Spectral Expansions for Credit Risk Modelling with Occupation Times
Abstract
:1. Introduction
2. Occupation Time Process for Underlying Diffusion
3. Occupation Time Process for $\mathit{F}$Diffusion
4. Occupation Time Model in the RiskNeutral Measure
5. Hazard Rate Model
6. Probability of Default
7. Implied Hazard Rate Function
8. Credit Default Swap (CDS) Spreads
9. Calibration of CDS Spreads: GBM Case
 The Black–Cox model Black and Cox (1976)
 The Alfonsi–Lelong model Alfonsi and Lelong (2012)
 The new hazard rate model (with $\lambda $ defined in (48))
10. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Explicit Expressions for a Drifted Brownian Motion
Appendix A.1. Default Probabilities
 If $x\in [\ell ,b),y\in (a,\ell ],{\tilde{\lambda}}_{n}\in (0,\alpha )$,$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle {\tilde{\varphi}}_{n,\alpha}^{\ell ,}(x){\tilde{\varphi}}_{n,\alpha}^{\ell ,}(y)=\left\{sin(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[(ba)sinh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a)+\frac{cosh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a))}{\sqrt{2(\alpha {\tilde{\lambda}}_{n})}}\right]\right.}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}{\left.cos(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[\frac{sinh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a))}{\sqrt{2{\tilde{\lambda}}_{n}}}+\left(\sqrt{\frac{\alpha {\tilde{\lambda}}_{n}}{{\tilde{\lambda}}_{n}}}(b\ell )\sqrt{\frac{{\tilde{\lambda}}_{n}}{\alpha {\tilde{\lambda}}_{n}}}(\ell a)\right)cosh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a))\right]\right\}}^{1}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}\times sin(\sqrt{2{\tilde{\lambda}}_{n}}(bx))sinh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(ya)).}\hfill \end{array}$$
 If $x\in [\ell ,b),y\in (a,\ell ],{\tilde{\lambda}}_{n}>\alpha $,$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle {\tilde{\varphi}}_{n,\alpha}^{\ell ,}(x){\tilde{\varphi}}_{n,\alpha}^{\ell ,}(y)=\left\{sin(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[(ba)sin(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a)\frac{cos(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a))}{\sqrt{2({\tilde{\lambda}}_{n}\alpha )}}\right]\right.}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}{\left.cos(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[\frac{sin(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a))}{\sqrt{2{\tilde{\lambda}}_{n}}}+\left(\sqrt{\frac{{\tilde{\lambda}}_{n}\alpha}{{\tilde{\lambda}}_{n}}}(b\ell )+\sqrt{\frac{{\tilde{\lambda}}_{n}}{{\tilde{\lambda}}_{n}\alpha}}(\ell a)\right)cos(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a))\right]\right\}}^{1}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}\times sin(\sqrt{2{\tilde{\lambda}}_{n}}(bx))sin(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(ya)).}\hfill \end{array}$$
 If $x\in [\ell ,b),y\in [\ell ,b),{\tilde{\lambda}}_{n}\in (0,\alpha )$,$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle {\tilde{\varphi}}_{n,\alpha}^{\ell ,}(x){\tilde{\varphi}}_{n,\alpha}^{\ell ,}(y)=\frac{\sqrt{2(\alpha {\tilde{\lambda}}_{n})}cosh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a))sin(\sqrt{2\tilde{\lambda}}(\ell a))\sqrt{2{\tilde{\lambda}}_{n}}cos(\sqrt{2{\tilde{\lambda}}_{n}}(\ell a))sinh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a)}{\sqrt{2{\tilde{\lambda}}_{n}}sin(\sqrt{2{\tilde{\lambda}}_{n}}(ba))}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}\times \left\{cos(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[\frac{sinh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a))}{\sqrt{2{\tilde{\lambda}}_{n}}}+\left(\sqrt{\frac{\alpha {\tilde{\lambda}}_{n}}{{\tilde{\lambda}}_{n}}}(b\ell )\sqrt{\frac{{\tilde{\lambda}}_{n}}{\alpha {\tilde{\lambda}}_{n}}}(\ell a)\right)cosh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a))\right]\right.}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}{\left.sin(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[(ba)sinh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a)+\frac{cosh(\sqrt{2(\alpha {\tilde{\lambda}}_{n})}(\ell a))}{\sqrt{2(\alpha {\tilde{\lambda}}_{n})}}\right]\right\}}^{1}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}\times sin(\sqrt{2{\tilde{\lambda}}_{n}}(bx))sin(\sqrt{2{\tilde{\lambda}}_{n}}(by)).}\hfill \end{array}$$
 If $x\in [\ell ,b),y\in [\ell ,b),{\tilde{\lambda}}_{n}>\alpha $,$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle {\tilde{\varphi}}_{n,\alpha}^{\ell ,}(x){\tilde{\varphi}}_{n,\alpha}^{\ell ,}(y)=\frac{\sqrt{2({\tilde{\lambda}}_{n}\alpha )}cos(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a))sin(\sqrt{2\tilde{\lambda}}(\ell a))\sqrt{2{\tilde{\lambda}}_{n}}cos(\sqrt{2{\tilde{\lambda}}_{n}}(\ell a))sin(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a)}{\sqrt{2{\tilde{\lambda}}_{n}}sin(\sqrt{2{\tilde{\lambda}}_{n}}(ba))}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}\times \left\{cos(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[\frac{sin(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a))}{\sqrt{2{\tilde{\lambda}}_{n}}}+\left(\sqrt{\frac{{\tilde{\lambda}}_{n}\alpha}{{\tilde{\lambda}}_{n}}}(b\ell )+\sqrt{\frac{{\tilde{\lambda}}_{n}}{{\tilde{\lambda}}_{n}\alpha}}(\ell a)\right)cos(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a))\right]\right.}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}{\left.sin(\sqrt{2{\tilde{\lambda}}_{n}}(b\ell ))\left[(ba)sin(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a)\frac{cos(\sqrt{2({\tilde{\lambda}}_{n}\alpha )}(\ell a))}{\sqrt{2({\tilde{\lambda}}_{n}\alpha )}}\right]\right\}}^{1}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \phantom{\rule{1.em}{0ex}}\times sin(\sqrt{2{\tilde{\lambda}}_{n}}(bx))sin(\sqrt{2{\tilde{\lambda}}_{n}}(by)).}\hfill \end{array}$$
Appendix A.2. CDS Spread
 If $y\in (a,\ell ]$,$$\begin{array}{cc}\hfill {\scriptstyle {\tilde{G}}_{(a,b),\alpha}^{\ell ,}(\lambda ;x,y)=}& {\scriptstyle \frac{2sinh(\sqrt{2\lambda}(bx))sinh(\sqrt{2\lambda +2\alpha}(ya))}{\sqrt{2\lambda}cosh(\sqrt{2\lambda}(b\ell ))sinh(\sqrt{2\lambda +2\alpha}(\ell a))+\sqrt{2\lambda +2\alpha}cosh(\sqrt{2\lambda +2\alpha}(\ell a))sinh(\sqrt{2\lambda}(b\ell ))},}\hfill \end{array}$$
 If $y\in [\ell ,b)$,$$\begin{array}{cc}\hfill {\scriptstyle {\tilde{G}}_{(a,b),\alpha}^{\ell ,}(\lambda ;x,y)=}& {\scriptstyle \frac{2sinh(\sqrt{2\lambda}(x\wedge ya))sinh(\sqrt{2\lambda}(bx\vee y))}{\sqrt{2\lambda}sinh(\sqrt{2\lambda}(ba))}+\frac{2sinh(\sqrt{2\lambda}(bx))sinh(\sqrt{2\lambda}(by))}{\sqrt{2\lambda}sinh(\sqrt{2\lambda}(ba))}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\scriptstyle \times \frac{\sqrt{2\lambda}cosh(\sqrt{2\lambda}(\ell a))sinh(\sqrt{2\lambda +2\alpha}(\ell a))\sqrt{2\lambda +2\alpha}cosh(\sqrt{2\lambda +2\alpha}(\ell a))sinh(\sqrt{2\lambda}(\ell a))}{\sqrt{2\lambda}cosh(\sqrt{2\lambda}(b\ell ))sinh(\sqrt{2\lambda +2\alpha}(\ell a))+\sqrt{2\lambda +2\alpha}cosh(\sqrt{2\lambda +2\alpha}(\ell a))sinh(\sqrt{2\lambda}(b\ell ))},}\hfill \end{array}$$
Notes
1  Here we distinguish the Xdiffusion from the Fdiffusion (i.e., the firm’s value process) $F:=\mathsf{F}(X)$ which is obtained through a smooth monotonic mapping $\mathsf{F}:\mathcal{I}\to \mathcal{D}$ (where $\mathcal{D}$ is the state space for the Fdiffusion) with unique inverse $\mathsf{X}={\mathsf{F}}^{1}$. 
2  The cemetery state ${\partial}^{\u2020}$ is not included in the interval $\mathcal{I}$. When the process is killed and immediately sent to the cemetery state, it stays there indefinitely. 
3  ${\mathbb{E}}_{x}[X;A]:={\mathbb{E}}_{x}\left[X{\mathbb{I}}_{A}\right]$ for any random variable X and event A. 
4  ${\mathbb{P}}_{x}\left({\tilde{X}}_{(a,b),t}\in dy\right):=\mathbb{P}\left({\tilde{X}}_{(a,b),t}\in dy{X}_{0}=x\right).$ 
5  When $\alpha =0$, we obtain the regular transition density of ${X}_{(a,b)}$ without an instantaneous killing:
$$\begin{array}{c}\hfill {p}_{(a,b)}(t;x,y)dy:={\mathbb{P}}_{x}\left({X}_{t}\in dy,{m}_{t}>a,{M}_{t}<b\right);\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{0.166667em}{0ex}}x,y\in (a,b),\end{array}$$

6  The transition and joint densities are pointwise convergent as $b\to r$:
$${\tilde{p}}_{(a,r),\alpha}^{\ell ,}(t;x,y):=\underset{b\to r}{lim}{\tilde{p}}_{(a,b),\alpha}^{\ell ,}(t;x,y),\phantom{\rule{2.em}{0ex}}{f}_{{\mathcal{A}}_{t}^{\ell ,},{X}_{t}}^{(a,r)}(u,yx):=\underset{b\to r}{lim}{f}_{{\mathcal{A}}_{t}^{\ell ,},{X}_{t}}^{(a,b)}(u,yx).$$

7  We assume there exists a riskneutral probability $\tilde{\mathbb{P}}$ equivalent to $\mathbb{P}$ so that the discounted price process of the defaultable claim is a Doob–Levy $(\tilde{\mathbb{P}},\mathbb{F})$martingale. In the GBM model, we clearly have the discounted firm’s value process ${e}^{{r}^{f}t}{F}_{t}={F}_{0}{e}^{\frac{{\sigma}^{2}}{2}t+\sigma {\tilde{W}}_{t}},t\ge 0$, as a $(\tilde{\mathbb{P}},\mathbb{F})$martingale. 
8  ${\tilde{\mathbb{E}}}_{t,\mathsf{A},x}\left[h(\tau \wedge T,{\tilde{X}}_{\tau \wedge T})\right]:=\tilde{\mathbb{E}}\left[h(\tau \wedge T,{\tilde{X}}_{\tau \wedge T}){\mathcal{A}}_{t}^{\ell ,}=\mathsf{A},{X}_{t}=x\right]$. 
9  For $0\le t<T$, Theorem 1 extends to the timet value of the credit derivative since ${D}_{t,\mathsf{A},x}^{\vartheta ,T}={D}_{0,x}^{\vartheta \mathsf{A},Tt}$, thanks to Lemma 1. 
10  ${\tau}^{(\vartheta )}\wedge T={\tau}_{a}\wedge T\phantom{\rule{0.277778em}{0ex}}(\mathrm{a}.\mathrm{s}.)$, for $\vartheta \ge T$, where the Tmaturity credit derivative price ${D}_{0,x}^{\vartheta ,T}$ corresponds to that in the Black–Cox model with default barrier $A(t)$. 
11  The name “hazard rate model” comes from the fact that we are employing a hazard rate process to model default probabilities. 
12  For $0\le t<T$, Theorem 2 extends to the timet value of the credit derivative: ${D}_{t,x}^{\alpha ,T}={D}_{0,x}^{\alpha ,Tt}$. 
13  We can easily extend it to the hazard rate model with $\lambda $ defined in (48), since
$${\tilde{\mathbb{P}}}_{x}({\tau}^{({\alpha}_{1},{\alpha}_{2})}>T)={e}^{{\alpha}_{1}T}{\int}_{a}^{\infty}{\tilde{p}}_{(a,\infty ),{\alpha}_{2}{\alpha}_{1}}^{\ell ,}(T;x,y).$$

14  Under the occupation time model, the implied hazard rate function can be computed by Laplace inverting, with respect to $\alpha $, of the numerator and denominator in (60) separately. 
15  We can easily extend it to the hazard rate model with $\lambda $ defined in (48), by sending ${r}^{f}\to {r}^{f}+{\alpha}_{1}$ and $\alpha \to {\alpha}_{2}{\alpha}_{1}$. 
16  Under the occupation time model, the implied hazard rate function can be computed by Laplace inverting, with respect to $\alpha $, of $\mathfrak{f}$ and $\mathfrak{g}$ in (64). 
17  The eigenvalues ${\{{\tilde{\lambda}}_{n}\}}_{n\ge 1}$ can be obtained numerically by the bisection or NewtonRaphson methods. 
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Variable Name  Description  Value 

${F}_{0}$  initial firm’s value  $55.59$ 
${r}^{f}$  constant riskfree rate  $5\%$ 
$\sigma $  constant volatility  $28\%$ 
$LGD$  constant Loss Given Default  $0.6$ 
Tenor  0.5  1  2  3  4  5  7  10 
CDS  11.86  15.13  21.29  28.79  37.21  45.83  60.03  73.17 
Variable  Description  Black–Cox  Occupation  A–L  Hazard Rate 

${A}_{0}$  default barrier (at time 0)  $18.96$  $13.47$  NA  $12.83$ 
$\gamma $  growth rate of barrier  $3.51\%$  $0.19\%$  11.22%  $0.32\%$ 
${L}_{0}$  occupation barrier (at time 0)  NA  23.29  28.78  38.42 
$\theta /T$  grace period relative to T  NA  0.2368  NA  NA 
${\alpha}_{1}$  killing rate (above L)  NA  NA  0.27%  $0.18\%$ 
${\alpha}_{2}$  killing rate (below L)  NA  NA  3.42%  $2.33\%$ 
loss function value  $5.70\times {10}^{6}$  $5.16\times {10}^{6}$  $3.10\times {10}^{8}$  $5.77\times {10}^{10}$ 
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Campolieti, G.; Kato, H.; Makarov, R.N. Spectral Expansions for Credit Risk Modelling with Occupation Times. Risks 2022, 10, 228. https://doi.org/10.3390/risks10120228
Campolieti G, Kato H, Makarov RN. Spectral Expansions for Credit Risk Modelling with Occupation Times. Risks. 2022; 10(12):228. https://doi.org/10.3390/risks10120228
Chicago/Turabian StyleCampolieti, Giuseppe, Hiromichi Kato, and Roman N. Makarov. 2022. "Spectral Expansions for Credit Risk Modelling with Occupation Times" Risks 10, no. 12: 228. https://doi.org/10.3390/risks10120228