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I want to be able to determine the probability of a short option position (call or put) expiring worthless.

Don't know where to start but I see probabilities derived from the greeks on some web sites?

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    $\begingroup$ Are you familiar with $N(d_2)$ ? see ltnielsen.com/wp-content/uploads/Understanding.pdf $\endgroup$
    – Alex C
    Commented Jun 19, 2016 at 19:20
  • $\begingroup$ Can you calculate arbitrary priced options? If so just Calc. The price of a digital at the same strike. $\endgroup$
    – will
    Commented Jun 19, 2016 at 20:41

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As I have often heard this theory of "delta == probability of being ITM", I just put on some wise words from Paul Wilmott ;)

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    $\begingroup$ Agreed. Although one could argue that it is the probability of the option being ITM. But under a probability measure using the underlying asset price as numéraire, which although being mathematically equivalent to the real world measure, is a pure mathematical construct, hence not very useful. $\endgroup$
    – Quantuple
    Commented Nov 27, 2019 at 15:06
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You can think of delta for calls (-delta for puts) as the first order approximation to probability of expiring in the money. If you subtract this probability from 100%, you'll have the probability of expiring worthless.

If you want more exact probability, there are algorithms to construct a pdf from IV skew, and calculate probability from there.

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