Partial Derivatives: Computation, Higher Partials, and Schwarz's Theorem

Every pricing model differentiates one variable at a time. The Black-Scholes delta $\Delta = \partial V/\partial S$ is the derivative of the option price with respect to the underlying price, holding strike, rate, volatility, and time fixed. The gamma $\Gamma = \partial^2 V/\partial S^2$ is the second partial derivative. Vega is $\partial V/\partial \sigma$, theta $\partial V/\partial T$, rho $\partial V/\partial r$ — the entire Greek alphabet is a table of partial derivatives. The central result of this lesson is Schwarz's theorem: the order of mixed partials is irrelevant when both are continuous, so $\partial^2 V/\partial S\,\partial\sigma = \partial^2 V/\partial\sigma\,\partial S$ — cross-Greeks commute.

Learning objectives. By the end of this note you should be able to: