Suppose that the payoff of some contract is $V_{T}=S_{T}-S_{T'}$ where $T'<T$ and we want to value the contract at time $t<T'$ (the situation where this arises could be a total return swap, where a fixing is anticipated at time $T'$ for the performance period $[T',T]$). Then presumably from risk-neutral pricing $$M_{t}V_{t}=E_{t}[M_{T}(S_{T}-S_{T'})]$$ for a choice of nuumeraire $M$. If we use the $T$-forward measure, then $$V_{t}=P_{t}^{T}E^{T}_{t}[S_{T}]-P_{t}^{T}E_{t}^{T}[S_{T'}]$$ where $P_{t}^{T}$ is the discount factor observed at time $t$ for the period $[t,T]$.

The first expectation is just $S_{t}-I_{t}^{T}$ where $I_{t}^{T}$ is the present value of any income $S$ pays in $[t,T]$ (we are using the fact that in the $T$-forward measure $E_{t}^{T}[S_{T}]=Fwd_{t}^{T}[S]=\frac{S_{t}-I_{t}^{T}}{P_{t}^{T}}$.

My question is how to properly calculate the second expectation by (presumably) switching to the $T'$-forward measure. Basically, the conceptual difficulty that I am having is how to make sense of the quantity $P_{T}^{T'}$ that shows up when you change numeraires: $$P_{t}^{T}E_{t}^{T}[S_{T'}]=\frac{P_{t}^{T}E_{t}^{T'}[\frac{S_{T'}}{P_{T}^{T'}}]P_{t}^{T'}}{P_{t}^{T}}=P_{t}^{T'}E_{t}^{T'}[\frac{S_{T'}}{P_{T}^{T'}}].$$

To me it seems like a misapplication of the change of numeraire formula to substitute $T'$ instead of $T$ into the new numeraire process under the expectation, even though $S$ is being evaluated at $T'$ (the overall payoff is still at time $T$). Barring that possibility, I proceed formally like $$P_{t}^{T'}E_{t}^{T'}[\frac{S_{T'}}{P_{T}^{T'}}]=P_{t}^{T'}E_{t}^{T'}[S_{T'}P_{T'}^{T}]=P_{t}^{T'}E_{t}^{T'}[\frac{S_{T'}P_{t}^{T}}{P_{t}^{T'}}],$$ but it's not clear to me at the moment how to treat this. If we just take everything out of the expectation, we get $$P_{t}^{T}E_{t}^{T'}[S_{T'}]=P_{t}^{T}\frac{S_{t}-I_{t}^{T'}}{P_{t}^{T'}}$$


The last step (taking everything outside the expectation ) is invalid. The expression $E[\frac{S_{T'}}{P^{T'}_T}]$ is the expectation of the product of 2 random variables. As such, you need a model to describe the joint evolution of the stock and the discount factor before you can go any further.


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