Liquidity risk and the term structure of interest rates

Abstract

This paper develops an arbitrage-free pricing theory for a term structure of fixed income securities that incorporates liquidity risk. In our model, there is a quantity impact on the term structure of zero-coupon bond prices from the trading of any single zero-coupon bond. We derive a set of conditions under which the term structure evolution is arbitrage-free. These no arbitrage conditions constrain both the risk premia and the term structure’s volatility. In addition, we also provide conditions under which the market is complete, and we show that the replication cost of an interest rate derivative is the solution to a backward stochastic differential equation.

Appendix

Lemma 1

Under (C1), there exist $\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^{d})$-measurable functions $S^{-1}$ and $\varPhi '$ that satisfy

$$\begin{aligned} \varPhi '_{t}(z;\omega )=\varPsi _{t}(\mathbf {x};\omega ) \text{ and } S^{-1}(t,z,\omega )=\mathbf {x} \text{ when } z=S(t,\mathbf {x};\omega ). \end{aligned}$$

Proof

By (C1), there exists an inverse of $S$. By the Kuratowski-Ryll-Nardzewski Theorem, the inverse $S^{-1}$ and the function $\varPhi '$ can be chosen to be $\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^{d})$-measurable if it can be shown that the set-valued function

$$\begin{aligned} F(t,z;\omega )\mapsto \{(p,\mathbf {x})\in \mathbb {R} \times \mathbb {R}^{n}:S(t,\mathbf {x};\omega )=z \text{ and } p=\varPsi _{t}(\mathbf {x};\omega )\} \end{aligned}$$

is closed-valued and strongly measurable. For fixed $t,\omega$, the fact that $S(t,\cdot ;\omega )$ and $\varPsi _{t}(\cdot ;\omega )$ are continuous clearly implies that $F(t,z,\omega )$ is closed, and non-empty due to (C1). Furthermore, if $K\subset \mathbb {R}\times \mathbb {R}^{n}$ is a compact set and $K_{0}$ is a countable dense subset of $K$, the set

$$\begin{aligned} \{(t,z,\omega ):K\cap F(t,z,\omega )\ne \emptyset \}&= \{(t,z,\omega ):\exists (p,\mathbf {x})\in K,S(t,\mathbf {x};\omega )=z \text{ and } p=\varPsi _{t}(\mathbf {x};\omega )\}\\&= \bigcap _{n\ge 1}\bigcup _{(p,\mathbf {x})\in K_{0}}\left\{ (t,z,\omega ):|\varPsi _{t}(\mathbf {x};\omega )-p|+| S(t,\mathbf {x};\omega )-z|<\frac{1}{n}\right\} \end{aligned}$$

is measurable. By extension, if $C$ is a closed set, then $C=\bigcup _{n\ge 1}C\cap \overline{B}_{n}$, with $\overline{B}_{n}$ the closed ball of radius $n$ around $0$ in $\mathbb {R}\times \mathbb {R}^{n}$. Since $C\cap \overline{B}_{n}$ is compact for all $n$,

$$\begin{aligned} \{(t,z,\omega ):C\cap F(t,z;\omega )\ne \emptyset \}&= \bigcup _{n\ge 1}\{(t,z,\omega ):\exists (p,\mathbf {x})\in (C\cap \overline{B}_{n})\cap F(t,z;\omega )\} \end{aligned}$$

is also measurable, which by definition implies that $F$ is strongly measurable.$\square$

Lemma 2

Under (C1) and (C2’), $\widetilde{\varPhi }$ is $\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^{d})$-measurable and, for all $\epsilon >0$, there exists a $\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^{d})$-measurable function $\mathcal {X}:\mathbb {R}_{+}\times \varOmega \times \mathbb {R}^{d}\rightarrow \mathbb {R}^{n}$ such that

$$\begin{aligned} S(t,\mathcal {X}(t,z;\omega );\omega )=z \text{ and } \varPsi _{t}(\mathcal {X}(t,z;\omega );\omega )\le \widetilde{\varPhi }_{t}(z;\omega )+\epsilon \end{aligned}$$

Proof

We first show that $\widetilde{\varPhi }$ is measurable. For this, it suffices to prove that

$$\begin{aligned} \widetilde{\varPhi }_{t}(z;\omega )=\lim _{n\ge 1} \inf _{\mathbf {x}\in \mathbb {Q}^{n}} \left\{ \varPsi _{t}(\mathbf {x};\omega ):|S(t,\mathbf {x};\omega )- z|\le \frac{1}{n}\right\} . \end{aligned}$$

(35)

Indeed, the fact that $\varPsi (\mathbf {x};\cdot )$ and $S(\mathbf {x})$ are measurable for all $\mathbf {x}$ would then imply that $\widetilde{\varPhi }$ is measurable. To prove (35), we note that for fixed $\omega$ and $t, S(t,\mathbf {x};\omega )$ and $\varPsi _{t}(\mathbf {x};\omega )$ are continuous functions of $\mathbf {x}$, hence

$$\begin{aligned} \inf _{\mathbf {x}\in \mathbb {Q}^{n}}\{\varPsi _{t}(\mathbf {x};\omega ) :|S(t,\mathbf {x},\omega )-z|\le \frac{1}{n}\}=\inf _{\mathbf {x} \in \mathbb {R}^{n}}\{\varPsi _{t}(\mathbf {x};\omega ):|S(t,\mathbf {x},\omega ) -z|\le \frac{1}{n}\}\le \widetilde{\varPhi }_{t}(z;\omega ). \end{aligned}$$

However, for any $\mathbf {x}$ such that $|S(t,\mathbf {x},\omega )-z|\le \frac{1}{n}$, (C2’) implies that $|\varPsi _{t}(\mathbf {x};\omega )-\varPsi _{t}(\mathbf {x}';\omega )|\le \frac{1}{n}C$ when $z=S(t,\mathbf {x}',\omega ),$ in which $C$ is a positive constant independent of $\mathbf {x},\mathbf {x}'$ but dependent on $(z,\omega )$ and $t$. Hence,

$$\begin{aligned} \Big |\inf _{\mathbf {x}\in \mathbb {Q}^{n}}\left\{ \varPsi _{t} (\mathbf {x};\omega ):|S(t,\mathbf {x},\omega )-z|\le \frac{1}{n}\right\} -\widetilde{\varPhi }_{t}(z;\omega )\Big |\le \frac{1}{n}C. \end{aligned}$$

By taking the limit as $n\rightarrow \infty$, we conclude that

$$\begin{aligned} \widetilde{\varPhi }_{t}(z;\omega )=\lim _{n\ge 1} \inf _{\mathbf {x}\in \mathbb {Q}^{n}}\left\{ \varPsi _{t} (\mathbf {x};\omega ):|S(t,\mathbf {x},\omega )-{z}|\le \frac{1}{n}\right\} . \end{aligned}$$

In fact, we can also prove that $\widetilde{\varPhi }_{t}^{K}(z;\omega ):=\inf _{\mathbf {x}\in K}\{\varPsi _{t}(\mathbf {x};\omega ):{z}=S(t,\mathbf {x};\omega )\}$ is also measurable when $K$ is a compact subset of $\mathbb {R}^{n}$, by replacing $\mathbb {Q}^{n}$ in the above argument by a countable dense subset of $K$. This implies that the set

$$\begin{aligned} \{(t,{z},\omega ):\exists \mathbf {x}\in K,S(t,\mathbf {x};\omega )={z} \text{ and } \varPsi _{t}(\mathbf {x};\omega )\le \widetilde{\varPhi }_{t}({z};\omega )+\epsilon \} \end{aligned}$$

is measurable for all $\epsilon >0$, which in turn implies that the set-valued function

$$\begin{aligned} F(t,{z},\omega )\mapsto \{\mathbf {x}\in \mathbb {R}^{n}:S(t,\mathbf {x};\omega )={z} \text{ and } \varPsi _{t}(\mathbf {x};\omega )\le \widetilde{\varPhi }_{t}({z};\omega )+\epsilon \} \end{aligned}$$

is strongly measurable. It is nonempty and closed-valued since $\varPsi _{t}(\mathbf {x};\omega )$ and $S(t,\mathbf {x};\omega )$ are continuous functions of $\mathbf {x}$. Therefore there exists a measurable function $\mathcal {X}(t,{z},\omega )$ such that $\mathcal {X}(t,{z},\omega )\in F(t,{z},\omega )$ by the Kuratowski–Ryll-Nardzewski Theorem. $\square$