Exercise: Computing Delta for a Vanilla Call and Put

Prerequisites: Delta

Problem

Under Black-Scholes with $S = 100, K = 100, T = 0.5, r = 0.05, q = 0$:

1. Compute $\Delta_{\text{call}}$ and $\Delta_{\text{put}}$ for three volatility regimes: $\sigma \in \{0.1, 0.2, 0.4\}$.

2. For $\sigma = 0.2$, compute $\Delta_{\text{call}}$ at three spot levels: $S \in \{80, 100, 120\}$. What happens to delta as the option moves deeper ITM or OTM?

3. Put-call parity check. For each $\sigma$ in part 1, verify $\Delta_{\text{call}} - \Delta_{\text{put}} = 1$ (for $q = 0$).
4. Numerical finite-difference. For $\sigma = 0.2, S = 100$, compute $\Delta_{\text{call}}$ by central finite difference: $(C(S + h) - C(S - h))/(2h)$ with $h = 0.01$. Compare to the closed-form answer.

Hint