# Pricing and Arbitrage of Inverse Asset Claim

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

## 1 Answer

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").

• 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 – Archetupon Oct 17 '17 at 13:09
• 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. – Daneel Olivaw Oct 17 '17 at 13:28