Two ways to a Greek
The Greeks measure how an option's price reacts to its inputs. Delta $\Delta=\frac{\partial C}{\partial S}$ is the change per unit of the stock price, gamma $\Gamma=\frac{\partial^2 C}{\partial S^2}$ is how fast delta itself changes, and vega $\mathcal{V}=\frac{\partial C}{\partial\sigma}$ is the change per unit of volatility. They are the partial derivatives and Taylor coefficients of this skill, applied to a price.
For a Black-Scholes call the derivatives have closed forms: $\Delta=N(d_1)$, $\Gamma=\frac{\varphi(d_1)}{S\sigma\sqrt{T}}$ and $\mathcal{V}=S\,\varphi(d_1)\sqrt{T}$, where $N$ is the standard normal distribution function and $\varphi$ its density. Closed forms are exact and fast, but most real pricers (exotics, Monte Carlo, calibrated models) have none.
That is where finite differences come in: bump an input up and down, reprice, and divide. Delta is a central difference $\frac{C(S+h)-C(S-h)}{2h}$, vega the same in $\sigma$, and gamma a second difference $\frac{C(S+h)-2C(S)+C(S-h)}{h^2}$. The only choice is the bump size $h$: too large and the slope is crude, too small and rounding noise dominates (see the limits topic). Comparing finite differences with a closed form is the standard test that a pricer and its Greeks are right.
Lab · what's graded
- The Black-Scholes price25 pts
- A central difference25 pts
- Greeks from formulas25 pts
- Greeks by finite differences25 pts