Exercise: Gamma P&L and Realised-vs-Implied Volatility

Prerequisites: Gamma, Delta, Geometric Brownian Motion

Problem

A trader buys an ATM call ( $S_0 = K = 100, T = 0.5, r = 0.05$ ) priced at implied volatility $\sigma_{\text{imp}} = 0.2$ . They delta-hedge with $N = 500$ rebalancings. Meanwhile, the underlying actually moves with realised volatility $\sigma_R$ .

The gamma-scalping formula says the expected P&L is

\mathbb{E}[\text{P\&L}] \approx \int_0^T \tfrac{1}{2}\Gamma_t S_t^2 (\sigma_R^2 - \sigma_{\text{imp}}^2)\,dt.

So delta-hedging is profitable iff realised vol exceeds implied.

Simulate $M = 5000$ GBM paths under three realised-vol regimes: $\sigma_R \in \{0.15, 0.20, 0.30\}$ . For each, compute the mean P&L and compare to the theoretical gamma-scalping estimate.
For $\sigma_R = 0.20$ (= $\sigma_{\text{imp}}$ ), verify P&L mean $\approx 0$ . What about P&L std?
Interpretation. A trader believes implied vol of 20% is mispriced and the true realised vol will be 30%. Compute the expected profit per contract over the 6 months.

Hint

At each rebalancing, the P&L contribution is (price change of option) − (delta × stock change). Sum over all rebalancings. Use the delta and gamma computations from previous exercises. For gamma scalping formula, you can approximate $\Gamma_t$ and $S_t$ by their $t = 0$ values for a rough scale check (ATM $\Gamma_0 \approx 0.0273$ , $S_0 = 100$ ).

Jump to the solution when you're ready.