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).

More on Assumptions 1 and 2

The purists would argue that assumptions 1 and 2 as we have presented them are not the underlying, but rather the implications or properties of the, assumptions made by Vasicek. The core assumption is that the asset value of a name i follows geometric Brownian motion, similar to Black-Scholes stock price dynamics but with the additional specification of equi-correlation, which is required because we are dealing with a portfolio of names instead of a single stock. Thus:

$$ d A_{i,t}=\mu_{i} A_{i,t} dt + \sigma_{i} A_{i,t} d W_{i,t} $$

The solution of which is:

$$ A_{i,T}=A_{i,0} e^{ \left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} W_{i,T}} $$ $$ A_{i,T}=A_{i,0} e^{ \left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} \sqrt{T} Z_{i,T}} $$

Again assuming that a name defaults when its asset falls below a threshold, which in this case is set to the value of liabilities/obligation,\( B_{i}\), we get:

$$ PD_{i,T}=Prob \left[ A_{i,T} < B_{i, T} \right] $$ $$ PD_{i,T}=Prob \left[ A_{i,0} e^{ \left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} \sqrt{T} Z_{i,T}}< B_{i,T} \right] $$ $$ PD_{i,T}=Prob \left[ \ln A_{i,0} + \left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} \sqrt{T} Z_{i,T}< \ln B{i,T} \right] $$ $$ PD_{i,T}=Prob \left[ Z_{i,T}< \frac{\ln B_{i,T}-\ln A_{i,0}-\left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T}{\sigma_{i} \sqrt{T}} \right] $$ $$ PD_{i,T}=Prob \left[ Z_{i,T}< Threshold_{i} \right] $$

Where we have substituted \( Threshold_{i}\) for the long expression in order to show how this alternative equates to the settings in the previous section.

Now, as per the assumption, the \( Z_{i,T} \) are correlated standard normal: each is standard normal \( N(0,1) \), but has a constant correlation \( \rho \) with the processes of the other names. And the \(A_{i} \)'s in our settings in the previous section are nothing but these correlated \( Z_{i,T}\)'s.

An implication of the above is that we can determine the \( Threshold_{i} \) either from the accounting information using \( Threshold_{i}=\frac{\ln B_{i,T}-\ln A_{i,0}-\left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T}{\sigma_{i} \sqrt{T}}\) or from PD because:

$$ PD_{i,T}=Prob \left[ Z_{i,T}< Threshold_{i} \right] $$ $$ PD_{i,T}=N \left[ Threshold_{i} \right] $$ $$ Threshold_{i} =N^{-1} \left[ PD_{i,T} \right] $$

Now the above derivation was performed under the physical or actual measure, but for pricing purposes, we will need to perform the calculations under the risk-neutral probability measure. One may recall that in the Black-Scholes world, replacing \( \mu \) with r in the stock price dynamics gives the risk-neutral dynamics, but the assets of the firms, unlike the stock, are not traded; and we thus need to make use of the market price of risk. As per the CAPM(Capital Asset Pricing Model), the expected return of any asset is given by:

$$ \mu_{i}=r + \beta_{i} \left( r_{m}-r\right)$$ $$ \mu_{i}=r + \frac{\sigma_{i,m}}{\sigma_{m}^{2}} \left( r_{m}-r\right)$$ $$ \mu_{i}=r + \frac{\rho_{i,m}\sigma_{i} \sigma_{m}}{\sigma_{m}^{2}} \left( r_{m}-r\right)$$ $$ \mu_{i}=r + \rho_{i,m} \sigma_{i} \frac{r_{m}-r }{ \sigma_{m}} $$ $$ \mu_{i}=r + \rho_{i,m} \sigma_{i} \lambda_{m} $$ $$ r=\mu_{i}- \rho_{i,m} \sigma_{i} \lambda_{m} $$

Thus the asset value dynamics under the risk neutral measure can be expressed as:

$$ d \widetilde{A}_{i,t}=\left( \mu_{i}-\rho_{i,m} \sigma_{i} \lambda_{m} \right) \widetilde{A}_{i,t} dt + \sigma_{i} \widetilde{A}_{i,t} d \widetilde{W}_{i,T} $$

The solution of which is:

$$ \tilde{A_{i,t}}=A_{i,0} e^{ \left( \mu_{i} -\rho_{i,m} \sigma_{i} \lambda_{m}- \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} \tilde{W_{i,T}}} $$ $$ \tilde{A_{i,t}}=A_{i,0} e^{ \left( \mu_{i} -\rho_{i,m} \sigma_{i} \lambda_{m}- \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} \sqrt{T} \tilde{Z_{i,T}}} $$

And the threshold is then given by:

$$ \widetilde{PD}_{i,T}=Prob \left[ \tilde{A_{i,t}} < B_{i, T} \right] $$ $$ \widetilde{PD}_{i,T}=Prob \left[ A_{i,0} e^{ \left( \mu_{i} -\rho_{i,m} \sigma_{i} \lambda_{m}- \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} \sqrt{T} \tilde{Z_{i,T}}}< B_{i,T} \right] $$ $$ \widetilde{PD}_{i,T}=Prob \left[ \ln A_{i,0} + \left( \mu_{i} -\rho_{i,m} \sigma_{i} \lambda_{m} - \frac{1}{2}\sigma_{i}^{2} \right)T + \sigma_{i} \sqrt{T} \tilde{Z_{i,T}}< \ln B{i,T} \right] $$ $$ \widetilde{PD}_{i,T}=Prob \left[ \tilde{Z_{i,T}}< \frac{\ln B_{i,T}-\ln A_{i,0}-\left( \mu_{i} -\rho_{i,m} \sigma_{i} \lambda_{m}- \frac{1}{2}\sigma_{i}^{2} \right)T}{\sigma_{i} \sqrt{T}} \right] $$ $$ \widetilde{PD}_{i,T}=N \left[ \frac{\ln B_{i,T}-\ln A_{i,0}-\left( \mu_{i} -\rho_{i,m} \sigma_{i} \lambda_{m}- \frac{1}{2}\sigma_{i}^{2} \right)T}{\sigma_{i} \sqrt{T}} \right] $$ $$ N^{-1} \left[ \widetilde{PD}_{i,T} \right]= \frac{\ln B_{i,T}-\ln A_{i,0}-\left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T}{\sigma_{i} \sqrt{T}} +\frac{\rho_{i,m} \sigma_{i} \lambda_{m}T}{\sigma_{i} \sqrt{T}} $$ $$ N^{-1} \left[ \widetilde{PD}_{i,T} \right]= \frac{\ln B_{i,T}-\ln A_{i,0}-\left( \mu_{i} - \frac{1}{2}\sigma_{i}^{2} \right)T}{\sigma_{i} \sqrt{T}} +{\rho_{i,m} \lambda_{m} \sqrt{T}} $$ $$ N^{-1} \left[ \widetilde{PD}_{i,T} \right]= N^{-1} \left[PD_{i,T} \right] +{\rho_{i,m} \lambda_{m} \sqrt{T}} $$ $$ \widetilde{PD}_{i,T} = N \left[ N^{-1} \left[PD_{i,T} \right] +{\rho_{i,m} \lambda_{m} \sqrt{T}} \right] $$