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

enter image description here

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?

EDIT

The problem is very visible in the following setup:

enter image description here

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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