I am calculating expected return for composite option strategies based on event probabilities provided by the broker. For example, consider the following spread

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On the left hand side we see:

  • maximal possible return $R_{max}=\$25$
  • probability of max return $P_{Rmax}=0.84$
  • maximal possible loss $L_{max}=\$225$
  • probability of max loss $P_{Lmax}=0.089$
  • expected commission $C=2\times \$3=\$6$ (times two, expecting the same on exit)

So basically, I am parametrizing all outcomes by taking a range on the real line $x\in(0,1)$ corresponding to outcome probabilities. On that range I construct a function as follows:

$$f(x)=\begin{cases} R_{\max }-C & x<P_{\text{Rmax}} \\ R_{\max }-C-\frac{\left(L_{\max }+R_{\max }\right) \left(x-P_{\text{Rmax}}\right)}{-P_{\text{Lmax}}-P_{\text{Rmax}}+1} & P_{\text{Rmax}}\leq x\leq 1-P_{\text{Lmax}}\\ -C-L_{\max } & x>1-P_{\text{Lmax}} \end{cases}$$

which linearly interpolates between max return and max loss assigning them the corresponding probability regions. Here a plot of $f(x)$:

enter image description here

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

$$I=\frac{\int_0^1 dx f(x)}{L_{max}+2C}$$

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?


The problem is very visible in the following setup:

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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

enter image description here

which crosses the x-axis at around $0.675$ instead of $0.41$, suggesting that there is definitely something wrong with the approach. How would one improve upon it?


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