Estimating When an Underlying First Reaches Either Option Strike

Summary

The question asks whether the average time for an underlying to touch either short strike in a strangle can be estimated, and whether that average would be consistent enough to guide trading. It raises historical data and Monte Carlo simulation as possible alternatives, using a strangle with specified deltas and time to expiration as an example.

The response says the problem depends on assumptions about the underlying’s process and distribution. Under a Black–Scholes framework, it is a two-boundary stopping-time problem; a density for that stopping time can be used to calculate its expectation. The answer cautions that the result may overestimate the expected time for far out-of-the-money strikes because the assumed underlying distribution is unrealistic. The exchange provides a modeling direction, but no derivation, numerical estimate, variability analysis, or empirical comparison, so it does not establish whether the estimate is practically reliable.

Key ideas

• The expected first time a price reaches either of two strikes depends on the assumed underlying process and distribution.
• Within Black–Scholes, the question can be framed as a stopping-time problem with two boundaries.
• A stopping-time density can be used to calculate the expected time to reach either strike.
• The answer cautions that Black–Scholes assumptions can overestimate the expected time for far out-of-the-money strikes.
• The discussion gives no empirical validation or coefficient of variation for the estimate.

Tags

# Average time it takes to test a strike?


# Average time it takes to test a strike?












My question can be confusing so it’s better I explain it with an example.

Let’s say I sell a strangle. That is with call at +27 delta and put at -27 delta. With 30 days to expiration.

Is it possible that there is a mathematical formula to estimate the average time it takes for the underlying to reach either of the short strikes? If so, is the result consistent/useful(i.e it’s coefficient of variation of the average is low)?

Example: maybe with the formula it shows that after 90% of the 30 days has passed then one of the strikes would have been reached.

If there is no maths formula, then I think use of historical data or Monte Carlo simulations could solve the problem. Is there any research paper, or article on this topic?

## Answer by MrLCh (score 1, accepted)

https://quant.stackexchange.com/a/77691

The question is still quite unclear. Assuming you want to solve it mathematically you need to provide at least some assumptions regarding the underlying process/distribution.

Assuming you want to solve it in a Black-Scholes framework (not applicable to real life trading) your questions boils down to stopping time (with two bounds). The density for this has been derived here. From this you could calculate the expected value of the stopping time, however this will be overestimated for far OTM strikes because of the underlying distribution assumption.

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