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)?

  • $\begingroup$ 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. $\endgroup$ – FinanceGuyThatCantCode May 22 '17 at 19:04
  • $\begingroup$ Right. More generally, the full risk-neutral distribution can be inferred using the so-called Breeden-Litzenberger identity. $\endgroup$ – Quantuple May 23 '17 at 8:29

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