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I have a question about the binomial correlation measure at page 530 in Hull(2009), Options futures and other derivatives (7th Edition) which is defined for the bivariate case as:

$\beta_{AB}(T)=\frac{P_{AB}(T)-Q_A(T)Q_B(T)}{\sqrt{[Q_A(T)-Q_A(T)^2][Q_B(T)-Q_B(T)^2]}}$

where $P_{AB}$ is the joint probability of $A$ and $B$ defaulting between $0$ and $T$, and $Q_i(T)$ is the cumulative probability that company $i$ will default by time $T$.

If I want to extend to the trivariate case, should I have something which is related to partial correlation? Something like this:

$\beta_{AB,C}(T)=\frac{P_{AB}(T)-Q_{AC}(T)Q_{BC}(T)}{\sqrt{[Q_{AC}(T)-Q_{AC}(T)^2][Q_{BC}(T)-Q_{BC}(T)^2]}}$

where $Q_{ij}(T)$ is the joint cumulative probability that company $i$ and $j$ will default by time $T$.

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