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"If at least 10% of an option's value is time value (ie. time value >= 0.1*call price), it has a delta less than 90".

In practice and after doing many tests with an option pricing calculator, this statement seems to hold true. Can anyone mathematically prove or disprove this?

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How could the proof (if it is true) work. The first lines: $$C - (S-K)^+ \ge 0.1 C$$ where C is the call price and the left hand side is the time value. If we substract the rhs we get $$0.9C \ge (S-K)^+$$ and on more step gives $$0.9 \ge (S-K)^+/C$$. But at the moment I can not conclude for the Delta ... –  Richard Aug 17 '12 at 20:46
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This claim is false. A deep in-the-money option with very high volatility can have both large time value and high delta. As a counterexample, consider a call option with:

  • K = 100 (strike price)
  • S = 300 (spot price)
  • r = 0
  • T = 1
  • vol = 150%

This gives a Black-Scholes value of approximately \$230, so the time value is \$30, but the delta is 93.1%.

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That's why the maths in my comment did not work out ... I didn't try myself but what about more realistic parameters such as vol around $30\%$ and not that deep in the money? –  Richard Aug 18 '12 at 19:42
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As far as I could tell in my tests, with more realistic parameters the claim is usually true (presumably why the OP inferred that it might be true in general). –  user2825 Aug 18 '12 at 19:46
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