Long-Term Yield in an Affine HJM Framework on $S_{d}^{+}$

Abstract

We develop the HJM framework for forward rates driven by affine processes on the state space of symmetric positive semidefinite matrices. In this setting we find an explicit representation for the long-term yield in terms of the model parameters. This generalises the results of El Karoui et al. (Rev Deriv Res 1(4):351–369, 1997) and Biagini and Härtel (Int J Theor Appl Financ 17(3):1–24, 2012), where the long-term yield is investigated under no-arbitrage assumptions in a HJM setting using Brownian motions and Lévy processes respectively.

Appendix 1

Here we provide the proofs of Proposition 3.1 and Theorem 3.1. The results follow by applying the Fubini theorem for integrable functions (cf.  Theorem 14.16 in Chap. 14 of [42]) and the stochastic Fubini theorem (cf.  Theorem 65 in Chap. IV of [49]). For further details on the following computations, we refer to [35].

Proof of Proposition 3.1

Let us introduce for every maturity $T > 0$ the quantity

$$\begin{aligned} Z\!\left( t,T\right) := -\int _{t}^{T} f\!\left( t,u\right) \, du, \end{aligned}$$

(5.7)

for all $0 \le t \le T$. From the dynamics of the forward rate (3.1) we deduce that for all $T > 0$

$$\begin{aligned} Z\!\left( t,T\right) \mathop {=}\limits ^{(5.7)} -\int _{t}^{T} \! f\!\left( 0,u\right) \, du - \int _{t}^{T} \! \int _{0}^{t} \! \alpha \!\left( s,u\right) \, ds \, du - \int _{t}^{T} \! \int _{0}^{t} \! {\text {Tr}}\!\left[ \sigma \!\left( s,u\right) dX_s \right] du, \end{aligned}$$

(5.8)

for all $0 \le t \le T$. By combining (2.11), (3.3), (5.8), and (3.13), we derive the following identity

$$\begin{aligned} Z\!\left( t,T\right)&\mathop {=}\limits ^{(5.8)} Z\!\left( 0,T\right) \!+\! \int _{0}^{t} \! f\!\left( 0,u\right) \, du \!-\! \int _{t}^{T} \! \int _{0}^{t} \! \alpha \!\left( s,u\right) \, ds \, du \!-\! \int _{t}^{T} \! \int _{0}^{t} \! {\text {Tr}}\!\left[ \sigma \!\left( s,u\right) dX_{s} \right] du\\&\mathop {=}\limits ^{(3.13)} Z\!\left( 0,T\right) + \int _{0}^{t} \! r_{s} \, ds -\int _{0}^{t} \! \int _{s}^{T}\! \alpha \!\left( s,u\right) \, du \, ds - \int _{0}^{t} \! \int _{s}^{T}\! {\text {Tr}}\!\left[ \sigma \!\left( s,u\right) \, du \, dX_{s} \right] \\&\mathop {=}\limits _{(3.3)}^{(2.11)} Z\!\left( 0,T\right) + \int _{0}^{t} \! r_{s} \, ds -\int _{0}^{t} \! \int _{s}^{T}\! \alpha \!\left( s,u\right) \, du \, ds \\&\quad +\! \int _{0}^{t} \!{\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \left( \sqrt{X_{s}}\,dW_{s}\,Q + Q^{\top } dW_{s}^{\top } \sqrt{X_{s}}\right) \right] \\&\quad +\! \int _{0}^{t} {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \left( b +\! B\!\left( X_{s}\right) \right) \right] ds +\! \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!{\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \xi \right] \mu ^{X\!}\!\left( ds,d\xi \right) . \end{aligned}$$

Note that in general for $A, B \in \mathcal {M}_{d}$ with A symmetric, i.e. $A \in S_{d}$, it holds

$$\begin{aligned} {\text {Tr}}\!\left[ A \left( B + B^{\top }\right) \right]&= {\text {Tr}}\!\left[ A B \right] + {\text {Tr}}\!\left[ A B^{\top } \right] = {\text {Tr}}\!\left[ A B \right] + {\text {Tr}}\!\left[ \left( B A\right) ^{\top } \right] \nonumber \\&= {\text {Tr}}\!\left[ A B \right] + {\text {Tr}}\left[ B A \right] = 2 {\text {Tr}}\!\left[ A B \right] . \end{aligned}$$

(5.9)

Therefore, we get due to $\sigma \!\left( s,t\right) \in S_{d}$ for all $s, t \ge 0$ that

$$\begin{aligned} Z\!\left( t,T\right)&\mathop {=}\limits ^{(5.9)} Z\!\left( 0,T\right) \!+\! \int _{0}^{t} \! r_{s} \, ds \!-\!\int _{0}^{t} \! \int _{s}^{T}\! \alpha \!\left( s,u\right) \, du \, ds \!+\! 2 \int _{0}^{t} \! {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW_{s}\, Q \right] \nonumber \\&\quad +\! \int _{0}^{t} {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \left( b +\! B\!\left( X_{s}\right) \right) \right] ds +\! \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!{\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \xi \right] \mu ^{X\!}\!\left( ds,d\xi \right) . \end{aligned}$$

(5.10)

Note that for all $0 \le t \le T$

$$\begin{aligned} \Delta Z\!\left( t,T\right) = {\text {Tr}}\left[ \Sigma \!\left( t,T\right) \Delta X_{t}\right] . \end{aligned}$$

(5.11)

With the help of (5.10) and the fact that

$$\begin{aligned} \left\langle W_{lm}, W_{ru} \right\rangle _{s} = \left\{ \begin{array}{ll} s &{}\quad \text {if}\; l = r\; \text {and}\; m = u,\\ 0 &{}\quad \text {else,} \end{array}\right. \end{aligned}$$

(5.12)

we can calculate the quadratic variation of Z for all $T > 0$ follows

$$\begin{aligned} \left\langle Z\!\left( \,\cdot ,T\right) \right\rangle _{t}^{c}&= \left\langle {\text {Tr}}\!\left[ \int _{0}^{\cdot }\! \Sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW_{s}\, Q\right] \right\rangle _{t} \mathop {=}\limits ^{(5.12)} 4 \int _{0}^{t}\! {\text {Tr}}\!\left[ Q\, \Sigma \!\left( s,T\right) X_{s}\, \Sigma \!\left( s,T\right) Q^{\top }\right] \, ds. \end{aligned}$$

(5.13)

Further, we see that due to Eq. (2.27) of [13] for all $u \in S_{d}^{+}$ and a process Y on $S_{d}^{+}$ it holds

$$\begin{aligned} {\text {Tr}}\!\left[ B^{\top }\!\left( u\right) Y \right] = {\text {Tr}}\!\left[ B\!\left( Y\right) u \right] , \end{aligned}$$

(5.14)

where B is defined according to (2.13) and therefore for all $0\le t\le T$ and $\alpha = Q^{\top }Q$

$$\begin{aligned}&\int _{0}^{t}\!{\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \left( b + B\!\left( X_{s}\right) \right) \right] + 2\, {\text {Tr}}\!\left[ Q\, \Sigma \!\left( s,T\right) X_{s}\, \Sigma \!\left( s,T\right) Q^{\top } \right] ds \nonumber \\&\qquad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!\left( e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \, \xi \right] } - 1\right) \, \nu \!\left( ds,d\xi \right) \nonumber \\&\quad \mathop {=}\limits _{(5.14)}^{(2.12)} -\int _{0}^{t}\!{\text {Tr}}\!\left[ -\Sigma \!\left( s,T\right) b - 2\, \Sigma \!\left( s,T\right) \alpha \Sigma \!\left( s,T\right) X_{s} + B^{\top }\!\left( -\Sigma \!\left( s,T\right) \right) X_{s}\right] ds \nonumber \\&\qquad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!\left( e^{-{\text {Tr}}\left[ -\Sigma \left( s,T\right) \, \xi \right] } - 1\right) \, m\!\left( d\xi \right) ds\nonumber \\&\qquad + \int _{0}^{t}\!{\text {Tr}}\!\left[ X_{s} \int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!\left( e^{-{\text {Tr}}\left[ -\Sigma \left( s,T\right) \, \xi \right] } - 1\right) \, \mu \!\left( d\xi \right) \right] ds \nonumber \\&\quad \mathop {=}\limits _{(2.9)}^{(2.8)} \int _{0}^{t}\!\left( -F\!\left( -\Sigma \!\left( s,T\right) \right) -{\text {Tr}}\!\left[ R\!\left( -\Sigma \!\left( s,T\right) \right) X_{s}\right] \right) ds. \end{aligned}$$

(5.15)

Now, we apply Itô’s formula on $P\!\left( t,T\right) := \exp \!\left( Z\!\left( t,T\right) \right)$ for every maturity $T > 0$ (cf.  Definition 1.4.2 of [8]) and obtain the following representation, where we use Proposition 1.28 of Chap. II in [39] to combine the measures $\mu ^{X\!}\!\left( ds,d\xi \right)$ and $\nu \!\left( ds,d\xi \right)$, since the affine process X has only jumps of finite variation (cf.  (2.10)).

$$\begin{aligned} P\!\left( t,T\right)&= P\!\left( 0,T\right) + \int _{0}^{t} \! P\!\left( s-,T\right) \, dZ\!\left( s,T\right) + \frac{1}{2}\int _{0}^{t} P\!\left( s,T\right) \, d\left\langle Z\!\left( \,\cdot ,T\right) \right\rangle ^{c}_{s}\\&\quad + \sum _{0 < s \le t}^{\Delta Z\left( s,T\right) \ne 0} \left[ e^{Z\left( s,T\right) } - e^{Z\left( s-,T\right) } - \Delta Z\!\left( s,T\right) e^{Z\left( s-,T\right) }\right] \\&\mathop {=}\limits _{(5.11)}^{(5.13)} P\!\left( 0,T\right) + \int _{0}^{t} \! P\!\left( s-,T\right) \, dZ\!\left( s,T\right) \\&\quad + 2 \int _{0}^{t} \! P\!\left( s,T\right) {\text {Tr}}\!\left[ Q\, \Sigma \!\left( s,T\right) X_{s}\, \Sigma \!\left( s,T\right) Q^{\top } \right] \, ds\\&\quad + \sum _{0 \le s \le t}^{\Delta X_{s}\ne 0} \left[ e^{Z\left( s,T\right) } - e^{Z\left( s-,T\right) } - {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \Delta X_{s}\right] e^{Z\left( s-,T\right) }\right] \\&\mathop {=}\limits _{(5.10)}^{(2.1)} P\!\left( 0,T\right) + 2 \int _{0}^{t} \! P\!\left( s,T\right) {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW_{s}\, Q \right] \nonumber \\&\quad + \int _{0}^{t}\! P(s,T)\left( r_{s} - \int _{s}^{T}\!\alpha \!\left( s,u\right) du\right) \, ds \\&\quad + \int _{0}^{t}\! P(s,T) {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \left( b + B\!\left( X_{s}\right) \right) \right] \, ds\\&\quad + \int _{0}^{t}\! P(s-,T) \int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!{\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \xi \right] \, \mu ^{X\!}\!\left( ds,d\xi \right) \\&\quad + 2 \int _{0}^{t} \! P\!\left( s,T\right) {\text {Tr}}\!\left[ Q\, \Sigma \!\left( s,T\right) X_{s}\, \Sigma \!\left( s,T\right) Q^{\top } \right] \, ds \\&\quad + \sum _{0 \le s \le t}^{\Delta X_{s}\ne 0} \left[ e^{\Delta Z\left( s,T\right) }P\!\left( s-,T\right) \! -\! P\!\left( s-,T\right) \! -\! {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \Delta X_{s}\right] P\!\left( s-,T\right) \right] \\&\mathop {=}\limits _{(5.11)}^{(5.15)} P\!\left( 0,T\right) + \int _{0}^{t}\! P\!\left( s-,T\right) \left( r_{s} + A\!\left( s,T\right) \right) ds\\&+ 2 \int _{0}^{t}\! P\!\left( s,T\right) {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \sqrt{X_{s}} \, dW_{s} Q\right] \\&\quad + \int _{0}^{t}\! P\!\left( s-,T\right) \int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\! \left( e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \,\xi \right] } - 1\right) \left( \mu ^{X} - \nu \right) \left( ds,d\xi \right) . \end{aligned}$$

Assumption 1 guarantees that all integrals above are finite. $\square$

Proof of Theorem 3.1

By using (3.7) we see that the discounted bond price process under $\mathbb {Q}$ is

$$\begin{aligned} \frac{P\!\left( t,T\right) }{\beta _{t}}&\mathop {=}\limits ^{(3.8)} P\!\left( 0,T\right) + \int _{0}^{t}\! \frac{P\!\left( s,T\right) }{\beta _{s}}\left( A\!\left( s,T\right) + 2 {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \! \sqrt{X_{s}}\, \gamma _{s}\, Q\right] \right) ds \nonumber \\&\quad + 2 \int _{0}^{t}\! \frac{P\!\left( s,T\right) }{\beta _{s}} {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW^{*}_{s} \, Q\right] \nonumber \\&\quad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\! \frac{P\!\left( s-,T\right) }{\beta _{s}}\left( e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \,\xi \right] }\! -\! 1\right) \left( \mu ^{X\!} - \nu ^{*\!}\right) \!\left( ds,d\xi \right) \nonumber \\&\quad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\! \frac{P\!\left( s-,T\right) }{\beta _{s}}\left( e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \,\xi \right] }\! -\! 1\right) \left( K\!\left( s,\xi \right) \!-\!1\right) \nu \!\left( ds,d\xi \right) \end{aligned}$$

(5.16)

for all $0\le t\le T$. Since $\frac{P\left( t,T\right) }{\beta _{t}}$, $t \le T$, has to be a local martingale under $\mathbb {Q}$, the drift in (5.16) must disappear, i.e.  for all $0\le t\le T$

$$\begin{aligned} 0&\mathop {=}\limits ^{(2.12)} \int _{0}^{t}\! \frac{P\!\left( s,T\right) }{\beta _{s}} A\!\left( s,T\right) \, ds + 2 \int _{0}^{t}\! \frac{P\!\left( s,T\right) }{\beta _{s}} {\text {Tr}}\!\left[ \Sigma \!\left( s,T\right) \! \sqrt{X_{s}}\, \gamma _{s}\, Q\right] ds \nonumber \\&\quad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\! \frac{P\!\left( s-,T\right) }{\beta _{s}}\left( e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \,\xi \right] }\! -\! 1\right) \left( K\!\left( s,\xi \right) \!-\!1\right) \left( m\!\left( d\xi \right) \! +\! {\text {Tr}}\!\left[ X_{s} \mu \!\left( d\xi \right) \right] \right) ds. \end{aligned}$$

It follows for all $0\le t\le T$ that

$$\begin{aligned} A\!\left( t,T\right)&= - 2 {\text {Tr}}\!\left[ \Sigma \!\left( t,T\right) \! \sqrt{X_{t}}\, \gamma _{t}\, Q\right] \nonumber \\&\quad - \int _{S_{d}^{+}\! \setminus \left\{ 0\right\} \!}\!\left( e^{{\text {Tr}}\left[ \Sigma \left( t,T\right) \,\xi \right] }\! -\! 1\right) \left( K\!\left( t,\xi \right) \!-\!1\right) \left( m\!\left( d\xi \right) \! +\! {\text {Tr}}\!\left[ X_{t} \mu \!\left( d\xi \right) \right] \right) \end{aligned}$$

$dt \otimes d\mathbb {P}$-a.s.

Consequently we get for all $0\le t\le T$ by Satz 6.28 in [42]

$$\begin{aligned} \alpha \!\left( t,T\right)&\mathop {=}\limits ^{(3.4)} - \partial _{T} A\!\left( t,T\right) - \partial _{T} F\!\left( -\Sigma \!\left( t,T\right) \right) - \partial _{T} {\text {Tr}}\!\left[ R\!\left( -\Sigma \!\left( t,T\right) \right) X_{t}\right] \\&\mathop {=}\limits ^{(5.9)} -\! {\text {Tr}}\!\left[ \sigma \!\left( t,T\right) \left( b + B\!\left( X_{t}\right) + 2 \sqrt{X_{t}}\, \gamma _{t}\, Q\right) \right] \! -\! 4 {\text {Tr}}\!\left[ Q\, \sigma \!\left( t,T\right) X_{t}\, \Sigma \!\left( t,T\right) Q^{\top }\right] \\&\quad - \int _{S_{d}^{+}\! \setminus \left\{ 0\right\} \!}\! {\text {Tr}}\left[ \sigma \!\left( t,T\right) \xi \right] e^{{\text {Tr}}\left[ \Sigma \left( t,T\right) \,\xi \right] } K\!\left( t,\xi \right) \left( m\!\left( d\xi \right) \! +\! {\text {Tr}}\!\left[ X_{t} \mu \!\left( d\xi \right) \right] \right) \end{aligned}$$

$dt \otimes d\mathbb {P}$-a.s.

Hence, $\frac{P\left( t,T\right) }{\beta _{t}}$, $t \le T$, is a $\mathbb {Q}$-local martingale if and only if equation (3.10) is fulfilled $dt \otimes d\mathbb {P}$-a.s. Eq. (3.10) represents the HJM condition on the drift in the affine setting on $S_{d}^{+}$. Then, the forward rate under $\mathbb {Q}$ follows a process of the form

$$\begin{aligned} f\!\left( t,T\right)&\mathop {=}\limits _{(3.1)}^{(2.11)} f\!\left( 0,T\right) + \int _{0}^{t}\! \alpha \!\left( s,T\right) ds + \int _{0}^{t}\! {\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \left( b + B\!\left( X_{s}\right) + 2 \sqrt{X_{s}}\, \gamma _{s}\, Q\right) \right] ds \\&\quad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\! {\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \xi \right] \, \mu ^{X\!}\!\left( ds,d\xi \right) + 2 \int _{0}^{t}\! {\text {Tr}}\left[ \sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW^{*}_{s}\, Q\right] \\&\mathop {=}\limits ^{(3.10)} f\!\left( 0,T\right) - 4 \int _{0}^{t}\! {\text {Tr}}\left[ Q\, \sigma \!\left( s,T\right) \, X_{s}\, \Sigma \!\left( s,T\right) \, Q^{\top }\right] \,ds\\&\quad - \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!{\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \xi \right] e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \,\xi \right] }K\!\left( s,\xi \right) \nu \!\left( ds,d\xi \right) \\&\quad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\! {\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \xi \right] \, \mu ^{X\!}\!\left( ds,d\xi \right) + 2 \int _{0}^{t}\! {\text {Tr}}\left[ \sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW^{*}_{s}\, Q\right] \nonumber \\&\mathop {=}\limits ^{(3.3)} f\!\left( 0,T\right) + 4 \int _{0}^{t}\! {\text {Tr}}\left[ Q\, \sigma \!\left( s,T\right) X_{s}\, \int _{s}^{T}\!\sigma \!\left( s,u\right) du\ Q^{\top }\right] \,ds\\&\quad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!{\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \xi \right] \left( \mu ^{X\!}-\nu ^{*\!}\right) \!\left( ds,d\xi \right) \\&\quad - \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!{\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \xi \right] \left( e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \,\xi \right] } - 1\right) \nu ^{*\!}\!\left( ds,d\xi \right) \\&\quad + 2 \int _{0}^{t}\! {\text {Tr}}\left[ \sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW^{*}_{s}\, Q\right] \\&\mathop {=}\limits ^{(2.12)} f\!\left( 0,T\right) + \int _{0}^{t}\! \left\{ 4 {\text {Tr}}\left[ Q\, \sigma \!\left( s,T\right) X_{s}\, \int _{s}^{T}\!\sigma \!\left( s,u\right) du\ Q^{\top }\right] \right. \\&\quad \left. - \int _{S_{d}^{+}\! \setminus \left\{ 0\right\} \!}\! K\!\left( s,\xi \right) {\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \xi \right] \left( e^{{\text {Tr}}\left[ \Sigma \left( s,T\right) \,\xi \right] }\! -\! 1\right) \left( m\!\left( d\xi \right) \! +\! {\text {Tr}}\!\left[ X_{s} \mu \!\left( d\xi \right) \right] \right) \right\} ds \\&\quad + \int _{0}^{t}\!\int _{S_{d}^{+}\! \setminus \left\{ 0\right\} }\!{\text {Tr}}\!\left[ \sigma \!\left( s,T\right) \xi \right] \left( \mu ^{X\!}-\nu ^{*\!}\right) \!\left( ds,d\xi \right) \\&\quad + 2 \int _{0}^{t}\! {\text {Tr}}\left[ \sigma \!\left( s,T\right) \sqrt{X_{s}}\, dW^{*}_{s}\, Q\right] , \end{aligned}$$

where we have used again Proposition 1.28 of Chap. II in [39]. $\square$