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:


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:


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


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