Term structure modelling for multiple curves with stochastic discontinuities

Abstract

We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of multiple curve markets under minimal assumptions in an extended HJM framework and provide a fundamental theorem of asset pricing based on NAFLVR. The approach with stochastic discontinuities permits to embed market models directly, unifying seemingly different modelling philosophies. We also develop a tractable class of models, based on affine semimartingales, going beyond the requirement of stochastic continuity.

Appendices

Appendix A: Technical results

The following technical result on ratios and products of stochastic exponentials easily follows from Yor’s formula; see [33, § II.8.19].

Corollary A.1

For any semimartingales$X$, $Y$and$Z$with$\Delta Z > -1$, it holds that

$$\begin{aligned} \frac{\mathcal{E}(X)\mathcal{E}(Y)}{\mathcal{E}(Z)} &= \mathcal{E}\bigg( X + Y - Z + \langle X^{\mathrm{c}},Y^{\mathrm{c}}\rangle - \langle Y^{\mathrm{c}},Z^{\mathrm{c}}\rangle - \langle X^{\mathrm{c}},Z^{\mathrm{c}}\rangle +\langle Z^{\mathrm{c}},Z^{\mathrm{c}}\rangle \\ & \phantom{=: \mathcal{E}\bigg(} +\sum _{0< s\leq \cdot } \frac{\Delta Z_{s}(-\Delta X_{s} - \Delta Y_{s} + \Delta Z_{s}) + \Delta X_{s} \Delta Y_{s}}{1+\Delta Z_{s}}\bigg). \end{aligned}$$

Proof

of Lemma3.5 Due to Assumption 3.3, one can verify by Minkowski’s integral inequality and Hölder’s inequality that the stochastic integrals in (3.8) are well defined for every $T\in \mathbb{R}_{+}$ and $\delta \in \mathcal{D}_{0}$. Let $F(t, T,\delta ) : = \int _{(t, T]}f(t, u, \delta )\eta (du)$ for all $0\leq t\leq T<+\infty$. For $t< T$, (3.7) implies that

$$\begin{aligned} F(t,T,\delta ) & = \int _{(t, T]}\bigg( f(0, u, \delta ) + \int _{0}^{t} a(s,u,\delta ) ds +V(t,u,\delta ) + \int _{0}^{t} b(s,u,\delta ) dW_{s} \\ & \phantom{=:\int _{(t, T]}\bigg(} + \int _{0}^{t} \int _{E} g(s, x, u,\delta ) \bigl(\mu (ds,dx) - \nu (ds,dx)\bigr) \bigg)\eta (du)\\ & = \int _{0}^{T} f(0, u, \delta )\eta (du) + \int _{0}^{T} \int _{0}^{t} a(s, u, \delta ) ds\eta (du)\\ & \phantom{=:} + \int _{0}^{T} V(t,u,\delta ) \eta (du) + \int _{0}^{T} \int _{0}^{t} b(s, u, \delta ) dW_{s}\eta (du) \\ &\phantom{=:} + \int _{0}^{T} \int _{0}^{t} \int _{E} g(s, x, u,\delta ) \bigl(\mu (ds,dx) - \nu (ds,dx)\bigr) \eta (du) \\ & \phantom{=:} - \int _{0}^{t} f(0, u, \delta )\eta (du) - \int _{0}^{t} \int _{0}^{u} a(s, u, \delta ) ds\eta (du) \\ & \phantom{=:} - \int _{0}^{t}V(u,u,\delta ) \eta (du) - \int _{0}^{t} \int _{0}^{u} b(s, u, \delta ) dW_{s}\eta (du) \\ &\phantom{=:} - \int _{0}^{t} \int _{0}^{u} \int _{E} g(s, x, u,\delta ) \bigl(\mu (ds,dx) - \nu (ds,dx)\bigr) \eta (du). \end{aligned}$$

Due to Assumption 3.3, we can apply ordinary and stochastic Fubini theorems, in the versions of Veraar [52, Theorem 2.2] for the stochastic integral with respect to $W$ and in the version of Proposition A.2 in Björk et al. [4] for the stochastic integral with respect to the compensated random measure $\mu -\nu$. We therefore obtain

(A.1)

In (A.1), the finiteness of $\int _{0}^{\cdot }f(u,u,\delta )\eta (du)$ follows by Assumption 3.3 together with an analogous application of ordinary and stochastic Fubini theorems.

To complete the proof, it remains to establish (3.8) for $t = T\in \mathbb{R}_{+}$. To this effect, it suffices to show that $\Delta G(T,T,\delta )=\Delta F(T,T,\delta )$ for all $T\in \mathbb{R}_{+}$, where $\Delta G(T,T,\delta ):=G(T,T,\delta )-G(T-,T,\delta )$, and similarly for $\Delta F(T,T,\delta )$. By [33, Proposition II.1.17], the fact that $\nu (\{T\}\times E)=0$ implies that for every $T\in \mathbb{R}_{+}$, we have $\mathbb{Q}[\mu (\{T\}\times E)\neq 0]=0$. Therefore, $\mathbb{Q}[\Delta G(T,T,\delta )\neq 0]>0$ only if $T=T_{n}$ for some $n\in \mathbb{N}$. For $T=T_{1}$, (A.1) and (3.7) together imply that

$$\begin{aligned} \Delta G(T_{1},T_{1},\delta ) &= \bar{V}(T_{1},T_{1},\delta ) - f(T_{1},T_{1},\delta ) = -f(T_{1}-,T_{1},\delta ) \\ &= -F(T_{1}-,T_{1},\delta ) = \Delta F(T_{1},T_{1},\delta ), \end{aligned}$$

where the last equality follows from the convention $F(T_{1},T_{1},\delta )=0$. By induction over $n$, the same reasoning yields that $\Delta G(T_{n},T_{n},\delta )=\Delta F(T_{n},T_{n},\delta )$ for all $n\in \mathbb{N}$. Finally, the semimartingale property of $\delta$-tenor bond prices $(P(t,T,\delta ))_{0\leq t\leq T}$ follows from (A.1). □

Appendix B: Embedding of market models into the HJM framework

The general market model from Sect. 4 can be embedded into the extended HJM framework of Sect. 3. For simplicity of presentation, consider a market model for a single tenor ($\mathcal{D}=\{\delta \}$) and suppose that the forward Ibor rate $L(\cdot ,T,\delta )$ is given by (4.2) for all $T\in {\mathcal{T}}^{\delta }=\{T_{1},\ldots ,T_{N}\}$, with $T_{i+1}-T_{i}=\delta$ for $i=1,\ldots ,N-1$. For simplicity, assume that there is a fixed number $N+1$ of discontinuity dates, coinciding with the set of dates ${\mathcal{T}}^{0}:={\mathcal{T}}^{\delta }\cup \{T_{N+1}\}$, with $T_{N+1}:=T_{N}+\delta$. We say that $\{L(\cdot ,T,\delta ):T\in {\mathcal{T}}^{\delta }\}$ can be embedded into an extended HJM model if there exist a sigma-finite measure $\eta$ on $\mathbb{R}_{+}$, a spread process $S^{\delta }$ and a family of forward rates $\{f(\cdot ,T,\delta ):T\in {\mathcal{T}}^{\delta }\}$ such that

$$ L(t,T,\delta ) = \frac{1}{\delta }\left(S^{\delta }_{t}\frac{P(t,T,\delta )}{P(t,T+\delta )}-1\right) \qquad \text{for all }0\leq t\leq T\in {\mathcal{T}}^{\delta }, $$

(B.1)

where $P(t,T,\delta )$ is given by (3.5) for all $0\leq t\leq T\in {\mathcal{T}}^{\delta }$. In other words, in view of (2.2), the HJM model generates the same forward Ibor rates as the original market model, for every date $T\in {\mathcal{T}}^{\delta }$.

We remark that since a market model involves OIS bonds only for maturities ${\mathcal{T}}^{0}=\{T_{1},\ldots ,T_{N+1}\}$, there is no loss of generality in taking the measure $\eta$ in (3.5) as a purely atomic measure of the form

$$ \eta (du) = \sum _{i=1}^{N+1}\delta _{T_{i}}(du). $$

(B.2)

More specifically, if OIS bonds for maturities ${\mathcal{T}}^{0}$ are defined through (3.5) via a generic measure of the form (3.6), there always exists a measure $\eta$ as in (B.2) generating the same bond prices, up to a suitable specification of the forward rate process.

The following proposition explicitly shows how a general market model can be embedded into an HJM model. For $t\in [0,T_{N}]$, we define

$$ i(t):=\min \{j\in \{1,\ldots ,N\}:T_{j}\geq t\} $$

so that $T_{i(t)}$ is the smallest $T\in {\mathcal{T}}^{\delta }$ such that $T\geq t$.

Proposition B.1

Suppose that all the conditions of Theorem4.1are satisfied with respect to the measure$\eta$in (B.2), and assume furthermore that$L(t,T,\delta )>-1/\delta$a.s. for all$t\in [0,T]$and$T\in {\mathcal{T}}^{\delta }$. Then under the above assumptions, the market model$\{L(\cdot ,T,\delta ):T\in {\mathcal{T}}^{\delta }\}$can be embedded into an HJM model by choosing

(i) a family of forward rates$\{f(\cdot ,T,\delta ):T\in {\mathcal{T}}^{\delta }\}$with initial values

$$ f(0,T_{i},\delta ) = f(0,T_{i+1},0) - \log \frac{1+\delta L(0,T_{i},\delta )}{1+\delta L(0,T_{i-1},\delta )} \qquad \textit{ for }i=1,\ldots ,N $$

and satisfying (3.7), where for all$i=1,\ldots ,N$, the volatility process$b(\cdot ,T_{i},\delta )$, the jump function$g(\cdot ,\cdot ,T_{i},\delta )$and the random variables$(\Delta V(T_{n},T_{i},\delta ))_{n=1,\ldots ,N}$are respectively given by

$$ b(t,T_{i},\delta ) = \textstyle\begin{cases} b(t,T_{i},0) + b(t,T_{i+1},0) - \delta \frac{b^{L}(t,T_{i},\delta )}{1+\delta L(t-,T_{i},\delta )} &\quad \textit{if }i=i(t),\\ b(t,T_{i+1},0) - \delta (\frac{b^{L}(t,T_{i},\delta )}{1+\delta L(t-,T_{i},\delta )}-\frac{b^{L}(t,T_{i-1},\delta )}{1+\delta L(t-,T_{i-1},\delta )}) &\quad \textit{if }i>i(t), \end{cases} $$

$$ g(t,x,T_{i},\delta ) = \textstyle\begin{cases} \begin{aligned} &g(t,x,T_{i+1},0) + g(t,x,T_{i},0) \\ &- \log (1+\tfrac{\delta g^{L}(t,x,T_{i},\delta )}{1+\delta L(t-,T_{i},\delta )}) \end{aligned} & \quad \,\,\textit{if }i=i(t),\\ &\\ g(t,x,T_{i+1},0) - \log \frac{1+\tfrac{\delta g^{L}(t,x,T_{i},\delta )}{1+\delta L(t-,T_{i},\delta )}}{1 +\frac{\delta g^{L}(t,x,T_{i-1},\delta )}{1+\delta L(t-,T_{i-1},\delta )}} &\quad \,\, \textit{if }i>i(t), \end{cases} $$

$$ \Delta V(T_{n},T_{i},\delta ) = \Delta V(T_{n},T_{i+1},0) - \log \frac{\frac{1+\delta L(T_{n},T_{i},\delta )}{1+ \delta L(T_{n}-,T_{i},\delta )}}{\frac{1+ \delta L(T_{n},T_{i-1},\delta )}{1+\delta L(T_{n}-,T_{i-1},\delta )}} \qquad \textit{for }i\geq n+1, $$

and the process$a(\cdot ,T_{i},\delta )$is determined by condition (ii) of Theorem3.7;

(ii) a spread process$S^{\delta }$with initial value$S^{\delta }_{0} = (1+\delta L(0,0,\delta ))P(0,\delta )$and satisfying (3.3), (3.4), where the processes$\alpha ^{\delta }$, $H^{\delta }$, the function$L^{\delta }$and the random variables$(\Delta A^{\delta }_{T_{n}})_{n=1,\ldots ,N}$are respectively given by

$$\begin{aligned} \alpha ^{\delta }_{t} &= 0, \qquad H^{\delta }_{t} = 0, \qquad L^{\delta }(t,x) = 0,\\ \Delta A^{\delta }_{T_{n}} &= \frac{1+\delta L(T_{n},T_{n},\delta )}{1+\delta L(T_{n}-,T_{n},\delta )}e^{f(T_{n}-,T_{n},0)-f(T_{n}-,T_{n},\delta )-\Delta V(T_{n},T_{n+1},0)}-1. \end{aligned}$$

Moreover, the resulting HJM model satisfies all the conditions of Theorem3.7.

Proof

Since the proof involves rather lengthy computations, we only provide a sketch. For $T\in {\mathcal{T}}^{\delta }$, by Theorem 4.1 and the assumption $L(t,T,\delta )>-1/\delta$ a.s. for all $t\in [0,T]$, the process $(1+\delta L(\cdot ,T,\delta ))P(\cdot ,T+\delta )/X^{0}$ is a strictly positive ℚ-local martingale, so that $L(t-,T,\delta )>-1/\delta$ a.s. for all $t\in [0,T]$ and $T\in {\mathcal{T}}^{\delta }$. Define the process $Y(T,\delta )=(Y_{t}(T,\delta ))_{0\leq t\leq T}$ by $Y_{t}(T,\delta ):=S^{\delta }_{t}P(t,T,\delta )/P(t,T+\delta )$. An application of Corollary A.1, together with (3.3) and Corollary 3.6, yields a stochastic exponential representation and a semimartingale decomposition of $Y(T,\delta )$.

For the spread process $S^{\delta }$ given in (3.3), we start by imposing $H^{\delta }=0$ and $L^{\delta }=0$. We then proceed to determine the processes describing the forward rates $\{f(\cdot ,T,\delta ):T\in {\mathcal{T}}^{\delta }\}$ satisfying (3.7). In view of (B.1), for each $T\in {\mathcal{T}}^{\delta }$, we determine the process $b(\cdot ,T,\delta )$ by matching the Brownian part of $Y(T,\delta )$ with the Brownian part of $\delta L(\cdot ,T,\delta )$, while the jump function $g(\cdot ,\cdot ,T,\delta )$ is obtained in a similar way by matching the totally inaccessible jumps of $Y(T,\delta )$ with the totally inaccessible jumps of $\delta L(\cdot ,T,\delta )$. The drift process $a(\cdot ,T,\delta )$ is then uniquely determined by imposing condition (ii) of Theorem 3.7. As a next step, for each $n=1,\ldots ,N$, the random variable $\Delta A^{\delta }_{T_{n}}$ appearing in (3.3), (3.4) is determined by requiring that

$$ \Delta Y_{T_{n}}(T_{n},\delta ) = \delta \Delta L(T_{n},T_{n},\delta ). $$

(B.3)

Then for each $n=1,\ldots ,N-1$ and $T\in \{T_{n+1},\ldots ,T_{N}\}$, the random variable $\Delta V(T_{n},T,\delta )$ is determined by requiring that

$$ \Delta Y_{T_{n}}(T,\delta ) = \delta \Delta L(T_{n},T,\delta ), $$

(B.4)

while $\Delta V(T_{n},T,\delta ):=0$ for $T\leq T_{n}$. Note that $\Delta V(T_{n},T_{N+1},\delta ) = 0$ for $\delta \neq 0$ and $n= 1,\ldots , N+1$. At this stage, the forward rates $\{f(\cdot ,T,\delta ):T\in {\mathcal{T}}^{\delta }\}$ are completely specified. With this specification, it can be verified that (4.3) and (4.4) respectively imply that (3.9) and (3.10) are satisfied, using the fact that Assumption 3.3 as well as (3.9), (3.10) are satisfied for $\delta =0$ and $T\in {\mathcal{T}}^{0}$ by assumption. Moreover, it can be checked that if condition (ii) of Theorem 4.1 is satisfied, the random variables $\Delta A^{\delta }_{T_{n}}$ and $\Delta V(T_{n},T,\delta )$ resulting from (B.3), (B.4) satisfy conditions (iii), (iv) of Theorem 3.7 for every $n\in \mathbb{N}$ and $T\in {\mathcal{T}}^{\delta }$. It remains to specify the process $\alpha ^{\delta }$ appearing in (3.4). To this effect, an inspection of Lemma 3.5 and Corollary 3.6 reveals that since the measure $\eta$ is purely atomic, the terms $f(t,t,\delta )$ and $f(t,t,0)$ do not appear in condition (i) of Theorem 3.7 and in (3.11), respectively. Since (3.11) holds by assumption, $\alpha ^{\delta }=0$ follows by imposing condition (i) of Theorem 3.7. We have thus obtained that the two processes

$$ \bigl(1+\delta L(\cdot ,T,\delta )\bigr)\frac{P(\cdot ,T+\delta )}{X^{0}} \qquad \text{ and }\qquad \frac{S^{\delta }P(\cdot ,T,\delta )}{X^{0}} $$

are two local martingales starting from the same initial values, with the same continuous local martingale parts and with identical jumps. By [33, Theorem I.4.18 and Corollary I.4.19], we conclude that (B.1) holds for all $0\leq t\leq T\in {\mathcal{T}}^{\delta }$. □

We point out that the specification in Proposition B.1 is not the unique HJM model which allows embedding a given market model $\{L(\cdot ,T,\delta ):T\in {\mathcal{T}}^{\delta }\}$. Indeed, $b(t,T_{i(t)},\delta )$ and $H^{\delta }_{t}$ can be arbitrarily specified as long as they satisfy

$$ b(t,T_{i(t)},\delta ) - H^{\delta }_{t} = b(t,T_{i(t)},0) + b(t,T_{i(t)+1},0) - \delta \frac{b^{L}(t,T_{i(t)},\delta )}{1+\delta L(t-,T_{i(t)},\delta )}, $$

together with suitable integrability requirements. An analogous degree of freedom exists for the specification of the functions $g(t,x,T_{i(t)},\delta )$ and $L^{\delta }(t,x)$. Note also that the random variable $\Delta A^{\delta }_{T_{n}}$ in Proposition B.1 can be equivalently expressed as

$$ \Delta A^{\delta }_{T_{n}} = \frac{1+\delta L(T_{n},T_{n},\delta )}{1+\delta L(T_{n-1},T_{n-1},\delta )}\frac{P(T_{n},T_{n+1})}{P(T_{n-1},T_{n})}-1 \qquad \text{for }n=1,\ldots ,N. $$