What an option's delta secretly tells you about the odds

Question

A stock trades at 100. A 30-day call struck at 100 (implied vol 20\%) has delta 0.52. A further out-of-the-money call has delta 0.20. Read each delta two different ways. Roughly what is the chance each call finishes in the money, and why is delta only an approximation of that chance?

Accepted Answer

Delta =\partial C/\partial S has two everyday meanings: 1. Hedge ratio. To hedge one call you short \Delta shares. The 0.52 call moves like 52 shares of stock for a small wiggle. 2. Approximate probability of finishing in the money. For a call, \Delta \approx N(d1), while the true risk-neutral probability of finishing in the money is N(d2), with d2 = d1 - \sigma\sqrt{T}. Because d2 < d1, we always have N(d2) < N(d1), so delta slightly overstates the chance of finishing in the money. The gap is \sigma\sqrt{T}. Here \sigma\sqrt{T} = 0.20\sqrt{30/365} \approx 0.057. The 0.52-delta call has d1 \a…

What an option's delta secretly tells you about the odds

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