Derivation of the Black-Scholes-Merton PDE › Extending to Dividend-Paying Stocks

The basic BSM PDE assumes the stock pays no dividends. This assumption simplifies the analysis but excludes many important applications, including options on dividend-paying stocks, equity indices, and currencies. For a stock paying continuous dividends at rate $q$, the stock price dynamics become:

$$dS_t = (r - q) S_t \, dt + \sigma S_t \, dW_t^Q$$

where:

• $q$: continuous dividend yield
• $\sigma$: volatility of the underlying stock
• $r$: risk-free interest rate
• $S_t$: stock price
• $dW_t^Q$: increment of Wiener process under risk-neutral measure
• $dt$: small time increment

The key change is that the drift rate under the risk-neutral measure is now $r - q$ rather than $r$. The intuition is that a stock paying dividends at rate $q$ provides part of its return through dividend payments rather than price appreciation. In the risk-neutral world, the total expected return (price appreciation plus dividends) must equal $r$, so the expected price appreciation is only $r - q$.

The modified Black-Scholes PDE is:

$$\frac{\partial V}{\partial t} + (r-q)S\frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = rV$$

where:

• $V(S, t)$: price of the derivative
• $S$: stock price
• $r$: risk-free interest rate
• $q$: continuous dividend yield
• $\sigma$: volatility of the underlying stock

The intuition is that a stock paying dividends effectively grows slower (by the dividend yield $q$) than a non-dividend-paying stock, which reduces the drift term in the first-order spatial derivative. This modification is important for pricing options on dividend-paying stocks, equity indices, and currencies (where the foreign interest rate plays the role of a dividend yield, as we discussed in our chapter on foreign exchange markets).