Assume a portfolio that contains some asset A and that I am contemplating hedging my delta in A by taking a position in asset B. I would determine how much of B to buy/sell based on the linear correlation between the returns of the two assets. But, from a risk management perspective, the actual relationship between the returns of the two assets need not be linear (which is partly why rank correlation is preferable to linear correlation for simulated VaR). My question is: how do you reconcile the use of one correlation metric for trading with the use of another for measuring risk?
I think there's some semantics to be thought about first:
The word Hedging commonly implies that you want to hedge the changes in the present value of your total position ($\Pi=PV(A) +wPV(B)$), with $w$ the hedging weight. This statement can be understood locally:
I want to hedge ('immunize') my portfolio to local changes in the **underlying**
source of uncertainty.
or globally, usually understood as a hedge in the region of large overall losses:
I want to immunize my portfolio against large (negative) movements in total
If you think about these statements for a couple of seconds you may find a direct connection to risk measurement: We may measure risks locally as in case A, or in the tails of the pnl distribution, as in case B. Of course, it is usually very hard to identify the 'true' underlying generating process and its parameters, hence we apply some reasonable simplifications in the first case, or try to be extra careful ('super-hedge') in the second case.
I do not see a necessity to truly reconcile both views: One is maybe more day-to-day / trading oriented and gives some insights on how to manage your pnl (locally), and the other ansatz focuses on the tail of the pnl distribution, offering insights on how to, well, manage tail risks. In banks, I usually see three approaches being measured, limited and managed at the same time, two of them empirically driven and one supplementing them:
Simple correlation based hedging / pnl 'micro' management,
tail risk estimation based on some form of 'stressed VaR',
stress tests that cover some cases that were, and some that were not yet, observed historically.