The Black–Scholes Equation

A stock option is simply a contract that gives you the right, but not the duty, to buy (a "call") or sell (a "put") a stock later at a price fixed today, the strike. The equation is named after Fischer Black and Myron Scholes, who published it in 1973, and a "European" option has nothing to do with Europe: it just means you can only use it on the expiry date. It's good for answering the question every trader has: what's a fair price for that right today? The equation $V_t + \tfrac12\sigma^2S^2\,V_{SS} + rS\,V_S - rV = 0$ looks like a monster, but it's four simple terms: $V_t$ (how the option's value changes as time passes), $\tfrac12\sigma^2S^2\,V_{SS}$ (the spreading caused by the stock jiggling randomly, playing the role of heat diffusion), $rS\,V_S$ (the stock drifting upward with interest) and $-rV$ (money later being worth less than money now). Put together, they say the option's value has to balance out with what a safe bank account would earn. It's finance dressed up as the heat equation; bear with it.

The idea. Here's the clever part. Combine the option with just the right amount of stock ($\Delta = V_S$ shares) and the random part of the price cancels: when the stock jumps, the option and the shares move in opposite directions by the same amount. A portfolio with no risk can't earn more than the risk-free rate $r$, or everyone would borrow to buy it. Writing that out with Itô's formula gives the PDE.

A final-value problem. At expiry the call is worth its payoff, $V(S, T) = \max(S - K, 0)$, and there's no uncertainty about that. So instead of starting from an initial condition, we solve backward from $T$, which is why the time derivative has the "wrong" sign compared with the heat equation. Going back in time, the sharp corner of the payoff gets smoothed out, just as heat smooths a sharp initial temperature going forward.

Reduction to heat. With $x = \ln S$, $\tau = T - t$ and an exponential factor that removes the drift and decay terms, the equation becomes $u_\tau = \frac{\sigma^2}{2}u_{xx}$. The log makes sense because stock moves are about percentages, not dollars. Then it's the infinite rod again: convolving the payoff with the heat kernel gives the formula above. The normal CDF $N$ shows up because the heat kernel is a normal distribution.

The Greeks. Delta $= N(d_1)$ is how much the option's price moves per dollar of stock; it's also the hedge ratio, the number of shares from the paragraph on the idea.

A common mistake. It's tempting to think the option's price must depend on how fast you expect the stock to grow, since a stock that's heading up should make a call worth more. It doesn't. The expected growth rate never appears in the equation, only $\sigma$ and $r$. The hedging argument removes it: once the risk is cancelled, the portfolio has to grow at $r$, whatever you believe about the stock.

The model assumes constant volatility and frictionless trading; real markets violate both, which is why traders quote "implied volatility."

Options exchanges, banks and trading desks price and hedge trillions of dollars of contracts with Black–Scholes and its descendants; the 1997 Nobel Prize in economics recognized the work.