Delta Hedging for an At-The-Money Option

This question focuses on the mechanics of delta hedging for a plain-vanilla, at-the-money option on a single underlying asset. The candidate is asked to translate the intuitive idea of "hedging away small moves" into a concrete trading position in the underlying. It sits squarely in the core derivatives toolkit used in options market-making, equity derivatives, and structured products roles at banks and hedge funds. The setup is intentionally simple: a single option, no path dependence, and a snapshot in time, so the discussion can stay on how the hedge ratio relates to the option's moneyness and payoff shape, rather than on model calibration or complex risk factors.

To answer well, a candidate must connect option price sensitivity to a specific hedge quantity, invoking the standard derivative pricing framework. It leans on understanding delta as a first derivative with respect to the underlying price, and on how Black–Scholes or similar models link this to probability under a risk-neutral measure. Interviewers look for clarity on why the required share position is proportional to delta, awareness that this is a local, instantaneous hedge, and recognition of assumptions such as continuous rebalancing and frictionless markets.

What it tests

Delta-hedging is rooted in the principle that the price sensitivity of a derivative with respect to its underlying asset—the `delta`—dictates the proportion of the underlying needed to offset small price movements. For European options, this sensitivity is captured by the Black-Scholes formula, where `delta` for a call option is $N(d_1)$, the probability-weighted likelihood the option finishes in the money under a risk-neutral measure. At-the-money, the option's payoff is most responsive to small changes in the underlying, so the `delta` hovers near 0.5, reflecting a 50-50 chance of expiring in or out of the money. The reason this pattern holds is that, locally, the option's value changes about half as much as the underlying for small moves when the strike equals the spot price. The precise value shifts slightly with interest rates and dividends, but the intuition remains: delta is a measure of local linear approximation, and at the midpoint, the slope is about halfway between 0 and 1.