# Is it possible to detect a belief that a security will peak and then decline by analyzing American options pricing?

Please forgive me if this is a dumb question. I know only the basics of options and their valuation, and this is a question I've wondered for some time without being able to find a satisfactory answer on my own.

Owing to the nature of options contracts, given two contracts for the same security and at the same strike price, one strictly prefers a contract that expires later to one that expires earlier.

This is, I think trivially, true even if it is known that the underlying security will become worthless between the two expiration dates. But it is even more true if there is just a high probability of such a drop.

In general I wonder if it possible to construct a "best fit" of a stock's price between now and a particular expiration at a certain confidence interval (say, 50 percent) by assessing options trades - or if such a best fit is only possible if you make certain assumptions about the shape of the graph of the security price in question (for instance, maybe you can only say, if the price is strictly not decreasing below X between this and that date, then this is the line of best fit).

But my specific question in this case is: is it possible, by analyzing the publicly-available options trading information (prices, volatility, Greeks, volume, spreads, whatever), to make a statement like: "assuming efficient pricing, with 50 percent confidence, security XYZ will be priced at least X on this date, but priced Y < X on this later date." If so, how? If not, what kind of similar statement, if any, can be made?

• Hi @tacos_tacos_tacos, welcome to quant.SE! I am wondering why your statement should be true that one strictly prefers a contract that expires later to one that expires earlier, given you know for sure that the underlying security will become worthless between the two expiration dates...what is the justification of that? – muffin1974 Nov 5 '15 at 8:03
• @muffin1974 because you can exercise either on the earlier expiration date and thanks for warm welcome – tacos_tacos_tacos Nov 5 '15 at 8:15
• All right, then you should mention that you are talking about american option, because in the european style you are only allowed to exercise at expiration date. Regarding your question, are you thinking about measuring the impact of the option related variables onto future stock prices? I can not provide any citations, but why not simply regression future returns on the characteristics? This would allow to draw inference about the effects? – muffin1974 Nov 5 '15 at 8:21
• @muffin1974 I'm trying to get at a more fundamental question about whether options pricing can reveal the kind of belief that I describe (thanks for heads up on American distinction, yeah that is a big difference lol). So if we see all these long calls expiring after earnings or there is a big premium we tend to think that a lot of people think its going to have a great quarter or maybe people are hedging short positions - but we can look at short interest to clarify. But can options pricing reveal more complicated shapes of prices? – tacos_tacos_tacos Nov 5 '15 at 9:13

If you have many strikes of european-exercise options for two dates $T_1$ and $T_2$, then the option skew $\sigma_{1,2}(x)$ implies model-free risk-neutral probability distributions $p_1, p_2$ for each of these dates,
$$p_i(x) = {\left. \frac{\partial^2 }{\partial x^2}\right|} BS_{\text{Call}}(S_0, x, \sigma_i(x), r, T_i, q)$$
You can therefore assign confidence intervals to the stock, say exceeding $B_1$ at $T_1$ and to being below $B_2$ at $T_2$. However, in order to link them to get the joint probability of passing both barriers it is necessary to assume a model, perhaps a local vol or stochastic vol model.