## Vasicek Homogeneous Portfolio

In this section, we first derive the distribution of portfolio loss under the Vasicek's assumptions. We then proceed, using the properties of the derived distribution and our knowledge of options from the Black Scholes section, to price CDOs and kth/nth to default swaps under the simplifying assumptions. The goal is to develop a good understanding of these portfolio products under the simplified settings, pretty much what we aimed to achieve in the Black Scholes sections but for options on single underlying (in contrast to portfolio).

### Simplified n-th to default swap

We now price a simplified n-th to default swap, which we will call k-th to default swap, which has just two cash flows - premium is paid at the start of contract, and the protection amount equal to $$LGD \; (= 1-R)$$ of the k-th default is paid at the maturity of the contract if the number of defaults have exceeded k. We assume homogeneous portfolio but we drop the large portfolio assumption, assuming instead that there are m credit/names in the portfolio/basket. In our settings, the conditional probability of defaults are independent, so we can use the familiar Binomial formula to compute the probabilities of the number of defaults greater than or equal to k:

$$\tilde{P} \left[ d\left( S \right)>=k \mid S \right]=\sum_{i=k}^{m}{ {m \choose i} {\left( \tilde{P} \left[ {D \mid S}\right] \right) }^{i} {\left( 1-\tilde{P} \left[ {D \mid S}\right] \right)}^{m-i} }$$

The payoff of the swap at maturity, the protection amount, conditional on the realisation of the systemic factor is:

$$Payoff(S)=LGD \; \tilde{E} \left[ 1_{d\left( S \right)>=k} \mid S \right]= LGD \; \tilde{P} \left[ d\left( S \right)>=k \mid S \right]$$

And its value today is the expected value of the payoff under the risk neutral probability measure discounted to today:

$$V_{0}=e^{-r T}\tilde{E}\left[ LGD \; 1_{d\left( S \right)>=k} \right] =e^{-r T}LGD \; \tilde{E}\left[1_{d\left( S \right)>=k}\right]$$ $$V_{0}=e^{-r T}LGD \; \tilde{E}\left[ \tilde{E}\left[1_{d\left( S \right)>=k} \mid S \right] \right]$$ $$V_{0}=e^{-r T}LGD \; \tilde{E}\left[ \sum_{i=k}^{m}{ {m \choose i} {\left( \tilde{P} \left[ {D \mid S}\right] \right) }^{i} {\left( 1-\tilde{P} \left[ {D \mid S}\right] \right)}^{m-i} } \ \right]$$ $$V_{0}=e^{-r T}LGD \; \tilde{E}\left[ \sum_{i=k}^{m}{ {m \choose i} {\left( N\left[\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] \right) }^{i} {\left( N\left[-\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] \right) }^{m-i} } \ \right]$$

We can perform the calculations using Monte Carlo, or we can cast the computation into Gaussian-Hermite form

$$V_{0}=e^{-r T}LGD \;\int_{-\infty}^{\infty} { \sum_{i=k}^{m}{ {m \choose i} {\left( N\left[\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] \right) }^{i} {\left( N\left[-\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] \right) }^{m-i} } n \left( S \right)dS }$$ $$V_{0}=e^{-r T}LGD \;\int_{-\infty}^{\infty} { \sum_{i=k}^{m}{ {m \choose i} {\left( N\left[\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] \right) }^{i} {\left( N\left[-\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] \right) }^{m-i} } \frac{e^{-\frac{S^2}{2}}}{\sqrt{2\pi}} dS }$$

Letting $$y=\frac{S}{\sqrt{2}} \;\Rightarrow S = \sqrt{2} y \;\Rightarrow d S= \sqrt{2} d y$$ , we get

$$V_{0}=e^{-r T}LGD \;\int_{-\infty}^{\infty} { \sum_{i=k}^{m}{ {m \choose i} {\left( N\left[\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{2\rho} y}{\sqrt{1-\rho} } \right] \right) }^{i} {\left( N\left[-\frac{N^{-1}\left[ \tilde{PD}_{T} \right]-\sqrt{2 \rho} y}{\sqrt{1-\rho} } \right] \right) }^{m-i} } \frac{e^{-y^2}}{\sqrt{\pi}} d y }$$

### K-th to default swap

As we did for the Single Tranche CDO, we now remove the simplifying assumption, and now assume that the premium is paid on a generic set of dates $$T_{1}, T_{2},, T_{N}$$. We will again assume that the protection amount is also paid at one of these dates if the k-th to default happen in the period preceding the date.

The key to pricing this again is the risk-neutral survival curve. Again, we have already done most of the work needed to derive the risk-neutral survival curve. We can write the basket survival probability at time $$T_{i}$$ as:

$$Q_{T_i}= 1- \tilde{P} \left[ d\left( S_{T_i} \right)>=k \mid S_{T_i} \right]$$

And its expected value under the risk neutral measure is:

$$\tilde{E}\left[ Q_{T_i} \right]= 1- \tilde{E} \left[ \tilde{P} \left[ d\left( S_{T_i} \right)>=k \mid S_{T_i} \right] \right]$$ $$\tilde{E}\left[ Q_{T_i} \right]= 1- \tilde{E} \left[ \sum_{i=k}^{m}{ {m \choose i} {\left( N\left[\frac{N^{-1}\left[ \tilde{PD}_{T_i} \right]-\sqrt{\rho} S_{T_i}}{\sqrt{1-\rho} } \right] \right) }^{i} {\left( N\left[-\frac{N^{-1}\left[ \tilde{PD}_{T_i} \right]-\sqrt{\rho} S_{T_i}}{\sqrt{1-\rho} } \right] \right) }^{m-i} } \right]$$

We can cast the computation into Gaussian-Hermite form by letting $$y=\frac{S}{\sqrt{2}} \;\Rightarrow S = \sqrt{2} y \;\Rightarrow d S= \sqrt{2} d y$$ , we get

$$\tilde{E}\left[ Q_{T_i} \right]= 1- \;\int_{-\infty}^{\infty} { \sum_{i=k}^{m}{ {m \choose i} {\left( N\left[\frac{N^{-1}\left[ \tilde{PD}_{T_i} \right]-\sqrt{2\rho} y}{\sqrt{1-\rho} } \right] \right) }^{i} {\left( N\left[-\frac{N^{-1}\left[ \tilde{PD}_{T_i} \right]-\sqrt{2 \rho} y}{\sqrt{1-\rho} } \right] \right) }^{m-i} } \frac{e^{-y^2}}{\sqrt{\pi}} d y }$$