# 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?

• @ Dragan Chupacabric : By probability you mean risk-neutral Probability (associated to stock-numéraire), right ? This is not comlpetely clear from your question when you say "I use delta as a proxy for this probability of success for single options". Regards Mar 24, 2011 at 10:18
• @ TheBridge: Correct, I'm looking for a risk-neutral probability, which makes an option (combo) a fair bet. Mar 24, 2011 at 13:51

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)

• @ glyphard: What you are saying is correct, but I'm interested in the prob of success of a trade (breaking even or better at expiration). I'd like to heuristically imply this prob for a straddle/strangle. I should be able to imply it from the attributes (prices of diff strikes etc.) of the surrounding options, since they can be used to imply the risk-neutral distribution for the underlying. I'm wondering if there is a heuristic, just like in my example for spreads. Regards Mar 24, 2011 at 14:45
• @Dragan: A heuristic for that is to calculate the breakeven, at price of the underlying for a given position, then take the delta for an option at that strike. that is your probability of success for the trade. (the probability that the underlying reaches that level by expiration it works for any options position) Mar 25, 2011 at 21:53

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.

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)