### 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:

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