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It's shown in a research paper that: $$ E\left[ \Phi\left(\frac{\Phi^{-1}(PD)-\sqrt{\rho_{1}}Y}{\sqrt{1-\rho_1}}\right) \times \Phi\left(\frac{\Phi^{-1}(LGD)-\sqrt{\rho_{2}}Z}{\sqrt{1-\rho_2}}\right)\right]= \Phi_2\left( \Phi^{-1}(PD), \Phi^{-1}(LGD); \rho\sqrt{\rho_1\rho_2}\right) $$ where $Y$ and $Z$ are both standard normal distributed random variables and their correlation is $\rho$.

I was trying to derive this equation, but no success. Is there anyone could help? Thank you very much.

bozhao
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  • What is PD? What s LGD? What are $\Phi$ and $\Phi_2$? What are $\rho_1$ and $\rho_2$? That's a lot of whats ... Moreover, the question contradicts itself: it states: It's shown in a research paper that blah" and then you ask for help to show the same result. If it is shown in the paper, why would you want to show it again. And which paper? – wolfies Jun 27 '17 at 16:55
  • @wolfies sorry for the confusion. PD, LGD, $\rho_1$ and $\rho_2$ are both constant, $\Phi$ is CDF of standard normal distribution and $\Phi_2$ is bivariant standard normal distribution. It is from a internal research paper, and in the paper it just gives out this equation without detailed derivation. Hope there extra information could help. Thanks. – bozhao Jun 28 '17 at 11:52
  • If they are all constants, why do you need such clumsy and unnecessary notation? $\Phi^{-1}(PD)$ is just a constant $c_1$, $\Phi^{-1}(LGD)$ is just a constant $c_2$ etc. Next, have you checked whether it actually holds for some numerical values? – wolfies Jun 28 '17 at 15:40

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