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Pricing cross-currency interest rate swaps under the Levy market model

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Abstract

This paper derives a pricing model for interest rate swaps when the underlying markets and settlement currency can be set arbitrarily. Using the risk-neutral valuation method developed by Musiela and Rutkowski (Martingale methods in financing modelling, 2nd edn, Springer, New York, 2005), the authors generate arbitrage-free prices for a Levy market. The Levy processes are attractive because they support better statistical fits than a Gaussian economy. A closed-form solution for the swap value results from replicating the payment at each settlement date. The results then show that the domestic and foreign term structures are important factors in the pricing model; the swap value contains a correction term that reflects the currency hedging cost for the correlation between interest rates and the exchange rate.

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Acknowledgements

The authors would like to thank the editors and the anonymous referee for their helpful comments and suggestions that substantially improve the manuscript. Financial support from the Ministry of Science and Technology of Taiwan is also acknowledged (Grant No. 96-2416-H-260-010).

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Correspondence to Ming-Chieh Wang.

Appendices

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.

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Wang, MC., Huang, LJ. Pricing cross-currency interest rate swaps under the Levy market model. Rev Deriv Res 22, 329–355 (2019). https://doi.org/10.1007/s11147-018-9150-1

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