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I'm totally lost with this derivation of deterministic time dependent volatility in the BS equation. The screenshot shows the Appendix from Glenn Swindle (Valuation and Risk Mgmt in Energy...). Can anyone shed light on what's going on with the substitution in A.15 and how we get the expression for ds/dt? Or even point me to a reference that might have more details on how ds/dt was derived...would be deeply appreciated.

Since no one else is biting ...

Cheating a little bit by recognizing that where we want to end up is with the last sentence from the screenshot, it looks like s(t) is introduced as a change in variable to weight the first term in A.14 by the average of the deterministic variance remaining until expiration.

Hope that helps, and someone should check whether I've interpreted the arithmetic correctly.

Hi Marsden - I should post what I tried to do....before giving up and turning to Chat GPT. This thing is insane....it solved the whole thing out (yes, it interpreted the screenshot!!!). There's only one point of confusion left and I'll take some time to figure it out. The conversation with chat GPT is too long to post here...but you can give it a try too. 
And thanks a ton for responding, and yes your interpretation is correct - at-least ChatG thinks so!