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).

Assumptions

We will use the index i to represent a generic name/credit in a portfolio of n names/credits.

Assumption 1: The uncertainty (or innovation) in the asset value of a name i is assumed to follow the process:

$$ A_{i}=\sqrt{1-\rho} \epsilon_{i}+ \sqrt{\rho} S $$

Where both S and \( \epsilon_{i} \) are assumed to be standard normal \( N(0,1) \), and the \( \epsilon_{i} \) are assumed to be independent of of each other and of S. This equation simply means that the uncertainty in the asset value of a name is driven by an idiosyncratic/specific factor, and a systemic/common factor that drives the correlation between the names. This particular form with the square root and all that looks complicated, but this representation enables a very simple interpretation of the uncertainty in the value of an individual asset and the dependence between the assets of different names. For example, under the assumed dynamics, the uncertainty in the asset value of a name is also normal with mean and variance:

$$ E\left[ A_{i} \right]=E\left[\sqrt{1-\rho} \epsilon_{i}+ \sqrt{\rho} S \right]=\sqrt{1-\rho}E\left[ \epsilon_{i} \right]+ \sqrt{\rho} E\left[ S \right]=0 $$ $$ Var \left[ A_{i} \right]=Var \left[\sqrt{1-\rho} \epsilon_{i}+ \sqrt{\rho} S \right]=\left( 1-\rho \right) Var \left[ \epsilon_{i} \right]+ \rho Var \left[ S \right]=1 $$

and the correlation between any two names, say i and j, is:

$$ Cor \left[ A_{i}, A_{j} \right]=\frac{Cov \left[ A_{i}, A_{j} \right]}{Std\left[A_{i}\right] Std\left[A_{j}\right]}=Cov \left[ A_{i}, A_{j} \right]=Cov \left[\sqrt{1-\rho} \epsilon_{i}+ \sqrt{\rho}S,\sqrt{1-\rho} \epsilon_{j}+ \sqrt{\rho}S \right]=Cov \left[\sqrt{\rho}S, \sqrt{\rho}S \right] =\rho \, Var \left[ S\right] =\rho $$

You can see the assumed form gives very parsimonious dynamics of the assets of individual names and the dependence among names.

Assumption 2: Now we move to the second assumption of the model, which says, in the spirit of Merton, that a name defaults when the innovation in its asset is below a threshold:

$$ PD_{i}=Prob \left[ A_{i} < Threshold_{i} \right] $$ $$ PD_{i}=Prob \left[ \sqrt{1-\rho} \epsilon_{i}+ \sqrt{\rho} S < Threshold_{i} \right] $$

We will need two additional assumptions, which we state below, but the role that they play in the framework will become apparent when we use these assumptions to simplify the formula, so we merely state the assumptions here for completeness:

Assumption 3: The individual names are homogeneous (interchangeable).

Assumption 4: The portfolio is very large

Before we proceed to the derivation of the loss distribution, it is important to discuss the relationship between PD and the systemic factor. As this is a portfolio model, we should not be surprised to see that the systemic factor plays the most important part. After all this is the factor that drives the joint defaults dynamics because it is the only source of correlation/dependence among names. Conditional on the realisation of the systemic factor, the individual names are independent. The conditional PD is given by:

$$ P\left[ {D_{i} \mid S} \right]=Prob \left[ \sqrt{1-\rho} \epsilon_{i}+ \sqrt{\rho} S < Threshold_{i} \right] $$ $$ P\left[ {D_{i} \mid S} \right]=Prob \left[\epsilon_{i}< \frac{Threshold_{i}-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] $$ $$ P\left[ {D_{i} \mid S} \right]=N\left[\frac{Threshold_{i}-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] $$

We can also invert this relationship to give S in terms of PD:

$$ P\left[ {D_{i} \mid S} \right]=N\left[ \frac{Threshold_{i}-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] $$ $$ N^{-1} \left[ P\left[ {D_{i} \mid S} \right] \right]=\frac{Threshold_{i}-\sqrt{\rho} S}{\sqrt{1-\rho} }$$ $$ S =\frac{Threshold_{i}-\sqrt{1-\rho} N^{-1} \left[ P\left[ {D_{i} \mid S} \right] \right]}{\sqrt{\rho}}$$ $$ S =-\frac{\sqrt{1-\rho} N^{-1} \left[ P\left[ {D_{i} \mid S} \right] \right]-Threshold_{i}}{\sqrt{\rho}}$$