A Unified View of LIBOR Models

Abstract

We provide a unified framework for modeling LIBOR rates using general semimartingales as driving processes and generic functional forms to describe the evolution of the dynamics. We derive sufficient conditions for the model to be arbitrage-free which are easily verifiable, and for the LIBOR rates to be true martingales under the respective forward measures. We discuss when the conditions are also necessary and comment on further desirable properties such as those leading to analytical tractability and positivity of rates. This framework allows to consider several popular models in the literature, such as LIBOR market models driven by Brownian motion or jump processes, the Lévy forward price model as well as the affine LIBOR model, under one umbrella. Moreover, we derive structural results about LIBOR models and show, in particular, that only models where the forward price is an exponentially affine function of the driving process preserve their structure under different forward measures.

Financial support from the PROCOPE project “Financial markets in transition: mathematical models and challenges” and the Europlace Institute of Finance project “Post-crisis models for interest rate markets” is gratefully acknowledged.

Appendix A: Semimartingale Characteristics and Martingales

Let $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in [0,T_*]},{\mathbb {P}})$ denote a complete stochastic basis and $T_*$ denote a finite time horizon. Let X be an ${\mathbb {R}^{d}}$-valued semimartingale on this basis whose characteristics are absolutely continuous, i.e. its local characteristics are given by (b, c, F; A) with $A_{t}=t$, for some truncation function h; cf. Jacod and Shiryaev [15, Proposition II.2.9]. Moreover, let $f:{\mathbb {R}}_{+}\times {\mathbb {R}^{d}}\rightarrow {\mathbb {R}}$ be a function of class $C^{1,2}({\mathbb {R}}_{+}\times {\mathbb {R}^{d}})$.

The process $f(\cdot , X)$ is a real-valued semimartingale which has again absolutely continuous characteristics. Let us denote its local characteristics by $(b^{f}, c^{f}, F^{f})$ for a truncation function $h^{f}$. Then, noting that Itô’s formula holds for the function $f \in C^{1,2}({\mathbb {R}}_{+} \times {\mathbb {R}^{d}})$ and reasoning as in the proof of Corollary A.6 from Goll and Kallsen [13], we have that

$$\begin{aligned} b^{f}_{t}&= \frac{{{\mathrm {d}}}}{{{{\mathrm {d}}}}t} f(t, X_{t-}) + \langle {\mathrm {D}}f(t, X_{t-}), b_{t} \rangle + \frac{1}{2} \sum _{i,j=1}^{d} {\mathrm {D}}_{ij}^{2} f(t, X_{t-}) c_{t}^{ij} \nonumber \\ \nonumber&\quad + \int \limits _{{\mathbb {R}^{d}}} \left( h^{f}\big (f(t, X_{t-}+x)-f(t, X_{t-})\big ) - \langle {\mathrm {D}}f(t, X_{t-}), h(x)\rangle \right) F_{t}({{\mathrm {d}}}x)\\ c^{f}_{t}&= \big \langle {\mathrm {D}}f(t, X_{t-}), c_{t} {\mathrm {D}}f(t, X_{t-})\big \rangle \\ \nonumber F^{f}_t(G)&= \int \limits _{{\mathbb {R}^{d}}} {1}_{G} \big (f(t, X_{t-}+x)-f(t, X_{t-})\big ) F_{t}({{\mathrm {d}}}x), \quad \quad G \in {\mathcal {B}}({\mathbb {R}}\setminus \{0\}). \end{aligned}$$

(A.1)

Proposition A.1

Let X be an ${\mathbb {R}^{d}}$-valued semimartingale with absolutely continuous characteristics (b, c, F) and let $f: {\mathbb {R}}_{+} \times {\mathbb {R}^{d}}\rightarrow {\mathbb {R}}$ be a function of class $C^{1, 2}$ such that the process Y defined by

$$\begin{aligned} Y_{t} := \mathrm {e}^{f(t, X_{t})} \end{aligned}$$

(A.2)

is exponentially special. If the following condition holds

$$\begin{aligned} \langle {\mathrm {D}}f(t, X_{t-}), b_t \rangle&= - \frac{{{\mathrm {d}}}}{{{{\mathrm {d}}}}t} f(t, X_{t-}) - \frac{1}{2} \sum _{i,j=1}^{d} {\mathrm {D}}_{ij}^{2} f(t,X_{t-}) c_{t}^{ij} \nonumber \\&\quad - \frac{1}{2} \big \langle {\mathrm {D}}f(t, X_{t-}), c_t {\mathrm {D}}f(t, X_{t-})\big \rangle \\&- \int \limits _{{\mathbb {R}^{d}}} \Big ( \mathrm {e}^{f(t, X_{t-}+x)-f(t, X_{t-})} - 1 - \langle {\mathrm {D}}f(t, X_{t-}), h(x)\rangle \Big ) F_{t}({{\mathrm {d}}}x), \nonumber \end{aligned}$$

(A.3)

then Y is a local martingale.

Proof

The proof follows from Theorem 2.18 in Kallsen and Shiryaev [19]: set $\theta =1$ and apply the theorem to the semimartingale $f(\cdot , X)$. Indeed, since $f(\cdot , X)$ has absolutely continuous characteristics it is also quasi-left continuous, hence assertions (6) and (1) of Theorem 2.18. yield

$$ K^{f(\cdot , X)} (1) = \widetilde{K}^{f(\cdot , X)}(1) = \int \limits _{0}^{\cdot } \Big ( b^{f}_{t} + \frac{1}{2} c^{f}_{t} + \int \limits _{{\mathbb {R}}} \big ( \mathrm {e}^{x}-1-h^{f}(x)\big ) F^{f}_{t}({{\mathrm {d}}}x) \Big ) {{\mathrm {d}}}t. $$

By definition of the exponential compensator and Theorem 2.19 in Kallsen and Shiryaev [19] it follows that

$$ \mathrm {e}^{f(\cdot , X)- K^{f(\cdot , X)}(1)} \in {\mathcal {M}}_{\mathrm {loc}}. $$

Therefore, $\mathrm {e}^{f(\cdot , X)} \in {\mathcal {M}}_{\mathrm {loc}}$ if and only if $K^{f(\cdot , X)}(1)=0$ up to indistinguishability. Equivalently,

$$ b^{f}_{t} + \frac{1}{2} c^{f}_{t} + \int \limits _{{\mathbb {R}}} \big ( \mathrm {e}^{x}-1-h^{f}(x)\big ) F^{f}_{t}({{\mathrm {d}}}x) = 0 $$

for every t. Inserting the expressions for $b^{f}, c^{f}$ and $F^{f}$, cf. (A.1), into the above equality yields condition (A.3). $\square$

Proposition A.2

Let X be an ${\mathbb {R}^{d}}$-valued semimartingale with absolutely continuous characteristics (b, c, F) such that

$$\begin{aligned} \int \limits _{0}^{T_*} \int \limits _{{\mathbb {R}^{d}}} (|x|^{2} \wedge 1) F_{t}({{\mathrm {d}}}x) {{\mathrm {d}}}t + \int \limits _{0}^{T_*} \int \limits _{|x|>1} |x| \mathrm {e}^{K |x|} F_{t}({{\mathrm {d}}}x) {{\mathrm {d}}}t < C_{1} \end{aligned}$$

(A.4)

and

$$\begin{aligned} \int \limits _{0}^{T_*} \Vert c_{t}\Vert {{\mathrm {d}}}t < C_{2}, \end{aligned}$$

(A.5)

for some deterministic constants $C_{1},C_{2}>0$. Moreover, let $f: {\mathbb {R}}_{+} \times {\mathbb {R}^{d}}\rightarrow {\mathbb {R}}$ be a function of class $C^{1, 2}$ and globally Lipschitz with constant $K>0$ such that

$$ |f(t, x)-f(t,y)| \le K |x-y|, \qquad t \ge 0, \ \ x,y \in {\mathbb {R}^{d}}. $$

Then, the process $f(\cdot , X)$ is exponentially special, while the process Y defined by (A.2) and satisfying (A.3) is a uniformly integrable martingale.

Proof

The process $f(\cdot , X)$ is exponentially special if and only if

$$ {1}_{\{|x|>1\}} \mathrm {e}^{x} *\nu ^{f} \in {\mathcal {V}}. $$

Hence, it suffices to show that ${1}_{\{|x|>1\}} \mathrm {e}^{x} *\nu ^{f}_{T_*} < \infty$, as the integrand is positive. Since f is globally Lipschitz, we have

$$\begin{aligned} {1}_{\{|x|>1\}} \mathrm {e}^{x} *\nu ^{f}_{T_*}&= \int \limits _{0}^{T_*} \int \limits _{|x|>1} \mathrm {e}^{x} F_t^{f} ({{\mathrm {d}}}x) {{\mathrm {d}}}t \\&\mathop {=}\limits ^{(A.1)} \int \limits _{0}^{T_*} \int \limits _{{\mathbb {R}^{d}}} {1}_{\{|f(t, X_{t-}+x)-f(t, X_{t-})|>1\}} \mathrm {e}^{f(t, X_{t-}+x) - f(t, X_{t-})} F_t ({{\mathrm {d}}}x) {{\mathrm {d}}}t \\&\le \int \limits _{0}^{T_*} \int \limits _{K |x|>1} \mathrm {e}^{K |x|} F_t ({{\mathrm {d}}}x) {{\mathrm {d}}}t < \infty , \end{aligned}$$

which holds by the Lipschitz property and (A.4).

Moreover, if $F=\mathrm {e}^{f(\cdot , X)} \in {\mathcal {M}}_{\mathrm {loc}}$, applying Proposition 3.1 in Criens et al. [6] it is also a uniformly integrable martingale if the following condition holds:

$$\begin{aligned} \int \limits _{0}^{T_*} \bigg ( c^{f}_{t} + \int \limits _{{\mathbb {R}^{d}}} \Big [ (|x|^{2} \wedge 1) + |x| \mathrm {e}^{x} {1}_{\{|x| > 1\}} \Big ] F^{f}_t({{{\text {d}}}{x}}) \bigg ) {{{\text {d}}}{t}}< C^{f}, \end{aligned}$$

(A.6)

for some constant $C^{f} >0$. We first check the condition for the diffusion coefficient

$$\begin{aligned} \int \limits _{0}^{T_*} c^{f}_{t} {{\mathrm {d}}}t&= \int \limits _{0}^{T_*} \langle {\mathrm {D}}f(t, X_{t-}), c_{t} {\mathrm {D}}f(t, X_{t-}) \rangle {{\mathrm {d}}}t \le \int \limits _{0}^{T_*} \Vert c_{t}\Vert |{\mathrm {D}}f(t, X_{t-})|^2 {{\mathrm {d}}}t < C_{1}^{f}, \end{aligned}$$

which follows from (A.5) and the fact that ${\mathrm {D}}f(\cdot , X_{-})$ is bounded as a consequence of f being globally Lipschitz. As for the jump part, we have that

$$\begin{aligned} \int \limits _{0}^{T_*} \int \limits _{{\mathbb {R}}} (|x|^{2}&\wedge 1) F_t^{f} ({{\mathrm {d}}}x) {{\mathrm {d}}}t + \int \limits _{0}^{T_*} \int \limits _{|x|>1} |x| \mathrm {e}^{x} F_t^{f} ({{\mathrm {d}}}x) {{\mathrm {d}}}t \\&\mathop {=}\limits ^{(A.1)} \int \limits _{0}^{T_*} \int \limits _{{\mathbb {R}^{d}}} (|f(t, X_{t-}+x)-f(t, X_{t-})|^{2} \wedge 1) F_t({{\mathrm {d}}}x) {{\mathrm {d}}}t\\&\qquad \quad \qquad \qquad + \int \limits _{0}^{T_*} \int \limits _{{\mathbb {R}^{d}}} {1}_{\{|f(t,X_{t-}+x)-f(t,X_{t-})|>1\}} \\&\qquad \,\, \times |f(t, X_{t-}+x) - f(t, X_{t-})| \, \mathrm {e}^{f(t, X_{t-}+x) - f(t, X_{t-})} F_t({{\mathrm {d}}}x) {{\mathrm {d}}}t\\&\le \int \limits _{0}^{T_*} \int \limits _{{\mathbb {R}^{d}}} (K^{2} |x|^{2} \wedge 1) F({{\mathrm {d}}}x) {{\mathrm {d}}}t + \int \limits _{0}^{T_*} \int \limits _{K |x|> 1} K |x| \mathrm {e}^{K |x|} F^{f} ({{\mathrm {d}}}x) {{\mathrm {d}}}t < C_{2}^{f}, \end{aligned}$$

using again the Lipschitz property and (A.4). $\square$

Next, we provide the representation of Y as a stochastic exponential.

Lemma A.3

Let X be an ${\mathbb {R}^{d}}$-valued semimartingale with absolutely continuous characteristics $(B,C,\nu )$ and let $f:{\mathbb {R}}_{+} \times {\mathbb {R}^{d}}\rightarrow {\mathbb {R}}_{+}$ be a function of class $C^{1,2} ({\mathbb {R}}_{+} \times {\mathbb {R}^{d}})$. Define a real-valued semimartingale Y via (A.2). If $Y \in {\mathcal {M}}_{\mathrm {loc}}$, then it can be written as

$$ Y = {\mathcal {E}}\big ( {\mathrm {D}}f(\cdot , X_{-}) \cdot X^{c} + W(\cdot , x) *(\mu ^{X} - \nu ) \big ), $$

where $X^{c}$ is the continuous martingale part of X, $\mu ^{X}$ is the random measure of jumps of X with compensator $\nu$ and

$$ W(\cdot , x):= \mathrm {e}^{f(\cdot , X_{-} +x) - f(\cdot , X_{-})} -1. $$

Proof

Theorem 2.19 in Kallsen and Shiryaev [19] yields that

$$ \mathrm {e}^{f(\cdot , X)} = {\mathcal {E}}\left( f(\cdot , X)^{c} + (\mathrm {e}^{x}-1) *(\mu ^{f}-\nu ^{f})\right) , $$

using that $f(\cdot , X)$ is quasi-left continuous since X is also quasi-left continuous. Here $f(\cdot , X)^{c}$ denotes the continuous martingale part of $f(\cdot , X)$ and $\mu ^{f}$ its random measure of jumps. The result now follows using the form of the local characteristics $c^f, F^f$ of the process $f(\cdot ,X)$ in (A.1); see also the proof of Corollary A.6 in Goll and Kallsen [13].

Lemma A.4

Let X be an ${\mathbb {R}^{d}}$-valued semimartingale with absolutely continuous characteristics (b, c, F) with respect to the truncation function h. Let $f:{\mathbb {R}}_{+} \times {\mathbb {R}^{d}}\rightarrow {\mathbb {R}}_{+}$ be a function of class $C^{1,2}({\mathbb {R}}_{+} \times {\mathbb {R}^{d}})$ and globally Lipschitz. Assume that conditions (A.3), (A.4) and (A.5) are satisfied.

Define the probability measure ${\mathbb {P}}'\sim {\mathbb {P}}$ via

$$ \frac{{{\mathrm {d}}}{\mathbb {P}}'}{{{\mathrm {d}}}{\mathbb {P}}}\Big |_{{\mathcal {F}}_\cdot } := \mathrm {e}^{f(\cdot , X)}\,. $$

Then, the ${\mathbb {P}}'$-characteristics of the semimartingale X are absolutely continuous and provided by $(b',c',F')$, where

$$\begin{aligned} b'_t&= b_t + c_t \beta _t + \int \limits _{{\mathbb {R}^{d}}} ( Y_t(x) - 1 ) h(x) F_t({{\mathrm {d}}}x) \\ c'_t&= c_t \\ F'_t({{\mathrm {d}}}x)&= Y_t(x) F_t({{\mathrm {d}}}x), \end{aligned}$$

with $\beta _t = {\mathrm {D}}f(t,X_{t-})$ and $Y_t(x)= \mathrm {e}^{f(t,X_{t-}+x) - f(t,X_{t-})}$, for $t\in {\mathbb {R}}_{+}$ and $x\in {\mathbb {R}^{d}}$.

Proof

The result follows directly from the previous lemma and Proposition 2.6 in Kallsen [18]. $\square$