Derivation: Black-Scholes PDE with Continuous Dividend Yield

A stock that pays dividends leaks value to its holder: arbitrage forces the price to fall by the amount paid, and the holder of the hedge receives the payment. Both effects must enter the hedged-portfolio argument, and they conspire to produce a single, clean change to the Black–Scholes PDE — the drift coefficient $rS\,V_S$ becomes $(r-D)S\,V_S$ while everything else survives. The same PDE then prices index options and (with $D = r_f$) short-dated FX options. This is the exact derivation pattern of the no-dividend case in section 4.1, with one extra cash-flow term to track.