Functions of Random Variables

An option payoff is $\max(S_T-K,0)$ — a function of the terminal stock price $S_T$, not $S_T$ itself. A P&L is a function of a position size and a price move. Every derivative instrument in the vault's learn/program/options-pricing/ unit is, formally, a function of a random variable: if $X$ is a random variable and $g$ is an ordinary function, $Y=g(X)$ is itself a new random variable, and this note shows exactly how to get $Y$'s PMF from $X$'s. The payoff is the expected value rule (also called LOTUS, the "Law of the Unconscious Statistician"): a shortcut that computes $\mathbb{E}[g(X)]$ directly from $X$'s own PMF, without ever constructing $Y$'s PMF — resolving the tedium flagged at the end of Expectation, Mean, and Variance.

Learning objectives. By the end of this note you should be able to: