Hi VolSkewTrader
Our method is less of a formula and more of a process.
Strike Slope is a measure of the amount that implied volatility changes for every increase of 10 call delta points within the intra-month skew. It measures how lopsided the 'smile' or 'smirk' is. The derivative is a measure of the rate at which the strike slope changes for every increase of 10 call delta points within the intra-month skew. It measures the curvature of the intra-month skew or 'smile.' We chose just two parameters to describe the skew to get a reasonable fit for the fewest assumptions.
We start with lining up the calls and puts IVs using residual yields. We use the 85 to 15 call deltas in the study. We have more weightings to the call an puts for closeness to the 50 delta and we weight the call vs put the more OTM we go, meaning the 20 delta call IV will get more weighting than the same strike 80 delta put: The IVs will be slightly different even after our residual yield process. Then on to estimating the slope with a best fit, and we are left with errors from the slope line to the actual mid market IVs. We apply the derivative to minimize those errors.