Why an at-the-money option decays faster as expiry nears

Question

An at-the-money option's value is roughly $0.4\,S\,\sigma\sqrt{T}$ . Take $S = 100$ and $\sigma = 20\%$ .

Compare the daily decay of a $40$ -day option with a $4$ -day option. Why does decay accelerate as expiry approaches?

Accepted Answer

Value grows with \sqrt{T}, so its rate of decay, theta, grows like 1/\sqrt{T}, which blows up as T 	o 0. Let's see it in numbers. With 0.4 	imes 100 	imes 0.20 = 8 out front, the value is 8\sqrt{T} (with T in years). At 40 days: \sqrt{40/365} = 0.331, value = 8 	imes 0.331 = \2.65. One day later (39 days): \sqrt{39/365} = 0.327, value = \2.61. Decay \approx \0.04 that day. At 4 days: \sqrt{4/365} = 0.105, value = 8 	imes 0.105 = \0.84. One day later (3 days): \sqrt{3/365} = 0.0906, value = \0.72. Decay \approx \0.12 that day. So the near-expiry option loses about three times as much per …

Why an at-the-money option decays faster as expiry nears

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