Mathematics of Options Trading is a good book for the expected return modeling. Reehl explains a numerical, practical approach in estimating expected returns off option trades with respect to your typical option related variables. While the model is no where near perfect, it could definitely give the trader an edge.

Reehl's models basically chop an option trade's future payoff into separate regions with respect to future underlying values, e.g. a long call

Region 1: max loss (x-4 standard deviations) to break even

Region 2: break even to Strike Price

Region 3: Strike Price to max Profit (x+4 standard deviations)

where x = future underlying value.

then with the probability density function (PDF), expected returns for each region is derived. Their sum then becomes the net expected value for the trade.

All models have weaknesses. It's about understanding, and remedying their weaknesses to utilize them successfully. Reehl's approach assumes a lognormal distribution for PDF of the underlying expected returns, with respect to volatility and days til expiration. In times of panic, we've all learned that volatility could spike significantly more than the expected range(s).

So knowing the model may underestimate future volatility, how do we remedy this?

1) Take only long volatility/gamma trades where Reehl's model gives a +Expected Value.

2) Over-estimate future volatility significantly in the model for short vol/gamma trades.

3) Replace the lognormal with a power law distributions to suit fatter tails.

I'm sure there're more ways around it, these're just my immediate thoughts.

**Applying mathematical models**Reehl's models basically chop an option trade's future payoff into separate regions with respect to future underlying values, e.g. a long call

Region 1: max loss (x-4 standard deviations) to break even

Region 2: break even to Strike Price

Region 3: Strike Price to max Profit (x+4 standard deviations)

where x = future underlying value.

then with the probability density function (PDF), expected returns for each region is derived. Their sum then becomes the net expected value for the trade.

All models have weaknesses. It's about understanding, and remedying their weaknesses to utilize them successfully. Reehl's approach assumes a lognormal distribution for PDF of the underlying expected returns, with respect to volatility and days til expiration. In times of panic, we've all learned that volatility could spike significantly more than the expected range(s).

So knowing the model may underestimate future volatility, how do we remedy this?

1) Take only long volatility/gamma trades where Reehl's model gives a +Expected Value.

2) Over-estimate future volatility significantly in the model for short vol/gamma trades.

3) Replace the lognormal with a power law distributions to suit fatter tails.

I'm sure there're more ways around it, these're just my immediate thoughts.

## 1 Reflections:

Of course there are more ways around it, but your thoughts helped me understand how to maximize the weaknesses of some models in my favor. Understanding more about an option trade's future is an important binary options strategy

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