# Options trade - statistically expected return calculation?

I also asked this question here, for it to get more visibility.

I am calculating expected return for composite option strategies based on event probabilities provided by the broker. For example, consider the following spread On the left hand side we see: So basically, I am parametrizing all outcomes by taking a range on the real line x in the range (0,1), corresponding to outcome probabilities. On that range I construct a function as follows: which linearly interpolates between max return and max loss assigning them the corresponding probability regions. Here a plot of f(x): which basically visualizes the weight distribution of returns or losses based on respective probabilities. Now I take the expected overall % return/loss on risk to be given by the integral In the example above, the expected % return/loss on risk turns out to be I=-0.052, which means one can expect to lose on average about 5.2% of risked capital after a large number of such trades. (Very bad proposition.)

Is the above approach sound, or can you suggest a better way to calculate average expected return?

One thing that makes me skeptical, is that overall Profit Probability is stated by the broker to be 85%, while function f(x) crosses the x-axis at about 0.84 which is a slight deviation. Based on this I suspect that the above approach is not very precise?

Also, note that the function f(x) basically mimics the spread in this case and simply maps the outcomes onto the range x in (0,1). However, the stated probabilities for the outcomes are generated by the broker in the same fashion for any complicated composite strategies. Do you think this calculation may be less reliable for a more complicated strategy?

EDIT

The problem is very visible in the following setup: The overall Profit Probability in this case is stated by the broker to be 41%, while constructing function f(x) as above produces the weight curve which crosses the x-axis at around 0.675 instead of 0.41, suggesting that there is definitely something wrong with the approach.

• I have many reservations about what you are writing. Can you specify the terms of the contract? It can be without the name of the underlying. I would let you know that many classes of assets have distributions without a first moment. Brokers, for purposes of calculation, pretend they do have one. For example, equity securities have a distribution that is proportionate to a Cauchy distribution. en.wikipedia.org/wiki/Cauchy_distribution – Dave Harris Sep 8 '19 at 14:52
• @DaveHarris Thanks for taking the time to reply! What exactly do you mean by the terms of the contract? Both examples are combinations of simple options one week until expiration (bought and sold according to the respective diagrams, as usual). I actually recalculated the interpolation assuming gaussian distribution of deviations. Thanks for pointing out Cauchy distribution, I'll recalculate with that and compare. – Kagaratsch Sep 8 '19 at 15:49
• Easy enough to duplicate. The first position is a \$2.50 wide bearish vertical spread with a \$2.25 debit. Second is a short butterfly. – Bob Baerker Sep 8 '19 at 18:14
• @BobBaerker Oh, I see, yes that's right. I thought it was clear from the diagrams. – Kagaratsch Sep 8 '19 at 20:30