## 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).

### Introduction

The Black-Scholes' equivalent in the Credit portfolio or basket world is the Vasicek's Gaussian Homogeneous Portfolio model (HP), often used under the assumption of Large portfolio, and called LHP (Large Homogeneous Portfolio model). The model has created industries of its own, Basel's capital requirements (RWA) formula being one example. The model gives Black-Scholes type analytical formula for computing the price of CDOs (Credit Default Obligations), and k-th to default derivatives.

We give a detailed account of the assumptions underlying the Vasicek's LHP framework. We then derive the Vasicek's portfolio loss distribution (Black-Scholes' equivalent is the stock price distribution), followed by the loss density and loss quantile (famous Basel formula).

We then use these formulae in addition to the Risk Neutral Valuation to derive a formula for the price of a European call option on the portfolio loss, which then enables us to derive a formula for the price and survival rate of a simplified CDO tranche, and ultimately a CDO swap. We then differentiate these formula to compute the Greeks formulae.

We then proceed to derive the formulae for the price and Greeks of a k-th (or n-th) to default derivative. The derivation parallels the CDO section; however, here we have to drop the Large portfolio assumption, leaving us with just Homogeneous Portfolio. This is because the kth-to-default instruments reference a finite portfolio of names, and the assumption of large portfolio will not be meaningful. This changes the statistical distributions, and consequently the numerical computations, though the overall approach is similar to the CDO section.

As usual, before delving into the details, we list the relevant articles and books for references and further reading:

• Vasicek, O. (1987), Probability of loss on loan portfolio, KMV Corporation.
• Vasicek, O. (1991), Limiting loan loss probability distribution, KMV Corporation.
• Vasicek, O. (2002), Loan portfolio value, Risk.net.
• O'Kane, D. and Livesey, M. (2004), Base correlation explained, Lehman Brothers Fixed Income Research.
• O'Kane, D. (2008). Modelling single-name and multi-name credit derivatives. Chichester, West Sussex: John Wiley & Sons.