Inferring Risk-Neutral Price Distributions from Option Skew

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


# 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?

## Answer by Brian B (score 1)

https://quant.stackexchange.com/a/21591

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,

\begin{equation} p_i(x) = {\left. \frac{\partial^2 }{\partial x^2}\right|} BS_{\text{Call}}(S_0, x, \sigma_i(x), r, T_i, q) \end{equation}

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.

You would calibrate the model to available data, and then compute the probability via the usual methods (in this case, PDE schemes would be most efficient and Monte Carlo would likely be quickest to code).

American-exercise options do not make any part of this process simpler, and of course they lack the nice implied probability distribution.