Risk-Neutral Pricing of a Claim Paying the Inverse Asset Price
Summary
The exercise asks whether the reciprocal of an asset price can itself be the arbitrage-free price of a traded derivative, and how to value a claim paying the reciprocal at maturity. Its attempted valuation uses risk-neutral discounted expectation under geometric Brownian motion, then rewrites the terminal reciprocal using the current asset price and Brownian increments. Evaluating the exponential expectation produces a candidate price process.
The accepted response confirms the candidate pricing expression and points out that the conditional expectation is taken given information at the current time. That conditioning allows the current asset-price factor to be handled directly, without the attempted multiply-and-divide detour. The response does not finish the first question; it notes that the conclusion is not a simple yes or no. Thus, the document illustrates a valuation technique but leaves the traded-price and arbitrage argument incomplete.
Key ideas
- A maturity payoff equal to the reciprocal asset price can be valued by a discounted conditional expectation under the risk-neutral measure.
- The current asset price is known at the valuation time and can be factored outside the conditional expectation when rewriting the reciprocal payoff.
- The response accepts the candidate valuation but does not fully show its derivation.
- Whether the reciprocal process itself is an arbitrage-free traded price is left unresolved and requires a qualified answer.
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# Pricing and Arbitrage of Inverse Asset Claim
# 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 $$\pi_t[X]=\frac{1}{S_t}e^{(\sigma^2-2r)(T-t)}$$
## Answer by Daneel Olivaw (score 2, accepted)
https://quant.stackexchange.com/a/36480
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").Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.