Pricing Options

An option's price depends on the stock price and the time left. Its value spreads out over time the way heat spreads through metal, and the Black-Scholes formula is the solution.

How it works

Parabolic PDEs

In one sentence

In the Black–Scholes model an option's value $V(S, t)$ satisfies $V_t + \tfrac12\sigma^2S^2V_{SS} + rSV_S - rV = 0$ , with the payoff at expiry as the final condition.

General formula

{\htmlData{sym=0}{\frac{\partial V}{\partial t}}} + \tfrac12{\htmlData{sym=3}{\sigma}}^2 {\htmlData{sym=5}{S}}^2{\htmlData{sym=1}{\frac{\partial^2 V}{\partial S^2}}} + {\htmlData{sym=4}{r}}{\htmlData{sym=5}{S}}{\htmlData{sym=2}{\frac{\partial V}{\partial S}}} - {\htmlData{sym=4}{r}}{\htmlData{sym=6}{V}} = 0

$V_t$ is time decay, the $\sigma^2$ term is the value volatility adds through curvature, and $rSV_S - rV$ keeps the price consistent with risk-free borrowing.

What the symbols mean

$\frac{\partial V}{\partial t}$
Theta
How much value the option loses as time passes.
$\frac{\partial^2 V}{\partial S^2}$
Gamma
How fast delta itself changes as the stock moves.
$\frac{\partial V}{\partial S}$
Delta
How much the option's value moves when the stock moves by $1.
$\sigma$
Volatility
How wildly the stock price swings.
$r$
Risk-free rate
The interest rate on a safe investment like government bonds.
$S$
Stock price
The current price of the underlying stock.
$V$
Option value
What the option is worth right now.

$\sigma$ is volatility, $r$ the interest rate and $S$ the stock price.
A change of variables turns this PDE into the heat equation.
The call price is $S\,N(d_1) - Ke^{-rT}N(d_2)$ , where $N$ is the normal CDF.

Learn the math

The Black–Scholes EquationPDE · Lesson 20, Beyond the Basics