I'm reading a book that discusses about derivatives pricing and have some doubts about a particular problem and really appreciate your advice:
Assume a non-dividend paying stock follows a geometric Brownian motion. What is the value of a contract that at maturity T pays the inverse of the stock price observed at the maturity?
Here is the solution:
Under risk-neutral measure, $dS = rSdt+σSdW(t)$. Apply Ito's lemma to $V=1/S$, we get $dV = (-r+σ^2)Vdt-σVdW(t)$. So V follows a geometric Brownian motion as well and we can apply Ito's lemma to $lnV$, we can get: $dln(V)=(-r+0.5*σ^2)dt-σdW(t)$. If I understand correctly, we don't need to find out the risk neutral measure for V because V is dependent on stock price S, and we already have the risk neutral measure for S and have $dS = rSdt+σSdW(t)$, due to fundamental theorem of asset pricing, the risk neutral measure is unique in this case and the unique risk neutral measure is derived from stock price S. I'm wondering if my understanding is correct?