In that case you are better off using a tight call or put spread as an approximation of a digital and then multiply that by 2. While it's still not perfect, it takes the implied skew into account.
No coding experience required Download R: https://cran.r-project.org/bin/windows/base/ Download Rstudio: https://rstudio.com/products/rstudio/download/ Open Rstudio Copy and paste Kev's code. Call the function with your inputs: bsProbOfTouch(S,K,T=1,r=0,q=0,sigma=0.1) Then press run. Voila Doing those steps is worth the time. Kevin and others have posted lots of useful R code on ET over the years
Thank you, I will download R later. I know I am jumping the gun before I download but first, a dumb question: VBA is tied to excel, so when I bring up excel and call up macro, I can start coding. MATLAB is tied to MATLAB. What platform do I need for R, and for that matter Python? Is the downloaded R the platform?
That's the same as the dual-delta (2*N[d2]) approximation, except arrived at via finite differences instead of analytically. Normalized to probability terms the spread price converges to dual-delta of the midpoint between the strikes as distance between the strikes approaches zero. All of the approximations discussed in this thread, 2*delta, 2*dualdelta, and the function I posted (as well as Peterffy's crude version of Breeden-Litzenberger), take the implied skew into account. This is because they are dealing with the greeks/ivols/prices at K (the strike/barrier, U, L,...) to make the estimates. However, for estimating real-world probabilities, taking the implied skew into account seems to hurt more than help. At least this is true historically for the SPX (the only name I have actually tested it on) since 2003 (as far as my options data goes back). Also I tested only the probabilities of ITM (the terminal distribution on the third Friday monthlies), not the touch probabilities.
That is a good idea. It will be a useful exercise even if you find that option implied probabilities aren't very good estimates of real-world probabilities. You should have no trouble translating the function I posted to VBA, as it was tranlated from Basic to R to begin with. A few tips for your test. Start with the monthlies on major indices like the SPX. Take each of 22 days to expiry as a separate panel. Convert probabilities at all strikes to standard scores, then pool within each panel. Within each panel the histogram of the pooled standard scores over the whole test time period should look like a normal distribution.Deviations from normal should point you towards systematic (perhaps tradeable) biases. For your second pass through the data, note that a density should integrate to one. Raw options implied densities tend (especially these past few weeks) to integrate to considerably more. You want to adjust for this. How you adjust your implied density to sum to one can make a huge difference in real-world forecast performance. Note that Pererffy's density method in a post above sums to one by construction, but achieves this piecewise in exactly 100% the wrong direction.
I am confused. How does 2*delta or 2*dualdelta take the skew into account if it only uses a single volatility? A replicated digital (i.e. call spread) would take two options, each traded with distinct implied volatility for the strike and thus will include the skewness between these two strikes. [edit] got it, you are taking the limit of the strike spread and thus converging to zero vol difference - sadly, that also brings you from "real" world to a pure BSM world Real world outcomes would include real-world drift, so naturally you'd never match your risk neutral probability that uses option prices. However, when managing a digital or one-touch exposure (real life expression of risk neutral probabilities) you'd certainly want to include the skew into your hedging/pricing model so you carry the right amount of delta.