I'm working through the following little exotic exercise and have some questions and curiosity as to whether I'm on the right track

Consider the claims $$Y_t=\frac{1}{S_t}$$ $$X=\frac{1}{S_T}$$ a) Can $Y_t$ be the arbitrage-free price of a traded derivative?

Answer?-- So this question is for some reason stumping me. I suppose it means the literal process $Y_t$ (that is, not under a risk-neutral expectation), which seems highly unlikely to be an arb free price process. I just can't seem to put it in any rigorous terms.

b) Derive an expression for the arbitrage free price process $\pi_t[X]$

Under risk-neutral valuation, we have $$\pi_t[X]=E^Q[\frac{X}{B_T}]=E^Q[\frac{\frac{1}{S_T}}{B_T}]=E^Q[\frac{1}{S_TB_T}]$$ So, here's where I had the idea to multiply both sides by $S_t$. Now, I've done a lot of problems with change of numeraire, but this really isn't that, so I'm now going to continue under the assumption that we are still under Q: $$\pi_t[X]=\frac{1}{S_t}E^Q[\frac{S_t}{S_TB_T}]<=>$$ $$\pi_t[X]=\frac{1}{S_t}E^Q[e^{-r(T-t)+(\frac{1}{2}\sigma^2-r)(T-t)-\sigma(W_T-W_t)}]<=>$$ $$\pi_t[X]=\frac{1}{S_t}E^Q[e^{(\frac{1}{2}\sigma^2-2r)(T-t)-\sigma(W_T-W_t)}]$$ Using the fact that $E[e^{\mu+\sigma Z}]=e^{\mu+\frac{1}{2}\sigma^2}$, we have $$\pi_t[X]=\frac{1}{S_t}e^{(\sigma^2-2r)(T-t)}$$


Concerning question $\text{b}$, your result is correct but you don't need to complicate things by dividing and multiplying by $S_t$: your expectation $E^Q[\cdot] = E^Q[\cdot|\mathcal{F}_t]$ is really conditional on infomation at $t$, hence you can simply take the $1/S_t$ factor from $1/S_T$ outside the conditional expectation without having to multiply and divide by $S_t$.

As for question $\text{a}$, once you have answered question $\text{b}$ it should be relatively straigthforward (hint: the answer is not a clear cut "yes" or "no").

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    $\begingroup$ Ah, so you're saying it is, if sigma and r are such that the exponential part of the risk-neutral formula figured out in part b = 1? Is this some known result that goes by some name (i.e. the _ condition)? Thanks $\endgroup$ – Archetupon Oct 17 '17 at 13:09
  • $\begingroup$ Exactly, when market parameters $r$ and $\sigma^2$ are such that: $2r = \sigma^2$, then the price $\pi_t[X]$ at $t$ of the claim is given by $Y_t$. This not a particular result and as such does not have a name (AFAIK). For a more general case where the payoff is given by $X'=S^{(1)}_T/S^{(2)}_T$ where $S^{(1)}$, $S^{(2)}$ are two different assets, you can check my answer to Replicating a portfolio with a certain payoff function, the insights are similar. $\endgroup$ – Daneel Olivaw Oct 17 '17 at 13:28

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