Delta of the ATM Call

Question

Roughly what is the delta of an at-the-money call option, and why? Give the intuition, then name the refinement (is it slightly above or below, and what drives it?).

Accepted Answer

About 0.5. Two complementary intuitions: 1. Probability view: delta of a call is close to the risk-neutral probability of expiring in the money (exactly N(d_2); delta is N(d_1), a near neighbor). At the money, up vs down is roughly a coin flip → ≈ 0.5. 2. Hedging view: an ATM option responds to a small stock move on one side of a near-symmetric distribution, it picks up about half of each 1 move. The refinement: ATM-spot call delta is slightly above 0.5, in Black–Scholes, Delta = N(d_1) with d_1 = fracln(S/K) + (r + tfracsigma^22)TsigmasqrtT = left(fracrsigma + fracsigma2right)sqrtT > 0 text at S = K, so Delta = N(d_1) > 0.5. Drivers: the fracsigma^22 term (lognormal drift of the hedged process) and the rate r (forward above spot). E.g. sigma = 20%, T = 1, r=0: d_1 = 0.1 → Delta approx 0.54. Worth volunteering: the option struck at the forward has delta closer to 0.5, and traders often quote "50-delta" rather than "ATM" for exactly this reason.

Delta of the ATM Call

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