Estimating Expiration Profit Probabilities for Option Combinations
Summary
This discussion asks how to estimate the chance that a non-defined-risk options combination, such as a straddle or strangle, will be profitable at expiration. It contrasts using delta as a rough probability proxy for an individual option with inferring probabilities from the payoff and price of a defined-risk spread. One response suggests summing the absolute deltas of a strangle’s legs to approximate the probability that at least one leg finishes in the money; it also notes that this event is distinct from both legs finishing in the money.
Another response proposes modeling the expiration value of each leg under Black–Scholes assumptions, combining the resulting distributions numerically, and evaluating the probability that the total position is profitable. These are heuristic suggestions, not a demonstrated, validated method. Delta-based estimates depend on distributional assumptions, and the document does not fully specify how to handle dependence, premiums, or the position’s exact profit threshold. Probability that a leg is in the money should not be confused with probability that the overall trade earns a profit.
Key ideas
- Delta is used as a rough proxy for an individual option’s probability of finishing in the money.
- One response estimates a strangle’s chance of having a leg in the money by adding the absolute deltas of its legs.
- The probability that a leg is in the money differs from the probability that the complete position is profitable.
- A distribution-based alternative models each option leg and combines the distributions numerically.
- The proposed estimates rely on assumptions and are not supported by validation in the discussion.
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# Heuristics for calculating theoretical probabilities of being ITM at time T for listed options
# Heuristics for calculating theoretical probabilities of being ITM at time T for listed options
I'm looking for a heuristic way to calculate the probabilities of being in the money at expiry for non-defined risk options combinations (listed options).
I use delta as a proxy for this probability of success for single options, which makes an implicit distributional assumption.
For spreads I use width of the spread (or the worst drawdown/largest possible gain for more complex defined risk combinations) and $ received/paid for it. I treat the options combos as if they were bets and I get the implied probabilities from the prices of those bets.
What is a good heuristic for estimating such probabilities for straddles and strangles (and other non-defined risk combinations)?
EDIT: To clarify the above: a straddle/strangle is a bet. What's the probability of this bet being profitable at expiration? How do I imply the probability of success of this bet?
## Answer by glyphard (score 3)
https://quant.stackexchange.com/a/807
For a straddle, the probability of both legs being in the money is zero :-) The probability of one of the legs being in the money is essentially 1.
For a strangle, the probability of one of the legs being in the money at expiration is the sum of the absolute values of the deltas of the two legs of the strangle. ( think about one side of the strandge close to the money, and the other side far out of the money... the total probability has to be greater than the probability of the near leg along)
## Answer by user59 (score 2)
https://quant.stackexchange.com/a/823
I'm probably missing something, but why not apply Black-Scholes to each leg and add the results to get the price distribution for the spread? You'll get a non-closed-form result, but can evaluate it to arbitrary precision using numerical methods.
To add probability distributions:
```
Suppose Z = X + Y where X and Y are independent probability
distributions. Then (PDF = probability distribution function, CDF =
cumulative distribution function):
P(Z=z) = P(X=x)*P(Y=z-x) integrated over all x, or (Mathematica format):
PDF[Z,z] = Integrate[PDF[X,x]*PDF[Y,z-x],{x,-Infinity,+Infinity}]
A mathematically equivalent form:
CDF[Z,z] = Integrate[CDF[X,x]*PDF[Y,y],{y,-Infinity,z-x},{x,-Infinity,+Infinity}]
(derivation left as exercise to the reader)
```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.