I jotted a time ago a Sage snippet for options pricing in elementary calculus terms, pasted here https://pastebin.com/tTMp6fPk https://pastebin.com/tTMp6fPk. T

I jotted a time ago a Sage snippet for options pricing in elementary calculus terms, pasted here https://pastebin.com/tTMp6fPk https://pastebin.com/tTMp6fPk. The idea is that the clean picture is done in terms of log-prices (not prices). Probability of log-prices follows a diffusion with an initial Dirac delta at-the-money. At expiration the profit function is deterministic (0 out of the money, a ramp if in the money) and the probability is certain gaussian. The expectancy of the value of a function applied to a random var of given density is like a weighted sum of the values, weighted by the frequency/density, as in a dot product (an integral here). Add to that the "time value of money" (see Investopedia) that works as linear drift, and you are done.