## 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).

### Loss distribution

Let $$L_{i} \left( S \right)$$ the loss of a name i given the realization of the systemic factor S. Then:

$$L_{i} \left( S \right)= P\left[ {D_{i} \mid S} \right] \; E_{i} \; LGD_{i}$$

Where E represents the exposure to the name and LGD (=1-Recovery Rate) represents the Loss Given Default. Assuming m names in a portfolio, the conditional loss of the portfolio is the sum of the losses of all names in the portfolio:

$$L \left( S \right)=\sum_{i=1}^{m} { P\left[ {D_{i} \mid S}\right] \; E_{i} \; LGD_{i}}$$

And assuming each name has an E of 1, a deterministic LGD, and all names have the same LGD and the $$P\left[ {D \mid S}\right]$$ (making use of the homogeneity assumption), the expression simplifies considerably:

$$L \left( S \right)=\sum_{i=1}^{m} { P\left[ {D \mid S}\right] \; 1 \; LGD }=LGD \sum_{i=1}^{m} { P\left[ {D \mid S} \right]}=LGD \; d \left( S \right)$$

Where $$d \left( S \right)$$ represents the number of defaults given S.

Let $$l \left( S \right)$$ represent the portfolio loss rate (loss/exposure, where exposure = m as each name has exposure of 1) given the realization of the systemic factor S:

$$l \left( S \right)=\frac{L \left( S \right)}{m}=\frac{LGD \; d \left( S \right)}{m}$$

We now proceed to calculate the probability distribution of the portfolio loss rate conditional on the realization of the systemic factor S:

$$P\left[ l \left( S \right)<=x \right]=P\left[ \frac{LGD \; d \left( S \right)}{m}<=x \right]= P\left[ d \left( S \right)<= \frac{ m x}{LGD} \right]$$

$$d \left( S \right)$$, number of defaults conditional on the realization of the systemic factor S follows a binomial distribution with p equal to $$P\left[ {D \mid S}\right]$$ because the defaults of individual names conditional on the systemic factors are independent. Thus:

$$P\left[ l \left( S \right)<=x \right]= P\left[ d \left( S \right)<= \frac{ m x}{LGD} \right]=\sum_{k=1}^{\frac{ m x}{LGD}}{ {m \choose k} {\left( P\left[ {D \mid S}\right] \right) }^{k} {\left( 1-P\left[ {D \mid S}\right] \right)}^{m-k} }$$

Now the assumption of large portfolio implies means $$m \rightarrow \infty$$, and we know that the binomial distribution converges to normal distribution with mean mp and variance mp(1-p)for large m, thus:

$$P\left[ l \left( S \right)<=x \right]= N\left[ \frac{\frac{ m x}{LGD}-m P\left[ {D \mid S}\right]} {\sqrt {m P\left[ {D \mid S}\right] \left(1-P\left[ {D \mid S}\right] \right)} } \right]$$ $$P\left[ l \left( S \right)<=x \right]= N\left[ \frac{\sqrt{m}}{\sqrt {P\left[ {D \mid S}\right] \left(1-P\left[ {D \mid S}\right] \right)} } \left( \frac{ x}{LGD}-P\left[ {D \mid S}\right] \right) \right]$$ 

Making use of the large portfolio assumption, we get:

$$\lim_{m \to \infty} P\left[ l \left( S \right)<=x \right] = \left\{ \begin{array}{rl} N\left[ \infty \right] = 1 & \mbox{if} & \frac{ x}{LGD} > P\left[ {D \mid S}\right]\\ N\left[ 0 \right] = \frac{1}{2} & \mbox{if} & \frac{x}{LGD}=P\left[ {D \mid S}\right]\\ N\left[ -\infty \right] = 0 & \mbox{if} & \frac{x}{LGD} < P\left[ {D \mid S}\right]\\ \end{array} \right.$$

Essentially it means that as the portfolio becomes very large, the idiosyncratic risk vanishes, and we are just left with the systemic risk. Gross loss conditional on S becomes equal to the conditional PD, and as result, the probability of loss less than (greater than) this conditional PD is 1 (0). In terms of the realised value of the systemic factor, the relevant domain (which gives non zero probability of loss) is:

$$\frac{x}{LGD}>P\left[ {D \mid S}\right] \Rightarrow \frac{x}{LGD}>N\left[ \frac{Threshold_{i}-\sqrt{\rho} S}{\sqrt{1-\rho} } \right]$$ $$\Rightarrow S >-\frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}}$$

We can now calculate the unconditional loss distribution by integrating out S, through an application of expectation under the probability density of S:

$$P\left[ l <=x \right]= E \left[ P\left[ l \left( S \right)<=x \right] \right] = \int_{-\infty}^{\infty}{ P\left[ l \left( S \right)<=x \right] d N\left[ S \right]}$$ $$P\left[ l <=x \right]= \int_{-\frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}}}^{\infty}{ 1 d N\left[ S \right]}$$ $$P\left[ l <=x \right]= N\left[ \infty \right]-N\left[ -\frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right]$$ $$P\left[ l <=x \right]= 1-1+N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right]$$ $$P\left[ l <=x \right]= N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right]$$

### Loss density and quantile

Having derived the loss distribution function, we can calculate the loss density using straight forward calculus:

$$P\left[ l =x \right]= \frac{d P\left[ l <=x \right] }{d x} =\frac{d}{d x} N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right]$$ $$= n\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{d}{d x} N^{-1} \left[ \frac{x}{LGD} \right]$$ $$= n\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{n\left[ N^{-1} \left[ \frac{x}{LGD} \right] \right] LGD }$$

And we can calculate quantile or Value at Risk at $$\alpha$$ level using straight forward algebra:

$$P\left[ l <=x \right]=\alpha$$ $$N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right]=\alpha$$ $$\frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} = N^{-1} \left[ \alpha \right]$$ $${ N^{-1} \left[ \frac{x}{LGD} \right]} = \frac{ Threshold_{i}+ \sqrt{\rho} N^{-1} \left[ \alpha \right]}{\sqrt{1-\rho}}$$ $$x = LGD \; N\left[\frac{ Threshold_{i}+ \sqrt{\rho} N^{-1} \left[ \alpha \right]}{\sqrt{1-\rho}}\right]$$