How do I calculate delta for a European call step by step?

Question

AcadiFi · Accepted Answer

For a European call on a non-dividend stock, delta = N(d1), where d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T). N(·) is the standard normal cumulative distribution. Delta lies between 0 and 1, approaches 1 for deep in-the-money calls, and approaches 0 for deep out-of-the-money.

Example: Brindle Motors trades at S = $82, the strike is K = $80, risk-free rate r = 4%, volatility σ = 28%, and T = 0.5 years. Compute the numerator: ln(82/80) = 0.02469; (0.04 + 0.5·0.0784)·0.5 = 0.0396; sum = 0.06429. Denominator σ√T = 0.28·0.7071 = 0.198. d1 = 0.06429 / 0.198 = 0.3247. From z-tables, N(0.32) ≈ 0.6255, so delta ≈ 0.626. The trader hedges 1,000 calls by shorting 626 shares.

With continuous dividend yield q, delta becomes e^(-qT)·N(d1). A 2% yield would scale delta to 0.626·e^(-0.01) ≈ 0.620. Key intuition: delta ≈ probability the option finishes ITM under the risk-neutral measure, adjusted by a discount factor. Exam traps include confusing d1 and d2, forgetting the dividend adjustment, and mixing up put delta (N(d1) - 1 for non-dividend puts). Practice three or four variations so the formula becomes reflexive before exam day.

How do I calculate delta for a European call step by step?

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