Spreads versus Butterfly

How can you ever make statements like these without considering a proper pdf function? (and that would probably be ‘90 style)
 
How can you ever make statements like these without considering a proper pdf function? (and that would probably be ‘90 style)

That's why we have standard deviations...its basically how options are priced.
 
How can you ever make statements like these without considering a proper pdf function? (and that would probably be ‘90 style)

However, the actual truth is PDF (probability density function ) is defined for continuous random variables, whereas PMF (probability mass function) is defined for discrete random variables.

The probability density function measures continuous variables—stock and investment returns are generally not continuous random variables; they are discrete.

Wouldn't a pmf be more suitable?
 
I like to start off a trade as a spread and then convert it to a butterfly if it moves in the desired direction to take the risk out of the trade
 
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