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?