Appendix 1: Proof of Lemma 1
In this appendix, we describe how we transformed dynamic of bond prices to follow the domestic martingale measure that we used for our model estimation.
By Eq. (3), the dynamics of the instantaneous forward rate are given by
$$ df^{i} \left( {t,T} \right) = \alpha^{i} \left( {t,T} \right)dt + \sigma^{i} \left( {t,T} \right)^{ \bot } dY_{t}, $$
and its integrated version is expressed by
$$ f^{i} \left( {t,T} \right) = f^{i} \left( {0,T} \right) + \int\nolimits_{0}^{t} {\alpha^{i} \left( {s,T} \right)ds} + \int\nolimits_{0}^{t} {\sigma^{i} \left( {s,T} \right)^{ \bot } dY_{s} } . $$
Therefore, we denote
$$ I_{t}^{i} = \ln B^{i} \left( {t,T} \right) = - \int\nolimits_{t}^{T} {f^{i} \left( {t,u} \right)du} . $$
We then have
$$ \begin{aligned} I_{t}^{i} & = - \int\nolimits_{t}^{T} {\left( {f^{i} \left( {u,u} \right) - f^{i} \left( {u,u} \right) + f^{i} \left( {t,u} \right)} \right)du} \\ & = - \int\nolimits_{t}^{T} {f^{i} \left( {u,u} \right)du} + \int\nolimits_{t}^{T} {f^{i} \left( {u,u} \right)du} - \int\nolimits_{t}^{T} {f^{i} \left( {t,u} \right)du} \\ & = - \int\nolimits_{t}^{T} {f^{i} \left( {u,u} \right)du} + \int\nolimits_{t}^{T} {\left( {f^{i} \left( {0,u} \right) + \int\nolimits_{0}^{u} {\alpha^{i} \left( {s,u} \right)ds} + \int\nolimits_{0}^{u} {\sigma^{i} \left( {s,u} \right)^{ \bot } dT_{s} } } \right)du} \\ & \quad - \,\int\nolimits_{t}^{T} {\left( {f^{i} \left( {0,u} \right) + \int\nolimits_{0}^{t} {\alpha^{i} \left( {s,u} \right)ds} + \int\nolimits_{0}^{t} {\sigma^{i} \left( {s,u} \right)^{ \bot } dY_{s} } } \right)du} \\ & = - \int\nolimits_{t}^{T} {f^{i} \left( {u,u} \right)du} + \int\nolimits_{t}^{T} {\int\nolimits_{t}^{u} {\alpha^{i} \left( {s,u} \right)dsdu} } + \int\nolimits_{t}^{T} {\int\nolimits_{t}^{u} {\sigma^{i} \left( {s,u} \right)^{ \bot } dY_{s} du} } . \\ \end{aligned} $$
Applying Fubini’s theorem to alter the order of integration, we attaining.
$$ I_{t}^{i} = - \int\nolimits_{t}^{T} {f^{i} \left( {u,u} \right)du} + \int\nolimits_{t}^{T} {\int\nolimits_{s}^{T} {\alpha^{i} \left( {s,u} \right)duds} } + \int\nolimits_{t}^{T} {\left( {\int\nolimits_{s}^{T} {\sigma^{i} \left( {s,u} \right)du} } \right)^{ \bot } dY_{s} }. $$
Consequently,
$$ \begin{aligned} I_{t}^{i} & = \int\nolimits_{0}^{t} {f^{i} \left( {u,u} \right)du} \\ & \quad - \int\nolimits_{0}^{T} {f^{i} \left( {u,u} \right)du} + \int\nolimits_{0}^{T} {\int\nolimits_{s}^{T} {\alpha^{i} \left( {s,u} \right)duds} } + \int\nolimits_{0}^{T} {\left( {\int\nolimits_{s}^{T} {\sigma^{i} \left( {s,u} \right)du} } \right)^{ \bot } dY_{s} } \\ & \quad - \int\nolimits_{0}^{t} {\int\nolimits_{s}^{T} {\alpha^{i} \left( {s,u} \right)duds} } - \int\nolimits_{0}^{t} {\left( {\int\nolimits_{s}^{T} {\sigma^{i} \left( {s,u} \right)du} } \right)^{ \bot } dY} \\ & = \int\nolimits_{0}^{t} {f^{i} \left( {u,u} \right)du} + I_{0}^{i} + \int\nolimits_{0}^{t} {A^{i} \left( {s,T} \right)ds} + \int\nolimits_{0}^{t} {\sum {^{i} \left( {s,T} \right)^{ \bot } } dY_{s} }. \\ \end{aligned} $$
Therefore, the bond price can be obtained
$$ \begin{aligned}
{{B}}^{i} \left( {{t,T}} \right) & = B^{i} \left( {0,T} \right)B_{t}^{i} \exp \left( {\int\nolimits_{0}^{t} {A^{i} \left( {s,T} \right)ds} + \int\nolimits_{0}^{t} {\sum {^{i} \left( {s,T} \right)}^{ \bot } dY_s} } \right). \\
{{B}}^{i} \left( {{t,T}} \right) & = B^{i} \left( {0,T} \right)B_{t}^{i} \exp \left( {\int\nolimits_{0}^{t} {A^{i} \left( {s,T} \right)ds} + \int\nolimits_{0}^{t} {\sum {^{i} \left( {s,T} \right)}^{ \bot }}}\right.\\
&\left.{ { \left( {b_{s} ds + c_{s} dW_{s} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {{{x}}\left( {\mu - v} \right)\left( {ds,dx} \right)} } \right)} } \right) \\
& = B^{i} \left( {0,T} \right)B_{t}^{i} \exp \left( {\int\nolimits_{0}^{t} {\left( {A^{i} \left( {s,T} \right) + \sum {^{i} \left( {s,T} \right)}^{ \bot } b_{s} } \right)ds} + \int\nolimits_{0}^{t} {S^{i} \left( {s,T} \right)^{ \bot } dW_{s} } } \right) \\
& \quad \exp \left( {\int\nolimits_{0}^{t} {\int\nolimits_{{{\mathbb{R}}^{{d}} }} {D^{i} \left( {s,x,T} \right)\left( {\mu - v} \right)\left( {ds,dx} \right)} } } \right). \end{aligned} $$
Furthermore, by Ito’s formula, we have:
$$ \begin{aligned}
\frac{{dB^{i} \left( {t,T} \right)}}{{B^{i} \left( {t - ,T} \right)}} & = \left( {r_{t}^{i} + A^{i} \left( {t,T} \right) + \sum {^{i} \left( {t,T} \right)^{ \bot } } b_{t} + \frac{1}{2}\left| {S^{i} \left( {t,T} \right)} \right|^{2} } \right)dt + S^{i} \left( {t,T} \right)^{ \bot } dW_{t} \\
& \quad + \int\nolimits_{{{\mathbb{R}}^{d} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)\left( {\mu - \nu } \right)\left( {dt,dx} \right)} \\
& \quad + \int\nolimits_{{{\mathbb{R}}^{d} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1 - D^{i} \left( {t,x,T} \right)} \right)\nu \left( {dt,dx} \right)} , \\ \end{aligned} $$
(A1)
where \( S^{i} \left( {t,T} \right) = c_{t}^{ \bot } \sum {^{i} \left( {t,T} \right)} ,\;D^{i} \left( {t,x,T} \right) = \sum {^{i} \left( {t,T} \right)^{ \bot } } x. \)
Thus we obtain the results.
Appendix 2: Proof of Lemma 2
In this appendix, we describe how we transformed the dynamic exchange rate to follow the domestic martingale measure.
By Eq. (2), the dynamics of the exchange rate are given by
$$ \frac{{dQ_{t}^{i} }}{{Q_{t - }^{i} }} = \mu_{t}^{Qi} dt + \sigma_{t}^{Qi \bot } dY_{t} = \mu_{t}^{Qi} dt + \sigma_{t}^{Qi \bot } \left( {b_{t} dt + c_{t} dW_{t} + \int\nolimits_{{{\mathbb{R}}^{d} }} {x\left( {\mu - v} \right)\left( {dt,dx} \right)} } \right). $$
The relative exchange rate \( Q_{t}^{*} \) is given by
$$ \begin{aligned} \frac{{dQ_{t}^{*} }}{{Q_{t - }^{*} }} & = \left( {r_{t}^{i} - r_{t}^{0} } \right)dt + \left[ {\left( {\mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} } \right)dt + \sigma_{t}^{Qi \bot } c_{t} dW_{t} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {\mu - \nu } \right)\left( {dt,dx} \right)} } \right] \\ & = \left[ {\left( {r_{t}^{i} - r_{t}^{0} + \mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} } \right)dt + \sigma_{t}^{Qi \bot } c_{t} dW_{t} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {\mu - \nu } \right)\left( {dt,dx} \right)} } \right] \\ & = \left[ {\left( {r_{t}^{i} - r_{t}^{0} + \mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} } \right)dt + \sigma_{t}^{Qi \bot } c_{t} \left( {dW_{t}^{0} + \beta_{t} dt} \right)} \right. \\ & \quad + \left. {\int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {\mu - v^{0} } \right)\left( {dt,dx} \right)} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\kappa \left( {t,x} \right)v\left( {dt,dx} \right)} } \right] \\ & = \left[ {\left( {r_{t}^{i} - r_{t}^{0} + \mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} + \sigma_{t}^{Qi \bot } c_{t} \beta_{t} } \right)dt + \sigma_{t}^{Qi \bot } c_{t} dW_{t}^{0} } \right. \\ & \quad + \left. {\int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {\mu - v^{0} } \right)\left( {dt,dx} \right)} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\kappa \left( {t,x} \right)v\left( {dt,dx} \right)} } \right]. \\ \end{aligned} $$
If \( \left( {r_{t}^{i} - r_{t}^{0} + \mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} + \sigma_{t}^{Qi \bot } c_{t} \beta_{t} } \right)dt + \int_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\kappa \left( {t,x} \right)v\left( {dt,dx} \right)} = 0 \), then
$$ \frac{{dQ_{t}^{*} }}{{Q_{t - }^{*} }} = \sigma_{t}^{Qi \bot } c_{t} dW_{t}^{0} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {\mu - v^{0} } \right)\left( {dt,dx} \right)} . $$
Therefore, the dynamics of the exchange rate for currency i are expressed by
$$ \begin{aligned} \frac{{dQ_{t}^{i} }}{{Q_{t - }^{i} }} & = \mu_{t}^{Qi} dt + \sigma_{t}^{Qi \bot } \left( {b_{t} dt + c_{t} dW_{t} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {x\left( {\mu - v} \right)\left( {dt,dx} \right)} } \right) \\ & = \left( {\mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} } \right)dt + \sigma_{t}^{Qi \bot } c_{t} dW_{t} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {\mu - v} \right)\left( {dt,dx} \right)} \\ & = \left( {\mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} + \sigma_{t}^{Qi \bot } c_{t} \beta_{t} } \right)dt + \sigma_{t}^{Qi \bot } c_{t} dW_{t}^{0} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {\mu - v^{0} } \right)} \\ & \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\kappa \left( {t,x} \right)v\left( {dt,dx} \right)} \\ & = \left( {r_{t}^{0} - r_{t}^{i} } \right)dt + \sigma_{t}^{Qi \bot } c_{t} dW_{t}^{0} + \int\nolimits_{{\mathbb{R}}} {\sigma_{t}^{Qi \bot } {{x}}\left( {\mu - \nu^{0} } \right)\left( {dt,dx} \right)} . \\ \end{aligned} $$
(A2)
This completes the proof of the lemma.
Appendix 3: Proof of Lemma 3
In this appendix, we detail processes by which the dynamics of the foreign bond price i, denominated in the domestic currency and currency i, that is \( Q_{t}^{i} B^{i} \left( {t,T} \right) \) and \( B^{i} \left( {t,T} \right) \), transform to follow the domestic martingale measure. The dynamics of \( Q_{t}^{i} B^{i} \left( {t,T} \right) \) are given by
$$ \begin{aligned} & \frac{{d\left( {{{Q_{t}^{i} B^{i} \left( {t,T} \right)} \mathord{\left/ {\vphantom {{Q_{t}^{i} B^{i} \left( {t,T} \right)} {B_{t}^{0} }}} \right. \kern-0pt} {B_{t}^{0} }}} \right)}}{{\left( {{{Q_{t - }^{i} B^{i} \left( {t - ,T} \right)} \mathord{\left/ {\vphantom {{Q_{t - }^{i} B^{i} \left( {t - ,T} \right)} {B_{t - }^{0} }}} \right. \kern-0pt} {B_{t - }^{0} }}} \right)}} \\ & \quad = \left( \mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} +
r_{t}^{i} + A^{i} \left( {t,T} \right) + b_{t}^{ \bot } \sum {^{i}
\left( {t,T} \right)} \right.\\ & \qquad\left.+\, \frac{1}{2}\left| {S^{i} \left( {t,T} \right)} \right|^{2} +
\sigma_{t}^{Qi \bot } c_{t} S^{i} \left( {t,T} \right) - r_{t}^{0}
\right)dt + \left( {\sigma_{t}^{Qi \bot } c_{t} + S^{i} \left( {t,T} \right)^{ \bot } } \right)dW_{t} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\sigma_{t}^{Qi \bot } x + \left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)\left( {\mu - v} \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1 - D^{i} \left( {t,x,T} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)\nu \left( {dt,dx} \right)} \\ & = \left( \mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} + r_{t}^{i} +
A^{i} \left( {t,T} \right) + b_{t}^{ \bot } \sum {^{i} \left( {t,T}
\right)} + \frac{1}{2}\left| {S^{i} \left( {t,T} \right)}
\right|^{2}\right.\\ &\quad\quad \left. +\, \sigma_{t}^{Qi \bot } c_{t} S^{i}
\left( {t,T} \right) - r_{t}^{0} \right)dt + \left( {\sigma_{t}^{Qi \bot } c_{t} + S^{i} \left( {t,T} \right)^{ \bot } } \right)\left( {dW_{t}^{0} + \beta_{t} dt} \right) \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\sigma_{t}^{Qi \bot } x + \left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)\left( {\mu - v^{0} } \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\sigma_{t}^{Qi \bot } x + \left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)\kappa \left( {t,x} \right)v\left( {dt,dx} \right)} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1 - D^{i} \left( {t,x,T} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)\nu \left( {dt,dx} \right)} . \\ \end{aligned} $$
$$ \begin{aligned}
& \text{If}\;\left( \mu_{t}^{Qi} + \sigma_{t}^{Qi \bot } b_{t} + r_{t}^{i} + A^{i} \left( {t,T} \right) + b_{t}^{ \bot } \sum {^{i} \left( {t,T} \right)} + \frac{1}{2}\left| {S^{i} \left( {t,T} \right)} \right|^{2} \right.\\
& \quad\left.+ \,\sigma_{t}^{Qi \bot } c_{t} S^{i} \left( {t,T} \right) - r_{t}^{0} \right) + \left( {\sigma_{t}^{Qi \bot } c_{t} + S^{i} \left( {t,T} \right)^{ \bot } } \right)\beta_{t} dt + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\sigma_{t}^{Qi \bot } x}\right.} \\
& \quad{ \left.{+ \left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)\kappa \left( {t,x} \right)v\left( {dt,dx} \right)} \\
& \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1 - D^{i} \left( {t,x,T} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)\nu \left( {dt,dx} \right) = 0} . \\ \end{aligned} $$
Then we have
$$ \begin{aligned} \frac{{d\left( {{{Q_{t}^{i} B^{i} \left( {t,T} \right)} \mathord{\left/ {\vphantom {{Q_{t}^{i} B^{i} \left( {t,T} \right)} {B_{t}^{0} }}} \right. \kern-0pt} {B_{t}^{0} }}} \right)}}{{\left( {{{Q_{t - }^{i} B^{i} \left( {t - ,T} \right)} \mathord{\left/ {\vphantom {{Q_{t - }^{i} B^{i} \left( {t - ,T} \right)} {B_{t - }^{0} }}} \right. \kern-0pt} {B_{t - }^{0} }}} \right)}} & = \left( {\sigma_{t}^{Qi \bot } c_{t} + S^{i} \left( {t,T} \right)} \right)dW_{t}^{0} \\ & \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\sigma_{t}^{Qi \bot } x + \left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right) + \sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)}\\ & \quad {\left( {\mu - v^{0} } \right)\left( {dt,dx} \right)} . \\ \end{aligned} $$
(A3)
Therefore,
$$ \begin{aligned} & \left( {A^{i} \left( {t,T} \right) + b_{t}^{ \bot } \sum {^{i} \left( {t,T} \right)} + \frac{1}{2}\left| {S^{i} \left( {t,T} \right)} \right|^{2} + S^{i} \left( {t,T} \right)^{ \bot } \beta_{t} } \right) \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1 - D^{i} \left( {t,x,T} \right) + \left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)\kappa \left( {t,x} \right)} \right)F\left( {t,dx} \right)} \\ & \quad = - \sigma_{t}^{Qi \bot } c_{t} S^{i} \left( {t,T} \right) - \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)\left( {1 + \kappa \left( {t,x} \right)} \right)F\left( {t,dx} \right)} . \\ \end{aligned} $$
Furthermore, the price dynamics of \( B^{i} \left( {t,T} \right) \) are given by
$$ \begin{aligned} \frac{{dB^{i} \left( {t,T} \right)}}{{B^{i} \left( {t - ,T} \right)}} & = \left( {r_{t}^{i} + A^{i} \left( {t,T} \right) + b_{t}^{ \bot } \sum {^{i} \left( {t,T} \right)} + \frac{1}{2}\left| {S^{i} \left( {t,T} \right)} \right|^{2} } \right)dt + S^{i} \left( {t,T} \right)^{ \bot } dW_{t} \\ & \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)\left( {\mu - \nu } \right)\left( {dt,dx} \right)}\\
&\quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1 - D^{i} \left( {t,x,T} \right)} \right)\nu \left( {dt,dx} \right)} \\ & = \left( {r_{t}^{i} + A^{i} \left( {t,T} \right) + b_{t}^{ \bot } \sum {^{i} \left( {t,T} \right)} + \frac{1}{2}\left| {S^{i} \left( {t,T} \right)} \right|^{2} + S^{i} \left( {t,T} \right)^{ \bot } \beta_{t} } \right)dt \\ & \quad + S^{i} \left( {t,T} \right)^{ \bot } dW_{t}^{0} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)\left( {\mu - \nu^{0} } \right)\left( {dt,dx} \right)} \\ & \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)\kappa \left( {t,x} \right)v\left( {dt,dx} \right)} \\ & \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1 - D^{i} \left( {t,x,T} \right)} \right)\nu \left( {dt,dx} \right)} \\ & = \left( {r_{t}^{i} - \sigma_{t}^{Qi \bot } c_{t} S^{i} \left( {t,T} \right)} \right)dt - \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\sigma_{t}^{Qi \bot } x\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)} \right)v^{0} \left( {dt,dx} \right)} \\ & \quad + S^{i} \left( {t,T} \right)^{ \bot } dW_{t}^{0} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{i} \left( {t,x,T} \right)}} - 1} \right)\left( {\mu - \nu^{0} } \right)\left( {dt,dx} \right)}. \\ \end{aligned} $$
(A4)
This completes the proof.
Appendix 4
In this appendix, we detail how to replicate the swap payments, as applied in Sect. 3. We use the forward measure approach to evaluate the expectation term in Eq. (12), which corresponds to Lemma 14.2.1 in Musiela and Rutkowski (2005), such that we modify the price processes and generalize them to include the jump terms.
To obtain these results, we use the dynamics of domestic and foreign bond prices in Eqs. (12) and (13), under the domestic martingale measure:
$$ \begin{aligned} \frac{{dB^{0} \left( {t,U} \right)}}{{B^{0} \left( {t - ,U} \right)}} & = r_{t}^{0} dt + S^{0} \left( {t,U} \right)^{ \bot } dW_{t}^{0} + \int\nolimits_{{{\mathbb{R}}^{d} }} {\left( {e^{{D^{0} \left( {t,x,U} \right)}} - 1} \right)\left( {\mu - \nu^{0} } \right)\left( {dt,dx} \right)}, \\ \frac{{dB^{k} \left( {t,U} \right)}}{{B^{k} \left( {t - ,U} \right)}} & = \left( {r_{t}^{k} - S^{k} \left( {t,U} \right)^{ \bot } c_{t} \zeta_{t}^{k} } \right)dt + S^{k} \left( {t,U} \right)^{ \bot } dW_{t}^{0} \\
&\quad+ \int\nolimits_{{{\mathbb{R}}^{d} }} {\left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)\left( {\mu - v^{0} } \right)\left( {dt,dx} \right)} \\ & \quad - \,\int\nolimits_{{{\mathbb{R}}^{d} }} {\left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)\zeta_{t}^{k \bot } xv^{0} \left( {dt,dx} \right)} . \\ \end{aligned} $$
The dynamics of the relative bond price \( B^{0} \left( {t,U} \right)/B^{k} \left( {t,U} \right) \) can be expressed as follows:
$$ \begin{aligned} & \frac{{d\left( {B^{0} \left( {t,U} \right)/B^{k} \left( {t,U} \right)} \right)}}{{B^{0} \left( {t - ,U} \right)/B^{k} \left( {t - ,U} \right)}} \\ & \quad = \left( {r_{t}^{0} - \left( {r_{t}^{k} - S^{k} \left( {t,U} \right)^{ \bot } c_{t} \zeta_{t}^{k} } \right)} \right)dt + \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } \left( {dW_{t}^{0} - S^{k} \left( {t,U} \right)dt} \right) \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\frac{{\left( {e^{{D^{0} \left( {t,x,U} \right)}} - 1} \right) - \left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)}}{{1 + \left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)}}} \right)\left( {\mu - \nu^{0} } \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} \left(
\frac{{\left( {e^{{D^{0} \left( {t,x,U} \right)}} - 1} \right) -
\left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)}}{{1 +
\left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)}}
\right.\\&\quad\left.- \left( \left( {e^{{D^{0} \left( {t,x,U} \right)}}
- 1} \right) - \left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1}
\right) \right) \vphantom{\frac{{\left( {e^{{D^{0} \left( {t,x,U} \right)}} - 1} \right) -
\left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)}}{{1 +
\left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)}}}\right)v^{0} \left( {dt,dx} \right) \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)\zeta_{t}^{k \bot } {{xv}}^{0} \left( {dt,dx} \right)} \\ & \quad = \left( {r_{t}^{0} - \left( {r_{t}^{k} - S^{k} \left( {t,U} \right)^{ \bot } c_{t} \zeta_{t}^{k} } \right) - \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } S^{k} \left( {t,U} \right)} \right)dt \\ & \quad \quad + \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } dW_{t}^{0} + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\mu - \nu^{0} } \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }} - \left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right) + \left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)\zeta_{t}^{k \bot } {{x}}} \right)v^{0} \left( {dt,dx} \right)} \\ & \quad = \left( {r_{t}^{0} - \left( {r_{t}^{k} - S^{k} \left( {t,U} \right)^{ \bot } c_{t} \zeta_{t}^{k} } \right) - \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } S^{k} \left( {t,U} \right)} \right)dt \\ & \quad \quad + \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } \left( {dW_{t}^{0,T} + S^{0} \left( {t,T} \right)} \right) \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\mu - e^{{ - D^{0} \left( {t,x,T} \right)}} \nu^{0,T} } \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\frac{{\left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }} - \left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right) + \left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)\zeta_{t}^{k \bot } {{x}}} \right)}}{{e^{{D^{0} \left( {t,x,T} \right)}} }}v^{0,T} \left( {dt,dx} \right)} \\ & \quad = \left( {r_{t}^{0} - \left( {r_{t}^{k} - S^{k} \left( {t,U} \right)^{ \bot } c_{t} \zeta_{t}^{k} } \right) - \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } S^{k} \left( {t,U} \right)} \right)dt \\ & \quad \quad + \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } S^{0} \left( {t,T} \right)dt + \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } dW_{t}^{0,T} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\mu - \nu^{0,T} } \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int\nolimits_{{{\mathbb{R}}^{{d}} }} \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }} + \frac{{\left( { - \left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right) + \left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)\zeta_{t}^{k \bot } {{x}}} \right)}}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right)\\
&\qquad v^{0,T} \left( {dt,dx} \right) . \\ \end{aligned} $$
(A5)
Similarly, by substituting the maturity date into this equation, we obtain the dynamics of the relative bond price \( B^{0} \left( {t,T} \right)/B^{k} \left( {t,T} \right) \). Combing the two relative bond price processes, the dynamics of \( \left( {B^{0} \left( {t,U} \right)B^{k} \left( {t,T} \right)} \right)/\left( {B^{k} \left( {t,U} \right)B^{0} \left( {t,T} \right)} \right) \) are given by
$$ \begin{aligned} & \frac{{d\left( {B^{0} \left( {t,U} \right)B^{k} \left( {t,T} \right)/\left( {B^{k} \left( {t,U} \right)B^{0} \left( {t,T} \right)} \right)} \right)}}{{B^{0} \left( {t - ,U} \right)B^{k} \left( {t - ,T} \right)/\left( {B^{k} \left( {t - ,U} \right)B^{0} \left( {t - ,T} \right)} \right)}} \\ & \quad = \left( \begin{aligned} & r_{t}^{0} - \left( {r_{t}^{k} - S^{k} \left( {t,U} \right)^{ \bot } c_{t} \zeta_{t}^{k} } \right) - \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } S^{k} \left( {t,U} \right) \\ &\quad + \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } S^{0} \left( {t,T} \right) + r_{t}^{k} - S^{k} \left( {t,T} \right)^{ \bot } c_{t} \zeta_{t}^{k} - r_{t}^{0} \\ &\quad + \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right)^{ \bot } \left( {S^{k} \left( {t,T} \right) - S^{0} \left( {t,T} \right)} \right) \hfill \\ \end{aligned} \right)dt \\ & \quad \quad + \left( {\left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right) + \left( {S^{k} \left( {t,T} \right) - S^{0} \left( {t,T} \right)} \right)} \right)^{ \bot } dW_{t}^{0,T} \\ & \quad \quad + \int_{{{\mathbb{R}}^{{d}} }} {\left( \begin{aligned} \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right) + \left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \hfill \\ \quad + \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \hfill \\ \end{aligned} \right)\left( {\mu - \nu^{0,T} } \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int_{{{\mathbb{R}}^{{d}} }} {\left( {\left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right)} \right)\nu^{0,T} \left( {dt,dx} \right)} \\ & \quad \quad + \int_{{{\mathbb{R}}^{{d}} }} \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }} + \frac{{\left( { - \left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right) + \left( {e^{{D^{k} \left( {t,x,U} \right)}} - 1} \right)\zeta_{t}^{k \bot } {{x}}} \right)}}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right)\\&\quad\quad v^{0,T} \left( {dt,dx} \right) \\ & \quad \quad - \int_{{{\mathbb{R}}^{{d}} }} {\left( {\frac{{\left( {e^{{D^{k} \left( {t,x,T} \right)}} - 1} \right)\zeta_{t}^{k \bot } {{x}}}}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right){{v}}^{0,T} \left( {dt,dx} \right)} \\ & \quad = \left( {\left( {S^{k} \left( {t,U} \right) - S^{k} \left( {t,T} \right)} \right)^{ \bot } \left( {c_{t} \zeta_{t}^{k} + S^{k} \left( {t,U} \right) - S^{0} \left( {t,U} \right)} \right)} \right)dt \\ & \quad \quad + \left( {\left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right) + \left( {S^{k} \left( {t,T} \right) - S^{0} \left( {t,T} \right)} \right)} \right)^{ \bot } dW_{t}^{0,T} \\ & \quad \quad + \int_{{{\mathbb{R}}^{{d}} }} {\left( \begin{aligned} \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right) + \left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \hfill \\ \quad + \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \hfill \\ \end{aligned} \right)\left( {\mu - \nu^{0,T} } \right)\left( {dt,dx} \right)} \\ & \quad \quad + \int_{{{\mathbb{R}}^{{d}} }} {\left( {\left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right)} \right)\nu^{0,T} \left( {dt,dx} \right)} \\ & \quad \quad + \int_{{{\mathbb{R}}^{{d}} }} {\left( \begin{aligned} &\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }} \hfill \\ & \quad + \frac{{\left( { - \left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right) + \left( {e^{{D^{k} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,T} \right)}} } \right)\zeta_{t}^{k \bot } {{x}}} \right)}}{{e^{{D^{0} \left( {t,x,T} \right)}} }} \hfill \\ \end{aligned} \right)v^{0,T} \left( {dt,dx} \right)} . \\ \end{aligned} $$
(A6)
Let
$$ \begin{aligned} g_{1}^{k} \left( {t,T,U} \right) & = \left( {S^{k} \left( {t,U} \right) - S^{k} \left( {t,T} \right)} \right)^{ \bot } \left( {c_{t} \zeta_{t}^{k} + S^{k} \left( {t,U} \right) - S^{0} \left( {t,U} \right)} \right), \\ g_{2}^{k} \left( {t,T,U} \right) & = \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \\&\quad+ \frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }} \\ & \quad + \frac{{\left( { - \left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right) + \left( {e^{{D^{k} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,T} \right)}} } \right)\zeta_{t}^{k \bot } {{x}}} \right)}}{{e^{{D^{0} \left( {t,x,T} \right)}} }} \\ & = \left( {\frac{{\left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right)\left( {e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} } \right)}}{{e^{{D^{k} \left( {t,x,U} \right)}} e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \\ & \quad + \frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} e^{{D^{0} \left( {t,x,T} \right)}} }}\left( {e^{{D^{0} \left( {t,x,T} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right) \\&\quad+ \frac{{\left( {e^{{D^{k} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,T} \right)}} } \right)\zeta_{t}^{k \bot } {{x}}}}{{e^{{D^{0} \left( {t,x,T} \right)}} }} \\ & = \left( {\frac{{\left( {e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right)\left( {e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} } \right)}}{{e^{{D^{k} \left( {t,x,U} \right)}} e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \\&\quad+ \frac{{\left( {e^{{D^{k} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,T} \right)}} } \right)\zeta_{t}^{k \bot } {{x}}}}{{e^{{D^{0} \left( {t,x,T} \right)}} }} \\ & = \frac{{\left( {e^{{D^{k} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,T} \right)}} } \right)}}{{e^{{D^{0} \left( {t,x,T} \right)}} }}\left( {e^{{D^{0} \left( {t,x,U} \right) - D^{k} \left( {t,x,U} \right)}} - 1 + \zeta_{t}^{k \bot } {{x}}} \right) \\ g_{3}^{k} \left( {t,T,U} \right) & = \left( {S^{0} \left( {t,U} \right) - S^{k} \left( {t,U} \right)} \right) + \left( {S^{k} \left( {t,T} \right) - S^{0} \left( {t,T} \right)} \right) \\ \end{aligned} $$
and
$$ \begin{aligned} g_{4}^{k} \left( {t,T,U} \right) & = \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right) + \left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right) \\ & \quad + \left( {\frac{{e^{{D^{0} \left( {t,x,U} \right)}} - e^{{D^{k} \left( {t,x,U} \right)}} }}{{e^{{D^{k} \left( {t,x,U} \right)}} }}} \right)\left( {\frac{{e^{{D^{k} \left( {t,x,T} \right)}} - e^{{D^{0} \left( {t,x,T} \right)}} }}{{e^{{D^{0} \left( {t,x,T} \right)}} }}} \right). \\ \end{aligned} $$
Then we can rewrite Eq. (A6) as
$$ \begin{aligned} & \frac{{d\left( {B^{0} \left( {t,U} \right)B^{k} \left( {t,T} \right)/\left( {B^{k} \left( {t,U} \right)B^{0} \left( {t,T} \right)} \right)} \right)}}{{B^{0} \left( {t - ,U} \right)B^{k} \left( {t - ,T} \right)/\left( {B^{k} \left( {t - ,U} \right)B^{0} \left( {t - ,T} \right)} \right)}} \\ & \quad = g_{1}^{k} \left( {t,T,U} \right)dt + g_{3}^{k} \left( {t,T,U} \right)^{ \bot } dW_{t}^{0,T} \\ & \quad \quad + \int_{{{\mathbb{R}}^{{d}} }} {g_{4}^{k} \left( {t,T,U} \right)\left( {\mu - \nu^{0,T} } \right)\left( {dt,dx} \right)} + \int_{{{\mathbb{R}}^{{d}} }} {g_{2}^{k} \left( {t,T,U} \right)\nu^{0,T} \left( {dt,dx} \right)} . \\ \end{aligned} $$
(A7)
By virtue of Eq. (A7), we get the following expression
$$ \begin{aligned} & \frac{{B^{0} \left( {T,U} \right)B^{k} \left( {T,T} \right)/\left( {B^{k} \left( {T,U} \right)B^{0} \left( {T,T} \right)} \right)}}{{B^{0} \left( {t,U} \right)B^{k} \left( {t,T} \right)/\left( {B^{k} \left( {t,U} \right)B^{0} \left( {t,T} \right)} \right)}} \\ & \quad =\displaystyle e^{{\int_{t}^{T} {g_{1}^{k} \left( {u,T,U} \right)du} + \int_{t}^{T} {\int_{{{\mathbb{R}}^{{d}} }} {g_{2}^{k} \left( {u,T,U} \right)\nu^{0,T} \left( {du,dx} \right)} } + \int_{t}^{T} {g_{3}^{k} \left( {u,T,U} \right)^{ \bot } dW_{u}^{0,T} }+ \int_{t}^{T} {\int_{{{\mathbb{R}}^{{d}} }} {g_{4}^{k} \left( {t,T,U} \right)\left( {\mu - \nu^{0,T} } \right)} \left( {du,dx} \right) - \frac{1}{2}\int_{t}^{T} {\left| {g_{3}^{k} \left( {u,T,U} \right)} \right|^{2} du} } }} \\ & \quad \quad +\displaystyle \,e^{{\int_{t}^{T} {\int_{{{\mathbb{R}}^{{d}} }} {\left( {{ \ln }\left( {1{{ + g}}_{ 4}^{{k}} \left( {t,T,U} \right)} \right) - g_{4}^{k} \left( {t,T,U} \right)} \right)} \mu \left( {du,dx} \right)} }} . \\ \end{aligned} $$
Therefore, we can evaluate a conditional expectation term and obtain the following equality:
$$ \begin{aligned} & B^{0} \left( {t,T_{j - 1} } \right)E_{{{\mathbb{P}}^{{0,T_{j - 1} }} }} \left[ {\left. {\frac{{B^{0} \left( {T_{j - 1} ,T_{j} } \right)B^{k} \left( {T_{j - 1} ,T_{j - 1} } \right)}}{{B^{0} \left( {T_{j - 1} ,T_{j - 1} } \right)B^{k} \left( {T_{j - 1} ,T_{j} } \right)}}} \right|{\mathcal{F}}_{{t}} } \right] \\ & \quad = B^{0} \left( {t,T_{j - 1} } \right)E_{{{\mathbb{P}}^{{0,T_{j - 1} }} }} \left[ {\frac{{B^{0} \left( {t,T_{j} } \right)B^{k} \left( {t,T_{j - 1} } \right)}}{{B^{k} \left( {t,T_{j} } \right)B^{0} \left( {t,T_{j - 1} } \right)}}}\right.\\ &\qquad e^{{\int\nolimits_{t}^{{T_{j - 1} }} {g_{1}^{k} \left( {u,T_{j
- 1} ,T_{j} } \right)du} + \int\nolimits_{t}^{{T_{j - 1} }}
{\int\nolimits_{{{\mathbb{R}}^{{d}} }} {g_{2}^{k} \left( {u,T_{j
- 1} ,T_{j} } \right)\nu^{{0,T_{j - 1} }} \left( {du,dx} \right)} }
+ \int\nolimits_{t}^{{T_{j - 1} }} {g_{3}^{k} \left( {u,T_{j - 1}
,T_{j} } \right)^{ \bot } dW_{u}^{{0,T_{j - 1} }} } }} \\ & \quad \quad e^{{\int\nolimits_{t}^{{T_{j - 1} }}
{\int\nolimits_{{{\mathbb{R}}^{{d}} }} {g_{4}^{k} \left( {t,T_{j
- 1} ,T_{j} } \right)\left( {\mu - \nu^{{0,T_{j - 1} }} }
\right)\left( {du,dx} \right) - \frac{1}{2}\int\nolimits_{t}^{{T_{j -
1} }} {\left| {g_{3}^{k} \left( {u,T_{j - 1} ,T_{j} } \right)}
\right|^{2} du} } } }} \\
&\qquad\left.\left.\left. {e^{{\int\nolimits_{t}^{{T_{j - 1} }}
{\int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {{ \ln }\left(
{1{{ + g}}_{ 4}^{{k}} \left( {t,T_{j - 1} ,T_{j} }
\right)} \right) - g_{4}^{k} \left( {t,T_{j - 1} ,T_{j} } \right)}
\right)\mu \left( {du,dx} \right)} } }} } \right|
\right|{\mathcal{F}}_{{t}} \vphantom{\frac{1^{1^{1}}}{2_1}} \right]\\ & \quad = B^{0} \left( {t,T_{j - 1} } \right)\frac{{B^{0} \left( {t,T_{j} } \right)B^{k} \left( {t,T_{j - 1} } \right)}}{{B^{k} \left( {t,T_{j} } \right)B^{0} \left( {t,T_{j - 1} } \right)}}e^{{\int\nolimits_{t}^{{T_{j - 1} }} {g_{1}^{k} \left( {u,T_{j - 1} ,T_{j} } \right)du} + \int\nolimits_{t}^{{T_{j - 1} }} {\int\nolimits_{{{\mathbb{R}}^{{d}} }} {g_{2}^{k} \left( {u,T_{j - 1} ,T_{j} } \right)\nu^{{0,T_{j - 1} }} \left( {du,dx} \right)} } }} \\ & \quad \quad E_{{{\mathbb{P}}^{{0,T_{j - 1} }} }} \left[ {e^{{\int\nolimits_{t}^{{T_{j - 1} }} {g_{3}^{k} \left( {u,T_{j - 1} ,T_{j} } \right)^{ \bot } dW_{u}^{{0,T_{j - 1} }} } }} e^{{\int\nolimits_{t}^{{T_{j - 1} }} {\int\nolimits_{{{\mathbb{R}}^{{d}} }} {g_{4}^{k} \left( {t,T_{j - 1} ,T_{j} } \right)\left( {\mu - \nu^{{0,T_{j - 1} }} } \right)} \left( {du,dx} \right)} - \frac{1}{2}\int\nolimits_{t}^{{T_{j - 1} }} {\left| {g_{3}^{k} \left( {u,T_{j - 1} ,T_{j} } \right)} \right|^{2} du} }} } \right. \\ & \left. {\left. {\quad \quad e^{{\int\nolimits_{t}^{{T_{j - 1} }} {\int\nolimits_{{{\mathbb{R}}^{{d}} }} {\left( {{ \ln }\left( {1{{ + g}}_{ 4}^{\text{k}} \left( {t,T_{j - 1} ,T_{j} } \right)} \right) - g_{4}^{k} \left( {t,T_{j - 1} ,T_{j} } \right)} \right)\mu \left( {du,dx} \right)} } }} } \right|{\mathcal{F}}_{{t}} } \vphantom{\frac{1^{1^{1}}}{2_1}} \right] \\ & \quad = \frac{{B^{0} \left( {t,T_{j} } \right)B^{k} \left( {t,T_{j - 1} } \right)}}{{B^{k} \left( {t,T_{j} } \right)}}e^{{\int_{t}^{{T_{j - 1} }} {g_{1}^{k} \left( {u,T_{j - 1} ,T_{j} } \right)du} + \int_{t}^{{T_{j - 1} }} {\int_{{{\mathbb{R}}^{{d}} }} {g_{2}^{k} \left( {u,T_{j - 1} ,T_{j} } \right)\nu^{{0,T_{j - 1} }} \left( {du,dx} \right)} } }}. \\ \end{aligned} $$
(A8)
Accordingly, we have obtained the solution of the expectation term under the forward martingale measure.
Appendix 5
An investor can acquire the foreign interest rate without any exchange rate risk by using financial instruments. The asset-linked foreign exchange option can be used to hedge the risk. In units of domestic currency, the payoffs of a European call option and put option at expiry date \( T \) are given by:
$$ \begin{aligned} C_{k} \left( T \right) & = \left( {Q_{k} \left( T \right) - \bar{Q}} \right)^{ + } {\text{Z}}_{k} \left( T \right), \\ P_{k} \left( T \right) & = \left( {\bar{Q} - Q_{k} \left( T \right)} \right)^{ + } {\text{Z}}_{k} \left( T \right), \\ \end{aligned} $$
where \( Q_{\,k} \left( T \right) \) is the exchange rate for market \( k \) at time \( T \), \( \bar{Q} \) is the strike price of exchange rate, and \( {\text{Z}}_{\,k} \left( T \right) \) is a financial asset for market \( k \) with its value denominated in currency \( k \). At time \( T - 1 \), with the goal of receiving the foreign asset return, the counterparty can invest one dollar in the foreign asset \( {\text{Z}}_{k} (T - 1) \) based on the exchange rate \( Q_{k} (T - 1) \). Additionally, to hedge currency risk, the party can purchase \( {1 \mathord{\left/ {\vphantom {1 {\left( {Q_{k} \left( {T - 1} \right){\text{Z}}_{k} \left( {T - 1} \right)} \right)}}} \right. \kern-0pt} {\left( {Q_{k} \left( {T - 1} \right){\text{Z}}_{k} \left( {T - 1} \right)} \right)}} \) units of put option and sell the same units of call option with the strike rate \( \overline{Q} = Q_{k} \left( {T - 1} \right) \). These options start at time \( T - 1 \) and mature at time \( T \), while the underlying asset is the foreign equity index \( {\text{Z}}_{k} \). Therefore, the investor can buy both the foreign asset and put asset-linked option, sell the call asset-linked option at time \( T - 1 \), and then obtain the foreign asset return without currency risk at time \( T \). The initial investment is
$$ \frac{{Q_{\,k} \left( {\,T - 1} \right)\,{\text{Z}}_{\,k} \left( {\,T - 1} \right) + P_{\,k} \left( {\,T - 1} \right) - C_{\,k} \left( {\,T - 1} \right)}}{{\mathop {Q_{\,k} \left( {\,T - 1} \right)\,{\text{Z}}_{\,k} \left( {\,T - 1} \right)}\limits^{{}} }}\; , $$
and the domestic-currency payoff is \( {{{\text{Z}}_{\,k} \left( {\,T} \right)\,} \mathord{\left/ {\vphantom {{{\text{Z}}_{\,k} \left( {\,T} \right)\,} {\,{\text{Z}}_{\,k} \left( {\,T - 1} \right)}}} \right. \kern-0pt} {\,{\text{Z}}_{\,k} \left( {\,T - 1} \right)}} \) at maturity.
This study expresses the foreign interest rate without currency risk as:
$$ \frac{1}{{B^{k} \left( {T_{j - 1} ,T_{j} } \right)}} = 1 + {{f^k}(T_{j - 1} ,T_{j} )}, $$
and then discounts it at time \( T_{j} - 1 \), that is
$$ \frac{{B^{0} \left( {T_{j - 1} ,T_{j} } \right)}}{{B^{k} \left( {T_{j - 1} ,T_{j} } \right)}}. $$
The value is Eq. (16) that we want to obtain.