Why is the BSM hedge called "dynamic"? What happens if I just set $\Delta$ once and never rebalance?

You will not have a hedge at all by the time you come back. The static "set-it-and-forget-it" delta hedge breaks within hours, not days.

Here is the failure mode:

Suppose you sell a call with $\Delta = 0.6$ at $S_0 = 100$ and buy 60 shares as a static hedge. One week later, the stock has risen to $105 and the new delta is, say, 0.75.

• Your option short has gained $5 × 100 = $500 of liability per contract (roughly, ignoring time decay)
• Your 60-share hedge has gained $60 \times \$5 = \$300$

Net P&L: $-\$200$ per contract. You are not delta-neutral anymore because the option's sensitivity grew faster than your hedge's — that gap is gamma at work.

The static hedge only works for infinitesimal stock moves and zero time. The instant either changes, the position has residual delta (and gamma, and theta…) that grows with the size of the move.

Why dynamic hedging is costly even when done perfectly:

Even a continuously rebalanced delta hedge does not perfectly replicate the option payoff because:

1. Gamma scalping cost. The trader buys high and sells low as $\Delta$ chases the underlying. The total P&L drag equals roughly $\tfrac{1}{2} \cdot \Gamma \cdot \sigma^2 \cdot S^2$ per unit time — exactly the theta of the option. So delta-hedging a short option bleeds at theta and is profitable only if realised vol $<$ implied vol.
2. Transaction costs. Each rebalance costs bid-ask spread and exchange fees.
3. Discrete vs. continuous time. BSM assumes continuous rebalancing; reality is at best per-second.
4. Stochastic vol. $\sigma$ in the formula is constant; reality is regime-shifting.

For the exam:

The "BSM replication" story assumes continuous rebalancing with no costs. Real-world hedging error is a small but persistent gap that traders price into the bid-ask spread. If a question asks "what is the cost of delta hedging in BSM theory?" the answer is zero — frictionless replication recovers the option payoff exactly. If a question asks "what is the cost in practice?" the answer is gamma scalping cost + transaction costs + vol-of-vol exposure.