Exercise: Computing Vega Across Strikes and Expiries

Prerequisites: Vega

Problem

Under Black-Scholes with $S = 100, r = 0.05, q = 0, \sigma = 0.2$:

1. Compute vega across strikes $K \in \{80, 90, 100, 110, 120\}$ for $T = 0.5$. Report in both raw units ($\partial V/\partial\sigma$) and per-1% units ($\partial V/\partial\sigma / 100$).

2. For $K = 100$, compute vega across expiries $T \in \{2.0, 1.0, 0.5, 0.1\}$. Verify that vega scales approximately as $\sqrt T$ for ATM options.

3. Gamma-vega identity check. For $S = 100, K = 100, T = 0.5$, verify numerically that $\mathcal{V} = \sigma S^2 T\Gamma$.
4. Finite-difference vega. Compute vega by central finite difference: $(V(S, K, T, r, \sigma + h) - V(S, K, T, r, \sigma - h))/(2h)$ with $h = 0.001$. Compare to closed form.

Hint