> [!tldr] > The Black-Scholes-Merton equation is a PDE that describes the fair price $V_{t}$ of an European derivative as a function of time $t$ and stock price $S_{t}$: $rV_{t}=\frac{ \partial V_{t} }{ \partial t } +rS\frac{ \partial V }{ \partial S_{t} }+ \frac{\sigma^{2}S^2}{2}\frac{ \partial^{2} V }{ \partial S_{t}^{2} }. $*It derives a solvable PDE from the SDE of stock movements*, and enforcing the boundary condition $V_{T}=f(S)$ of the product (for example $f(S)=(S-K)_{+}$ for an European call) gives the fair price at time $t$. > > It assumes that the stock price $S_{t}$ follows [[Geometric Brownian Motion]]. ### Deriving Black Scholes Consider the dynamically hedged portfolio: with $\Delta_{t}:= { \partial V_{t} } / { \partial S_{t} }$, $P_{t}:= V_{t} - \Delta_{t}S_{t}.$Then applying [[Ito's Lemma]] to $V_{t}$ and plug-in the geometric Brownian motion dynamics for $S_{t}$ gives: $\begin{align*} dP_{t}&= dV_{t} - \Delta_{t} ~dS_{t}\\ &= \underbrace{\left( \frac{ \partial V }{ \partial t } +\cancel{\mu S \frac{ \partial V }{ \partial S } } + \frac{\sigma^{2}S^{2}}{2}\frac{ \partial ^{2}V }{ \partial S^{2} } \right)dt + \cancel{\sigma S \frac{ \partial V }{ \partial S } ~dW_{t}}}_{dV_{t}} \\ &~ ~ ~ ~ ~ ~ ~- \Delta_{t}\underbrace{(\cancel{\mu S~dt} + \cancel{\sigma S~dW_{t})}}_{dS_{t}}\\[0.8em] &= \left( \frac{ \partial V }{ \partial t } + \frac{\sigma^{2}S^{2}}{2}\frac{ \partial ^{2}V }{ \partial S_{t}^{2} } \right)dt. \end{align*}$Here the cancellations happen because of the choice $\Delta_{t}:= { \partial V_{t} } / { \partial S_{t} }$. Therefore, *this portfolio is risk-free* as long as the hedge is maintained. By no-arb, [[Time Value of Money|its value must grow at the risk-free rate]]: $rP_{t}~dt=dP_{t}=\left( \frac{ \partial V }{ \partial t } + \frac{\sigma^{2}S^{2}}{2}\frac{ \partial ^{2}V }{ \partial S_{t}^{2} } \right)dt.$So equating the two sides gives: $r(V-\Delta S)= \frac{ \partial V }{ \partial t } + \frac{\sigma^{2}S^{2}}{2}\frac{ \partial ^{2}V }{ \partial S^{2} } , $the Black-Scholes-Merton PDE. ### BSM with Continuous Dividends