Pricing and Arbitrage of Inverse Asset Claim

Question

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 $$\pi_t[X]=\frac{1}{S_t}e^{(\sigma^2-2r)(T-t)}$$

Answer 1

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").