Is it possible to calculate implied probability of >=X% return based on implied volatilities from options

My question is: Is it possible to imply either the upside or downside (one sided) probability from looking at implied volatilities of stock options?

Let's take an example: say you had Stock A at $50, with a 3M ATM Option with an Imp. Vol of 20%. Is there a way, using perhaps non-ATM options, or some combination of options perhaps(?), to calculate the implied probability of Stock A being >10% by expiry (let's say 3m)? • You can get the risk neutral implied probability by pricing an undiscounted digital call or put - the distribution is only valid on expiration date. The pricing is based on the digital price with skew - for a digital put:$P=N(-d_2)+\frac{d \sigma}{dK}\frac{\partial C}{\partial \sigma}$where$C\$ is the price of a vanilla call - i.e. black scholes digital put + skew*vega. May 22 '17 at 19:04
• Right. More generally, the full risk-neutral distribution can be inferred using the so-called Breeden-Litzenberger identity. May 23 '17 at 8:29

1 Answer

You can find a variety of methods in this paper: Mizrach, Bruce, Estimating Implied Probabilities From Option Prices and the Underlying, in Cheng-few Lee and Alice C. Lee (eds.), Handbook of Quantitative Finance and Risk Management, New York: Springer-Verlag, 2010, 515-29.

MathWorks has an example of a method coded in Matlab: Estimating Option-Implied Probability Distributions for Asset Pricing.

You may also be interested in this simpler article: Option Prices Imply A Probability Distribution

To find other methods, you may need to search for papers on Google Scholar.