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Consider a derivative which depends on $n$ assets with price vector $X=(X^1,\dots,X^n)$. The derivative value $V_t$ is given by the function $v(t,X)$, so that the hedge ratios for the hedging portfolio are given by $\partial_iv(t,X)$ for each asset $X^i$ for $i=1,\dots,n$.

Is there anything we can say in general about the sign of $V_t-\sum_i\partial_iv(t,X)X^i_t$? If nothing, what additional structure do we need to equip the problem with in order to do so?

The motivation is that the quantity of interest is usually the value of the cash balance in a hedging portfolio. If its sign was known to be constant throughout, a potential differential between deposit and funding rates would not matter because we would only be paying either.

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