Exercise: Computing and Interpreting Gamma Across Strikes and Expiries

Prerequisites: Gamma, Delta

Problem

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

1. Compute $\Gamma$ across strikes $K \in \{80, 90, 100, 110, 120\}$ for $T = 0.5$. Plot (or tabulate) and identify where gamma is maximised.

2. For $K = 100$, compute $\Gamma$ across expiries $T \in \{1.0, 0.5, 0.1, 0.01\}$. What trend do you see?

3. Verify by finite difference. For $S = 100, K = 100, T = 0.5$, compute $\Gamma$ numerically via the central difference $(\Delta(S + h) - \Delta(S - h))/(2h)$ with $h = 0.01$. Compare to closed form.
4. Call–put gamma equality. Compute put gamma (by direct differentiation of put formula) at $S = K = 100, T = 0.5$. Verify it equals call gamma, as required by put-call parity.

Hint

Put formula: $P(S, t) = Ke^{-r(T-t)}\Phi(-d_2) - Se^{-q(T-t)}\Phi(-d_1)$.