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).

Simplified Single Tranche CDO: Greeks

As was shown in the price section, the simplified CDO is simply the linear combination of two options on portfolio loss. Thus, to simplify the presentation, we derive the formulae for the derivatives of the price of an option providing protection against losses in excess of a K, the price of which is given by:

$$ V_{0}=e^{-r T} LGD \; N_{2} \left[N^{-1} \left[ PD \right] , A_{K}, \sqrt{\rho} \right] -K e^{-r T}N\left[ A_{K} \right] $$

Where

$$ A_{K}=\frac{N^{-1} \left[ PD \right] -\sqrt{1-\rho} N^{-1} \left[ \frac{K}{LGD} \right]}{\sqrt{\rho}}$$

To lighten the notation, we rewrite it as:

$$ V_{0}=e^{-r T} LGD \; N_{2} \left[a , b, \sqrt{\rho} \right] -K e^{-r T}N\left[ b \right] $$

Where

$$ a=N^{-1} \left[ PD \right] $$ $$ b= A_{K}=\frac{N^{-1} \left[ PD \right] -\sqrt{1-\rho} N^{-1} \left[ \frac{K}{LGD} \right]}{\sqrt{\rho}}$$

Derivative of Bivariate Normal Distribution

Derivatives of \( N_{2}\) will be needed for every differentiation we attempt here, so we first list the derivatives of the Bivariate Normal with respect to its three parameters. The bivariate normal CDF formula is:

$$ N_{2} \left[a , b, \rho \right]=\int_{-\infty}^{a}{ \int_{-\infty}^{b} {f\left(x,y, \rho \right)dy}dx} $$ $$ f\left(x,y, \rho \right)=\frac{1}{2 \pi \sqrt{1-\rho^2}} e^{-\frac{x^2-2\rho x y +y^2}{2 \left(1-\rho^2 \right)}}$$

And its derivatives are (let us know if you are interested in their derivation because we have done the leg work):

$$ \frac{\partial N_{2} \left[a , b, \rho \right]}{\partial a}= n\left[ a \right] N \left[ \frac{b-\rho a}{\sqrt{1-\rho^2 )}} \right] $$ $$ \frac{\partial N_{2} \left[a , b, \rho \right]}{\partial b}= n\left[ b \right] N \left[ \frac{a-\rho b}{\sqrt{1-\rho^2 )}} \right] $$ $$ \frac{\partial N_{2} \left[a , b, \rho \right]}{\partial \rho}= f \left(a,b,\rho \right) $$

Derivative wrt \( \rho \)

Differentiating with respect to \( \rho \), we get:

$$ \frac{\partial V_{0}}{\partial \rho} = \frac{\partial}{\partial \rho} \left( e^{-r T} LGD \; N_{2} \left[a,b,\sqrt{\rho}\right] -K e^{-r T}N\left[ b \right] \right)$$ $$ = e^{-r T} LGD \left( \frac{\partial N_{2} \left[a,b,\sqrt{\rho}\right]}{\partial \sqrt{\rho}} \frac{\partial \sqrt{\rho}}{\partial \rho} + \frac{\partial N_{2} \left[a,b,\sqrt{\rho}\right]}{\partial b} \frac{\partial b}{\partial \rho} \right)-K e^{-r T} \frac{\partial N\left[ b \right]}{\partial b} \frac{\partial b }{\partial \rho} $$ $$ = e^{-r T} LGD \left( f\left( a,b,\sqrt{\rho} \right) \frac{\partial \sqrt{\rho}}{\partial \rho} + n\left[b\right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] \frac{\partial b}{\partial \rho} \right)-K e^{-r T} n\left[ b \right] \frac{\partial b }{\partial \rho} $$ $$ = e^{-r T} LGD \frac{1}{2 \sqrt{\rho}}f\left( a,b,\sqrt{\rho} \right)+ e^{-r T} \left( LGD \; n\left[b\right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] -K n\left[ b \right] \right) \frac{\partial b}{\partial \rho} $$ $$ = \frac{e^{-r T} LGD}{2 \sqrt{\rho}}f\left( a,b,\sqrt{\rho} \right)+ e^{-r T} \left( LGD \; n\left[b\right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] -K n\left[ b \right] \right) \left( -\frac{1}{{2 \rho \sqrt{\rho}}} \left( N^{-1} \left[PD\right] + N^{-1}\left[ \frac{K}{LGD}\right] \frac{2\rho-1}{\sqrt{1-\rho}} \right)\right) $$ $$ = \frac{e^{-r T} LGD}{2 \sqrt{\rho}}f\left( a,b,\sqrt{\rho} \right)-\frac{e^{-r T}}{{2 \rho \sqrt{\rho}}} \left( LGD \; n\left[b\right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] -K n\left[ b \right] \right) \left( N^{-1} \left[PD\right] + N^{-1}\left[ \frac{K}{LGD}\right] \frac{2\rho-1}{\sqrt{1-\rho}} \right) $$

Where we used

$$ b=\frac{N^{-1} \left[PD\right] -\sqrt{1-\rho} N^{-1} \left[ \frac{K}{LGD} \right]}{\sqrt{\rho}} =\frac{N^{-1} \left[PD\right]}{\sqrt{\rho}} - N^{-1}\left[ \frac{K}{LGD}\right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}}$$ $$ \frac{\partial b}{\partial \rho}=-\frac{N^{-1} \left[PD\right]}{2 \rho \sqrt{\rho}} - N^{-1}\left[ \frac{K}{LGD}\right] \frac{\sqrt{\rho} \frac{\partial}{\partial \rho} \sqrt{1-\rho} - \sqrt{1-\rho} \frac{\partial}{\partial \rho} \sqrt{\rho} }{\rho}$$ $$ =-\frac{N^{-1} \left[PD\right]}{2 \rho \sqrt{\rho}} - N^{-1}\left[ \frac{K}{LGD}\right] \frac{\frac{\sqrt{\rho}}{2 \sqrt{1-\rho}} - \frac{\sqrt{1-\rho}}{2\sqrt{\rho}}}{\rho}$$ $$ =-\frac{N^{-1} \left[PD\right]}{2 \rho \sqrt{\rho}} - N^{-1}\left[ \frac{K}{LGD}\right] \frac{2\rho-1}{2 \rho \sqrt{\rho} \sqrt{1-\rho}}$$ $$ =-\frac{1}{{2 \rho \sqrt{\rho}}} \left( N^{-1} \left[PD\right] + N^{-1}\left[ \frac{K}{LGD}\right] \frac{2\rho-1}{\sqrt{1-\rho}} \right)$$

Derivative with respect to PD

Now, differentiating with respect to PD, we get:

$$ \frac{\partial V_{0}}{\partial PD} = \frac{\partial}{\partial PD} \left( e^{-r T} LGD \; N_{2} \left[a,b,\sqrt{\rho}\right] -K e^{-r T}N\left[ b \right] \right)$$ $$ = e^{-r T} LGD \left( \frac{\partial N_{2} \left[a,b,\sqrt{\rho}\right]}{\partial a} \frac{\partial a}{\partial PD} + \frac{\partial N_{2} \left[a,b,\sqrt{\rho}\right]}{\partial b} \frac{\partial b}{\partial PD} \right)-K e^{-r T} \frac{\partial N\left[ b \right]}{\partial PD} \frac{\partial b }{\partial PD} $$ $$ = e^{-r T} LGD \left( n\left[ a \right] N \left[ \frac{b-\sqrt{\rho} a}{\sqrt{1-\rho}} \right] \frac{\partial a}{\partial PD} + n\left[ b \right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] \frac{\partial b}{\partial PD} \right)-K e^{-r T} n\left[ b \right] \frac{\partial b }{\partial PD} $$ $$ = e^{-r T} LGD \left( n\left[ a \right] N \left[ \frac{b-\sqrt{\rho} a}{\sqrt{1-\rho}} \right] \frac{1}{n\left[N^{-1} \left[ PD \right]\right]} + n\left[ b \right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] \frac{1}{\sqrt{\rho} n\left[N^{-1} \left[ PD \right]\right]} \right)-K e^{-r T} n\left[ b \right] \frac{1}{\sqrt{\rho} n\left[N^{-1} \left[ PD \right]\right]} $$ $$ = \frac{e^{-r T}}{n\left[N^{-1} \left[ PD \right]\right]} \left( LGD n\left[ a \right] N \left[ \frac{b-\sqrt{\rho} a}{\sqrt{1-\rho}} \right] + \frac{LGD}{\sqrt{\rho}} n\left[ b \right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] - \frac{K}{\sqrt{\rho}}n\left[ b \right] \right) $$

Where, we used

$$ b =\frac{N^{-1} \left[PD\right]}{\sqrt{\rho}} - N^{-1}\left[ \frac{K}{LGD}\right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}}$$ $$ \frac{\partial b}{\partial PD}= \frac{1}{\sqrt{\rho}} \frac{\partial }{\partial PD} N^{-1}\left[ PD \right] $$ $$ = \frac{1}{\sqrt{\rho} n\left[N^{-1} \left[ PD \right]\right]} $$

And

$$ a=N^{-1} \left[ PD \right] $$ $$ \frac{\partial a}{\partial PD}= \frac{\partial }{\partial PD} N^{-1} \left[ PD \right] $$ $$ = \frac{1}{n\left[ N^{-1} \left[ PD \right]\right]} $$

Derivative with respect to Strike

Differentiating with respect to K, we get:

$$ \frac{\partial V_{0}}{\partial K} = \frac{\partial}{\partial K} \left( e^{-r T} LGD \; N_{2} \left[a,b,\sqrt{\rho}\right] -K e^{-r T}N\left[ b \right] \right)$$ $$ = e^{-r T} LGD \frac{\partial N_{2} \left[a,b,\sqrt{\rho}\right]}{\partial b} \frac{\partial b}{\partial K} -K e^{-r T} \frac{\partial N\left[ b \right]}{\partial b} \frac{\partial b }{\partial K}-e^{-r T}N\left[ b \right] $$ $$ = -e^{-r T}N\left[ b \right] + e^{-r T} \left( LGD \; n\left[ b \right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] - K n\left[ b \right] \right) \frac{\partial b }{\partial K} $$ $$ = -e^{-r T}N\left[ b \right] + e^{-r T} \left( LGD \; n\left[ b \right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] - K n\left[ b \right] \right) \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{ n\left[ N^{-1}\left[ \frac{K}{LGD}\right] \right]} \frac{1}{LGD} $$

Where

$$ b =\frac{N^{-1} \left[PD\right]}{\sqrt{\rho}} - N^{-1}\left[ \frac{K}{LGD}\right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}}$$ $$ \frac{\partial b}{\partial K}= - \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{\partial }{\partial K} N^{-1}\left[ \frac{K}{LGD}\right] $$ $$ = - \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{ n\left[ N^{-1}\left[ \frac{K}{LGD}\right] \right]} \frac{\partial }{\partial K} \left( \frac{K}{LGD} \right) $$ $$ = \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{ n\left[ N^{-1}\left[ \frac{K}{LGD}\right] \right]} \frac{1}{LGD} $$

Derivative with respect to LGD

Finally, differentiating both sides with respect to LGD, we get:

$$ \frac{\partial V_{0}}{\partial LGD} = \frac{\partial}{\partial LGD} \left( e^{-r T} LGD \; N_{2} \left[a,b,\sqrt{\rho}\right] -K e^{-r T}N\left[ b \right] \right)$$ $$ = e^{-r T} N_{2} \left[a,b,\sqrt{\rho}\right]+ e^{-r T} LGD \frac{\partial N_{2} \left[a,b,\sqrt{\rho}\right]}{\partial b} \frac{\partial b}{\partial LGD} -K e^{-r T} \frac{\partial N\left[ b \right]}{\partial b} \frac{\partial b }{\partial LGD} $$ $$ = e^{-r T} N_{2} \left[a,b,\sqrt{\rho}\right]+ e^{-r T} \left( LGD \; n\left[ b \right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] -K n\left[ b \right] \right) \frac{\partial b }{\partial LGD} $$ $$ = e^{-r T} N_{2} \left[a,b,\sqrt{\rho}\right]+ e^{-r T} \left( LGD \; n\left[ b \right] N \left[ \frac{a-\sqrt{\rho} b}{\sqrt{1-\rho}} \right] -K n\left[ b \right] \right) \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{ n\left[ N^{-1}\left[ \frac{K}{LGD}\right] \right]} \frac{K}{LGD^2} $$

Where

$$ b =\frac{N^{-1} \left[PD\right]}{\sqrt{\rho}} - N^{-1}\left[ \frac{K}{LGD}\right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}}$$ $$ \frac{\partial b}{\partial LGD}= - \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{\partial }{\partial LGD} N^{-1}\left[ \frac{K}{LGD}\right] $$ $$ = - \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{ n\left[ N^{-1}\left[ \frac{K}{LGD}\right] \right]} \frac{\partial }{\partial LGD} \left( \frac{K}{LGD} \right) $$ $$ = \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{ n\left[ N^{-1}\left[ \frac{K}{LGD}\right] \right]} \frac{K}{LGD^2} $$