Exercise: Delta Hedging P&L Simulation

Prerequisites: Delta, Geometric Brownian Motion

Problem

A trader sells one ATM European call ($S_0 = K = 100$, $T = 0.5$ years, $\sigma = 0.2$, $r = 0.05$, $q = 0$) and dynamically delta-hedges. They discretise the hedging into $N$ equally-spaced rebalancing times over $[0, T]$.

1. Simulate $M = 10{,}000$ GBM paths of the underlying. For each path, compute the terminal hedged P&L:

   • Initially short 1 call, hold $\Delta_0$ shares, cash balance $C_0 - \Delta_0 S_0$.
   • At each rebalancing, update delta to $\Delta_t$ (buy/sell to match). Cash accrues at rate $r$.
   • At expiry, pay out $(S_T - K)_+$ on the short call, close the share position at $S_T$.

   The final P&L is the remaining cash balance minus the option payoff.

2. For three discretisation levels $N \in \{10, 50, 250\}$, report the mean and standard deviation of terminal P&L. Verify that mean is close to zero (cost of hedging absorbs the option premium) and std shrinks as $N$ grows.

3. Convergence rate. The theoretical hedging-error scaling is $\text{std}(\text{P\&L}) \sim 1/\sqrt N$. Verify this by plotting $\text{std}$ vs. $N$.
4. Discussion. What source of residual P&L remains at any finite $N$? (Hint: this is the gamma risk — the curvature that first-order delta hedging misses.)

Hint