Posted on StackExchange, but they have a nasty habit of deleting unanswered questions after a while, in an irrecoverable way, so I’m going to re-post all my random questions here as I go along in life. (You can see dead questions if you know the URL, they keep it around, but if you lose the URL, forget about it.) So….
Suppose we are holding a replicating portfolio of long an option and short some stock, so Suppose the stock follows geometric Brownian motion and pays continuous dividends at rate , so Naively, However, because the stock pays a dividend, common sense and the literature tell us that
Question: How do we rigorously arrive at the total derivative for which includes extra term , given that we know , without appeal to common sense i.e., from the equations, without recalling the mechanics of how the stock works? Because, naively, I would not include the extra term, if I just knew the equation defining and nothing about the mechanics of the stock.