i have some issues trying to prove this question.
Suppose we are in the Black-Scholes framework and that r = 0 so that the stock price dynamics, under the risk-neutral measure Q, is given by
d$S_t$ = $\sigma$$S_t$d$W_t$
Let Q' be defined by the likelihood ratio
$\frac{dQ'}{dQ}$ = $\frac{S_t}{S_0}$ on $F_t$, 0 <= t <= T
Show that $\frac{1}{S_0}$$E^Q$[$V_T$] = $E^{Q'}$[$(1-min $$P_t$$)^+$] where $P_t$= $\frac{S_0}{S_t}$ for all t => 0
I'm stuck trying to show that that two different functions will obtain the same minimum. Will appreciate if someone can help with this.
Thanks!