I jotted a time ago a Sage snippet for options pricing in elementary calculus terms, pasted here 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.
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.