Saddlepoint Approximation for Credit Portfolios

Abstract

We consider a portfolio of loans or bonds, where the loan borrowers or bond issuers may fail to meet the promised cashflows as stated in the loan contracts or bond indentures. These payment defaults lead to credit losses to the holder of the portfolio of these credit instruments or names (loans or bonds).

Appendix

5.1.1 Proof of formula (5.20)

The proof follows from Huang and Oosterlee (2011). Let $\mu =E[L]$ and consider

$$ E[L\mathbf {1}_{\{L\ge t\}}]=\int L \mathbf {1}_{\{L\ge t\}}\ \text {d}P=\mu \int {L\over \mu }\mathbf {1}_{\{L\ge t\}}\ \text {d}P. $$

Assume that L has a positive lower bound and we define a new measure Q on the same filtered probability space $(\varOmega ,\mathscr {F})$ such that

$$ Q(A)=\int _A{L\over \mu }\ \text {d}P\quad \text {for}\quad A\in \mathscr {F}. $$

The mgf of L under Q is given by

$$ M_L^Q(z)=\int e^{zL}{L\over \mu }\ \text {d}P={M'_L(z)\over \mu }={1\over \mu }{\text {d}\over \text {d}z}e^{\kappa _L(z)}={M_L(z)\over \mu }\kappa '_L(z). $$

The cgf of L under the two measures P and Q are related by

$$ \kappa _L^Q(z)=\log M_L^Q(z)=\kappa _L(z)+\log \kappa '_L(z)-\ln \mu . $$

For $\gamma \in (0,\alpha _+)$, we recall from (1.13b) that the conditional expectation of X in the tail region is given by

$$\begin{aligned} E[L\mathbf {1}_{\{L\ge t\}}]&=\mu Q[L\ge t]\\&={\mu \over 2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty }{e^{\kappa _L^Q(z)-zt}\over z}\ \text {d}z\\&={1\over 2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty }\kappa '_L(z){e^{\kappa _L(z)-zt}\over z}\ \text {d}z. \end{aligned}$$

Next, we show how to extend the result to the scenario where L has a negative lower bound $-B$, where $B>0$. Define $X=L+B$ so that X has a positive lower bound. We observe

$$ \kappa _X(z)=\kappa _L(z)+Bz\quad \text {and}\quad \kappa '_X(z)=\kappa '_L(z)+B $$

so that

$$ E\left[ L\mathbf {1}_{\{L\ge t\}}\right] =E\left[ X\mathbf {1}_{\{X-B\ge t\}}\right] -BP[X-B\ge t]. $$

On the other hand, we obtain

$$\begin{aligned} E\left[ X\mathbf {1}_{\{X-B\ge t\}}\right]&={1\over 2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty }[\kappa '_L(z)+B]{\exp (\kappa _L(z)+Bz-z(t+B))\over z}\ \text {d}z\\&={1\over 2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty }\kappa '_L(z){e^{\kappa _L(z)-zt}\over z}\ \text {d}z+BP[X-B\ge t]. \end{aligned}$$

Combining the last two equations, we obtain

$$ E\left[ L\mathbf {1}_{\{L\ge t\}}\right] ={1\over 2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty }\kappa '_L(z){e^{\kappa _L(z)-zt}\over z}\ \text {d}z. $$

When L is unbounded, we consider the truncated random variable $L_{K}=\max (L, K)$ so that L is bounded from below. We can show that $\displaystyle E\left[ L\mathbf {1}_{\{L\ge t\}}\right] =E\left[ L_K\mathbf {1}_{\{L_{K}\ge t\}}\right]$ using a similar procedure as above. Lastly, we take $L\rightarrow \infty$ and apply the monotone convergence theorem to show that $\displaystyle E\left[ L\mathbf {1}_{\{L\ge t\}}\right]$ has the same integral representation as that of bounded L.