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Estimating the Probability That a Short Option Expires Worthless

Article Quant Q&A · Author: Tension
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Summary

The document discusses estimating the chance that a short call or put expires worthless. It presents option delta, with the sign adjusted for puts, as a rough first-order proxy for the probability of finishing in the money; subtracting that estimate from one gives an approximate probability of expiring worthless.

For a more refined estimate, it points to reconstructing a probability density from the implied-volatility skew and integrating over the relevant outcomes. The discussion offers no derivation, implementation detail, or empirical comparison of these approaches. Delta is an approximation rather than an exact probability, and probability estimates depend on the assumptions and inputs used to infer the distribution.

Key ideas

  • Call delta, or the negative of put delta, can approximate the probability of expiring in the money.
  • The complementary probability gives an approximate chance of expiring worthless.
  • A probability density inferred from implied-volatility skew can provide a more detailed estimate.
  • The document gives no validation or implementation details for either approach.

Tags

OptionsDerivatives pricingVolatility
Full text
# How do I calculate the probability of a short option position expiring worthless?


# How do I calculate the probability of a short option position expiring worthless?












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?

## Answer by siou0107 (score 3)

https://quant.stackexchange.com/a/49959

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

## Answer by onlyvix.blogspot.com (score 1)

https://quant.stackexchange.com/a/27704

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.