I have some difficulty trying to show the answer to this question. Will be helpful if you can provide guidance on this.
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
$dS_{t} = \sigma dS_{t} dW_{t}$
Suppose the time now is 0. Consider the option whose payoff at time T is $V_{T} = ($$S_T$$ - min$$S_t$$)^+$
Let R be defined by the likelihood ratio $\frac{dR}{dQ}$ = $\frac{S_t}{S_0}$ on $F_t$, 0 <= t<= T
Show that $\frac{1}{S_0}$ $E^Q$[$V_T$] = $E^R$$[(1 - min$$P_t$$)^+$] where $P_t$ = $\frac{S_0}{S_t}$ for all t >= 0
I manage to show $E^R$$[$$\frac{1}{S_t}$$($$S_T$$ - min$$S_t$$)^+$] but i have no clue how to simplify from here.
Appreciate if someone can guide me on this.