What is delta in plain English, and how does it relate to the BSM hedge?

Delta has three equivalent interpretations, and all three are useful. Let me walk through each with your example of $\Delta = 0.6$.

Interpretation 1 — Price sensitivity:

If the stock moves up by $1, the call option price moves up by approximately $0.60. If you own 100 contracts (which equals 10,000 underlying shares notional), a $1 move in the stock translates to roughly $0.60 × 10,000 = $6,000 P&L.

This approximation is exact only for infinitesimal stock moves. For larger moves, gamma corrects the approximation.

Interpretation 2 — Hedge ratio:

If you are SHORT one call and want to be delta-neutral, you need to buy $0.6 \times 100 = 60$ shares of the underlying per option contract. The 60 shares offset the $0.60 of option price change. Per 100 contracts:

Interpretation 3 — Risk-neutral probability proxy:

Delta is not exactly $N(d_2)$, the risk-neutral probability that the call expires in-the-money — but for at-the-money calls with moderate $\sigma$ and $T$, delta and $N(d_2)$ are close. Many traders use delta as a rough heuristic for "probability of finishing ITM." A 0.6 delta call is roughly 55-65% likely to expire ITM under risk-neutral assumptions.

Why "dynamic" delta hedging is hard:

$\Delta$ itself changes as the stock moves and as time passes:

• If $S$ rises, $\Delta$ rises (the call moves deeper ITM and toward 1.0)
• If $S$ falls, $\Delta$ falls (the call moves OTM and toward 0)
• As $T \to 0$, $\Delta$ snaps to either 0 (OTM) or 1 (ITM)

A trader hedging a short call position must therefore continuously buy more shares as the stock rises and sell as it falls. The trader is, in effect, buying high and selling low — which means delta hedging a short option position bleeds money in proportion to realised volatility. That bleeding is exactly the option premium they collected.

For the exam:

$\Delta \approx N(d_1)$ for a non-dividend call. For a put, $\Delta = N(d_1) - 1$ (negative, between $-1$ and 0). You should be able to recite both signs.