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Asia-Pacific Financial Markets

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16.04.2020 | Original Research

Structural Pricing of CoCos and Deposit Insurance with Regime Switching and Jumps

verfasst von: Olivier Le Courtois, Xiaoshan Su

Erschienen in: Asia-Pacific Financial Markets | Ausgabe 4/2020

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Abstract

In this article, we develop a semi-analytical solution for a structural model that combines jump and regime switching risk. We use an Esscher transform that is applicable to regime switching double exponential jump diffusion to move from the historical world to the risk-neutral world. Further, we define and implement a matrix Wiener–Hopf factorization associated with the latter process, allowing us to price the various components of balance sheet. We illustrate the model with a study of a bank that issues contingent convertible bonds (CoCos). Thus, we obtain valuation formulas for the bank’s equity, debt, deposits, CoCos, and deposit insurance. We also show in an illustration the respective effects of the jump risk and of regime switching on the values of all of a bank’s balance sheet components.

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The probabilistic explanations of Eqs. (3.5) and (3.6) are the same. We demonstrate the eq. (3.6) in the case $\mathfrak {R}({\bar{\beta }}_j)\ne 0$. The equation $f(-{\bar{\beta }}_j)=0$ implies that

$$\begin{aligned} M^j_t=\langle \varvec{\gamma }_{2n+j},\varvec{Y}_t\rangle e^{{\bar{\beta }}_j A_t} \end{aligned}$$

is a martingale. Because $\mathfrak {R}({\bar{\beta }}_j)< 0$, $M^j_t$ is bounded on $[0,\tau _0^-]$. Then, the Doob’s optional sample theorem yields:

$$\begin{aligned} \gamma _{2n+j,k}=E_{\varvec{s}_k}(\langle \varvec{\gamma }_{2n+j},\varvec{{\widetilde{Y}}}^-_0 \rangle )=\left\{ \begin{aligned} (\zeta ^{(\varvec{a},-)} \varvec{{\bar{\vartheta }}}_j)_k&\qquad {\text {if }} k=1,2,\ldots ,n \\ {\bar{\vartheta }}_{j,k-n}&\qquad {\text {if }} k=n+1,n+2,\ldots ,3n \\ \end{aligned} \right. . \end{aligned}$$

The Swiss banking regulatory institution points out that for the minimum capital ratio 19% of Swiss banks, 9% can compose of contingent convertible capital. Thus, we chose 10% for the proportion of CoCos in the capital structure.

The Gaver–Stehfest algorithm does the inversion on the real line. For main advantages of this algorithm see in Kou and Wang (2003). Note that the definition of the matrix Wiener–Hopf factorization also works for complex $\varvec{{\widehat{a}}}$. Thus, other numerical Laplace inversion algorithms, such as the Abate-Whitt method, can also be used.

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Titel: Structural Pricing of CoCos and Deposit Insurance with Regime Switching and Jumps
verfasst von: Olivier Le Courtois
Xiaoshan Su
Publikationsdatum: 16.04.2020
Verlag: Springer Japan
Erschienen in: Asia-Pacific Financial Markets / Ausgabe 4/2020
Print ISSN: 1387-2834
Elektronische ISSN: 1573-6946
DOI: https://doi.org/10.1007/s10690-020-09304-6

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