists of two parts. We first prove that the processes on the right-hand side of (2.8) and (2.9) are well defined (the integrals in the exponential are well defined). Next we prove that

$$ \frac{{\mathbb {E}}^{\mathbb {P}}_{t}[S_{T}]}{{\mathbb {E}}^{\mathbb {P}}[S_{T}]} \stackrel{L^{1}}{\longrightarrow} M_{t}^{\infty} \qquad \text{as } T\rightarrow\infty $$

(A.1)

with $M_{t}^{\infty}$ defined by the right-hand side of (2.9). By Theorems 3.1 and 3.2 of Qin and Linetsky [34] and Proposition 2.1, the results in parts (i)–(iv) follow. The expression for $W^{\mathbb{Q}^{\infty}}$ then follows from Girsanov’s theorem (cf. Filipović [17, Theorem 2.3.3]), and the SDE for $(f_{t})$ under $\mathbb{Q}^{\infty}$ then follows immediately.

Since $M_{t}^{\infty}=S_{t}B_{t}^{\infty}$, we need to prove that the right-hand side of (2.8) is well defined. We first prove the following lemma which is central to all subsequent estimates.

Lemma A.1

For any function $h\in H_{\bar{w}}^{0}$, we have the estimate

$$\int_{T}^{\infty}|h(x)|\,dx\leq C(T)\|h\|_{\bar{w}},\qquad \textit{where } C(T)=K(T^{-\epsilon/2}\wedge1), $$

for some $K>0$ and $\epsilon>0$.

Proof

Since $\bar{w}(x)\geq1$, we can write for all $h\in H_{\bar{w}}^{0}$ that

$$\begin{aligned} |h(x)| &=\left|\int_{x}^{\infty}h'(s)\,ds\right|\leq\| h\|_{\bar{w}} \left (\int _{x}^{\infty}\frac{ds}{\bar{w}(s)}\right)^{1/2}\\ &\leq\|h\|_{\bar{w}}\left(\int_{x}^{\infty}K(s^{-(3+\epsilon)}\wedge 1)\,ds\right)^{1/2}\\ &\leq\|h\|_{\bar{w}} K(x^{-(1+\epsilon/2)}\wedge1), \end{aligned}$$

where the constant $K$ can change from line to line. Thus,

$$\int_{T}^{\infty}|h(x)|\,dx\leq\|h\|_{\bar{w}}K(T^{-\epsilon/2}\wedge1). $$

 □

Lemma A.1 ensures that each element of the vector $\sigma_{t}^{\infty}$ in (3.9) is well defined. The next lemma ensures that $\sigma_{t}^{\infty}\in{\ell^{2}}$ and the right-hand side of (2.8) is well defined.

Lemma A.2

$\int_{0}^{t}\|\sigma_{s}^{\infty}\|_{\ell^{2}}^{2}\,ds\leq C^{2}(0) tD_{2}^{2}$.

Proof

By Lemma A.1, $\int_{0}^{\infty}|\sigma_{s}^{j}(u)|\,du\leq C(0)\|\sigma_{s}^{j}\|_{\bar{w}}$. This implies

$$\begin{aligned} {\int_{0}^{t}\|\sigma_{s}^{\infty}\|_{\ell^{2}}^{2}\,ds} &\leq\int_{0}^{t}\sum_{j\in\mathbb{N}}\left(\int_{0}^{\infty}|\sigma_{s}^{j}(u)| \,du\right)^{2} \,ds\\ &\leq\int_{0}^{t}\sum_{j\in\mathbb{N}}C^{2}(0) \|\sigma_{s}^{j}\|_{\bar{w}}^{2} \,ds\\ &=C^{2}(0)\int_{0}^{t} \|\sigma_{s}\|^{2}_{L_{2}^{0}(H_{\bar{w}})}\,ds\\ &\leq C^{2}(0) tD_{2}^{2}, \end{aligned}$$

where $D_{2}$ is the volatility bound in (2.5). □

By the above lemma, the last integral in (2.8) is well defined. The stochastic integral $\int_{0}^{t} \sigma_{s}^{\infty}\cdot dW^{\mathbb{P}}_{s}$ is well defined due to Itô’s isometry. The first integral in (2.8) is bounded by

$$ \frac{1}{2}\int_{0}^{t} (\|\gamma_{s}\|_{\ell^{2}}^{2}+\|\sigma_{s}^{\infty}\| _{\ell ^{2}}^{2})\,ds\leq\frac{1}{2}\int_{0}^{t}\varGamma(s)^{2}\,ds+\frac{1}{2}C^{2}(0)tD_{2}^{2}, $$

which is well defined by the fact that $\varGamma\in L_{2}(\mathbb{R}_{+})$. Thus the right-hand side of (2.8) is well defined.

We now turn to the verification of (A.1). We first rewrite $P_{t}^{T}/P_{0}^{T}$ and $B_{t}^{\infty}$ defined by (2.8) in terms of the ℚ-Brownian motion $W^{\mathbb {Q}}$ as

$$\begin{aligned} \frac{P_{t}^{T}}{P_{0}^{T}}&=A_{t}\exp\bigg(- \int_{0}^{t} \sigma_{s}^{T}\cdot dW^{\mathbb {Q}}_{s}-\frac{1}{2} \int_{0}^{t} \|\sigma_{s}^{T}\|^{2}_{\ell^{2}}\,ds\bigg),\\ B_{t}^{\infty}&=A_{t}\exp\bigg(- \int_{0}^{t} \sigma_{s}^{\infty}\cdot dW^{\mathbb {Q}}_{s}-\frac{1}{2} \int_{0}^{t}\|\sigma_{s}^{\infty}\|_{\ell^{2}}^{2}\,ds\bigg). \end{aligned}$$

Fix $t\geq0$. We note that the condition (A.1) can be written under any locally equivalent probability measure ${\mathbb {Q}}^{V}$ associated with any valuation process $V$ as

$$ \lim_{T\rightarrow\infty}\mathbb{E}^{\mathbb{Q}^{V}}\bigg[\bigg|\frac {B_{t}^{T}}{V_{t}} - \frac{B_{t}^{\infty}}{V_{t}}\bigg|\bigg]=0. $$

We can use this freedom to choose the measure convenient for the setting at hand. Here we choose to verify it under ℚ, i.e., to show that

$$ \lim_{T\rightarrow\infty}\mathbb{E}^{\mathbb{Q}}\bigg[\bigg|\frac {P_{t}^{T}}{P_{0}^{T}A_{t}}- \frac{B_{t}^{\infty}}{A_{t}}\bigg|\bigg]=0. $$

(A.2)

We first introduce some notation. For $v\in[0,t]$ and $T\in[t,\infty]$, define

$$\begin{aligned} j_{v}^{T} &:= \int_{0}^{v} \sigma_{s}^{T}\cdot dW^{\mathbb{Q}}_{s},\\ k_{v}^{T} &:=\frac{1}{2} \int_{0}^{v} \|\sigma_{s}^{T}\|_{\ell^{2}}^{2}\,ds,\\ \bar{\sigma}_{v}^{T}&:=\sigma_{v}^{\infty}-\sigma_{v}^{T}=\int_{T-v}^{\infty}\sigma _{v}(u)\,du,\\ z_{v}^{T} &:=\frac{1}{2}\int_{0}^{v}\|\bar{\sigma}_{s}^{T}\|_{\ell^{2}}^{2}\,ds,\\ Y_{v}^{T} &:=e^{-(j_{v}^{T}-j_{v}^{\infty})-z_{v}^{T}}. \end{aligned}$$

For $p\geq1$ and a random variable $X$, we denote $\|X\|_{p}:=(\mathbb{E}^{\mathbb{Q}}[|X|^{p}])^{1/p}$, as long as the expectation is well defined. Then (A.2) can be rewritten as

$$ \lim_{T\rightarrow\infty} \|e^{-j_{t}^{T}-k_{t}^{T}}-e^{-j_{t}^{\infty}-k_{t}^{\infty}}\|_{1}=0. $$

By Hölder’s inequality,

$$ \limsup_{T\rightarrow\infty} \|e^{-j_{t}^{T}-k_{t}^{T}}-e^{-j_{t}^{\infty}-k_{t}^{\infty}}\|_{1} \leq\limsup_{T\rightarrow\infty}\|e^{-j_{t}^{\infty}-k_{t}^{\infty}}\|_{2}\| e^{-(j_{t}^{T}-j_{t}^{\infty})-(k_{t}^{T}-k_{t}^{\infty})}-1\|_{2}. $$

Lemma A.3 and A.4 below show that $\|e^{-j_{t}^{\infty}-k_{t}^{\infty}}\|_{2}$ is finite and

$$\lim_{T\rightarrow\infty} \|e^{-(j_{t}^{T}-j_{t}^{\infty})-(k_{t}^{T}-k_{t}^{\infty})}-1\|_{2}=0, $$

respectively.

Lemma A.3

For each $t>0$, there exists $C$ such that

$$ \sup_{v\leq t}\|Y_{v}^{T}\|_{2}\leq C < \infty \qquad \textit{and}\qquad \| e^{-j_{t}^{\infty}-k_{t}^{\infty}}\|_{2}\leq C< \infty. $$

Proof

We begin by considering the process $(Y_{v}^{T})^{2}=e^{-(2j_{v}^{T}-2j_{v}^{\infty})-4z_{v}^{T}+2z_{v}^{T}}$ for $t\in [0,T]$. By Itô’s formula, $(e^{-(2j_{v}^{T}-2j_{v}^{\infty})-4z_{v}^{T}})$ is a local martingale. Since it is also positive, it is a supermartingale (in fact, it is a true martingale due to Lemma A.2 and Novikov’s criterion). Therefore for all $v\leq t$,

$$ \mathbb{E}^{\mathbb{Q}}[e^{-(2j_{v}^{T}-2j_{v}^{\infty})-4z_{v}^{T}}]\leq1. $$

Similarly to Lemma A.2, $|z_{v}^{T}|\leq\frac {1}{2}C^{2}(T-v)vD_{2}^{2}$. Thus

$$\|Y_{v}^{T}\|_{2}^{2}=\mathbb{E}^{\mathbb{Q}}[e^{-(2j_{v}^{T}-2j_{v}^{\infty})-4z_{v}^{T}+2z_{v}^{T}}]\leq e^{C^{2}(0)vD_{2}^{2}}. $$

This implies

$$ \sup_{v\leq t} \|Y_{v}^{T}\|_{2}\leq e^{\frac{1}{2}C^{2}(0)tD_{2}^{2}}. $$

Similarly, $(e^{-j_{t}^{\infty}-k_{t}^{\infty}})^{2}=e^{-2j_{t}^{\infty}-4k_{t}^{\infty}+2k_{t}^{\infty}}$. The process $(e^{-2j_{t}^{\infty}-4k_{t}^{\infty}})$ is a supermartingale, and $k_{t}^{\infty}\leq\frac{1}{2}C^{2}(0)tD_{2}^{2}$ by Lemma A.2. Thus,

$$\|e^{-j_{t}^{\infty}-k_{t}^{\infty}}\|_{2}\leq e^{C^{2}(0)tD_{2}^{2}}. $$

 □

Lemma A.4

We have

$$ \lim_{T\rightarrow\infty} \|e^{-(j_{t}^{T}-j_{t}^{\infty})-(k_{t}^{T}-k_{t}^{\infty})}-1\|_{2}=0. $$

(A.3)

Proof

We need the following two intermediate lemmas. □

Lemma A.5

For $T\geq t$, $\sup_{v\leq t}|k_{v}^{T}-k_{v}^{\infty}|\leq C(0)C(T-t)tD_{2}^{2}$.

Proof

We estimate

$$\begin{aligned} {\sup_{v\leq t}|k_{v}^{T}{-}k_{v}^{\infty}|} &={\sup_{v\leq t}\bigg|\frac {1}{2}\sum_{j\in\mathbb{N}}\int_{0}^{v} \bigg(\Big(\int_{0}^{T-s} {+}\int _{0}^{\infty}\Big) \sigma_{s}^{j}(u)\,du\bigg)\bigg(\int_{T-s}^{\infty} \sigma _{s}^{j}(u)\,du\bigg)\,ds\bigg|}\\ &\leq{\sum_{j\in\mathbb{N}}\int_{0}^{t} \left(\int_{0}^{\infty}|\sigma _{s}^{j}(u)|\,du\right)\left(\int_{T-s}^{\infty}|\sigma_{s}^{j}(u)|\,du\right) \,ds}\\ &\leq{\sum_{j\in\mathbb{N}} \int_{0}^{t} C(0) \|\sigma_{s}^{j}\|_{\bar{w}} C(T-s)\|\sigma_{s}^{j}\|_{\bar{w}} \,ds}\\ &=C(0)C(T-t)\int_{0}^{t}\sum_{j\in\mathbb{N}} \|\sigma_{s}^{j}\| _{\bar {w}}^{2} \,ds\\ &={C(0)C(T-t)\int_{0}^{t} \|\sigma_{s}\|_{L_{2}^{0}(H_{\bar{w}})}^{2} \,ds}\\ &\leq C(0)C(T-t)t D_{2}^{2}. \end{aligned}$$

 □

Lemma A.6

We have

$$ \lim_{T\rightarrow\infty} \|Y_{t}^{T}-1\|_{2}=0. $$

(A.4)

Proof

By Itô’s formula,

$$ Y_{t}^{T}=1+ \int_{0}^{t} Y_{v}^{T}\bar{\sigma}_{v}^{T}\cdot dW^{\mathbb{Q}}_{v}. $$

By Itô’s isometry, we have

$$ \|Y_{t}^{T}-1\|_{2}^{2}=\mathbb{E}^{\mathbb{Q}}\bigg[\int_{0}^{t} \|Y_{v}^{T} \bar {\sigma}_{v}^{T}\|_{\ell^{2}}^{2} \,dv\bigg]. $$

By Lemma A.1, $|\bar{\sigma}_{v}^{T,j}|\leq C(T-v)\| \sigma_{v}^{j}\|_{\bar{w}}$. Thus, using Lemma A.3 for the last inequality, we get

$$\begin{aligned} {\|Y_{t}^{T}-1\|_{2}^{2}} & \leq\mathbb{E}^{\mathbb{Q}}\bigg[\sum_{j\in \mathbb {N}}\int_{0}^{t} |Y_{v}^{T}|^{2} C^{2}(T-v)\|\sigma_{v}^{j}\|^{2}_{\bar{w}} \,dv\bigg]\\ & \leq C^{2}(T-t)\mathbb{E}^{\mathbb{Q}}\bigg[\int_{0}^{t} |Y_{v}^{T}|^{2} \sum _{j\in\mathbb{N}}\|\sigma_{v}^{j}\|^{2}_{\bar{w}} \,dv\bigg]\\ &=C^{2}(T-t)\mathbb{E}^{\mathbb{Q}}\bigg[\int_{0}^{t} |Y_{v}^{T}|^{2} \|\sigma _{v}\| ^{2}_{L^{0}_{2}(H_{\bar{w}})} \,dv\bigg]\\ &\leq C^{2}(T-t)\mathbb{E}^{\mathbb{Q}}\bigg[\int_{0}^{t} |Y_{v}^{T}|^{2} D_{2}^{2}\,dv\bigg]\\ &\leq C^{2}(T-t)D_{2}^{2}\int_{0}^{t} \mathbb{E}^{\mathbb{Q}}[|Y_{v}^{T}|^{2}] \,dv\\ &\leq C^{2}(T-t)D_{2}^{2}\int_{0}^{t} C^{2} \,dv \\ &=C^{2}(T-t)D_{2}^{2}C^{2}t. \end{aligned}$$

Since $\lim_{T\rightarrow\infty}C(T-t)=0$, (A.4) is verified. □

Now we return to the proof of Lemma A.4. We have

$$\begin{aligned} \|e^{-(j_{t}^{T}-j_{t}^{\infty})-(k_{t}^{T}-k_{t}^{\infty})}-1\|_{2} & =\|Y_{t}^{T} e^{z_{t}^{T}-(k_{t}^{T}-k_{t}^{\infty})}-1\|_{2}\\ & \leq\|(Y_{t}^{T}-1)e^{z_{t}^{T}-(k_{t}^{T}-k_{t}^{\infty})}\|_{2}+\| e^{z_{t}^{T}-(k_{t}^{T}-k_{t}^{\infty})}-1\|_{2}. \end{aligned}$$

Recall that by Lemma A.5, $|k_{t}^{T}-k_{t}^{\infty}|\leq C(0)C(T-t)tD_{2}^{2}$. Using the same approach as in the proof of Lemma A.2, we can show that

$$|z_{t}^{T}|\leq\frac{1}{2}C^{2}(T-t)tD_{2}^{2}. $$

Thus we have

$$\begin{aligned} \|e^{-(j_{t}^{T}-j_{t}^{\infty})-(k_{t}^{T}-k_{t}^{\infty})}-1\|_{2} & \leq\|Y_{t}^{T}-1\|_{2} e^{\frac{1}{2}C^{2}(T-t)tD_{2}^{2}+C(0)C(T-t)tD_{2}^{2}}\\ &\quad{}+e^{\frac{1}{2}C^{2}(T-t)tD_{2}^{2}+C(0)C(T-t)tD_{2}^{2}}-1. \end{aligned}$$

Finally, (A.3) is verified by using Lemma A.6 and $\lim_{T\rightarrow\infty} C(T-t)=0$. □