How to Calculate Return Option with Forward Measure

Question

I am trying to computing the price of an option at time $t$, with payoff $X = \frac{S_{T_2}}{S_{T_1}}$, at time $T_2$, where $t < T_1 < T_2$.

Here how I compute it:

Using the forward measure $Q_{T_2}$ with settlement date $T_2$, price at $t$ is $$P_{t,T_2}E_{Q_{T_2}}\left(\frac{S_{T_2}}{S_{T_1}}| \mathcal{F}_t\right),$$ where $P_{t,T_2}$ is the price of a discount bond at $t$ maturing at $T_2$.

By the law of iterated expectation, and using the fact that price process $h_t/P_{t,T2}$ is a martingale for any contingent claim with payoff $H$ at $T_2$, I get, $$P_t = P_{t,T_2}E_{Q_{T_2}}\left(\frac{1}{S_{T_1}} \frac{S_{T_1}}{P_{T_1,T2}}| \mathcal{F}_t\right) = P_{t,T_2} E_{Q_{T_2}}\left(\frac{1}{P_{T_1,T2}}| \mathcal{F}_t\right) =P_{t,T_2} \frac{1}{P_{t,T2}} =1 $$

I find this result rather surprising (maybe because it is wrong?). If anybody has another answer or can come up with an intuitive justification of such result. I would be glad to get some insight.

Answer 1

There is a problem in your last step. Note that \begin{align*} P_{t, T_2}E_{Q_{T_2}}\left(\frac{1}{P_{T_1, T_2}} \mid \mathcal{F}_t \right) &= P_{t, T_2}E_{Q_{T_2}}\left(\frac{P_{T_1, T_1}}{P_{T_1, T_2}} \mid \mathcal{F}_t \right)\\ &=P_{t, T_2} \times \frac{P_{t, T_1}}{P_{t, T_2}}\\ &=P_{t, T_1}. \end{align*}