I have a simple equivalence to prove but I just can't seem to do it. Appreciate any form of help.
I want to show:
$$E^Q\left[\left(1-\frac{\min_{0 \le t \le T}S_t}{S_T} \right)^+]\right] = E^Q\left[\left(1- \min_{0 \le t \le T} \frac{S_0}{S_t}\right)^+\right]$$
These are values involving options payoff (if that helps). The payoff of the option at time T is $$V_T=\left(S_T-\min_{0 \le t \le T }S_t\right)^+$$
I know that I am supposed to use something like $$\min_{0\le t \le T} f(T-t) = \min_{0\le T-u\le T} f(u) = \min_{0\le u\le T}f(u)$$ (with $u =T-t$).