Risk-Neutral Pricing
No-arbitrage guarantees an equivalent martingale measure, so derivatives price as discounted expected payoffs under it.
Why can options be priced with an expectation, as if investors were risk neutral? The fundamental theorem of asset pricing answers: in a market without arbitrage there exists an equivalent probability measure under which discounted asset prices are martingales, and derivatives trade at the discounted expectation of their payoffs under that measure. Replication makes the expectation unique wherever payoffs are attainable. This is the hinge of the whole track: stochastic calculus supplies the dynamics, the martingale measure prices them, and the next three nodes are different machines, PDE, tree, and simulation, for computing the same expectation.