No-Arbitrage and Put-Call Parity

An arbitrage is a trade that costs nothing (or pays you) today, can never lose money, and makes money in at least one outcome. In liquid markets these don't last. The first trader to spot one does it in size, and that buying and selling pushes the prices back into line within seconds.

Pricing theory turns this around. Assume no arbitrage exists and ask what that forces prices to be. The main tool is the law of one price: if two portfolios pay exactly the same amount at time $T$ in every state of the world, they must cost the same today. If portfolio A costs 101 and portfolio B costs 100 with identical payoffs, you buy B, sell A, pocket 1 now, and at $T$ the two cash flows cancel.

The simplest case is a zero-coupon bond. A bond that pays $K$ at time $T$ costs $Ke^{-rT}$ today, where $r$ is the continuously compounded risk-free rate. At $r = 5\%$ and $T = 1$, a promise of 100 in a year is worth $100e^{-0.05} = 95.12$ today. If it traded at 94, you would buy it and borrow 94 at $r$, owing $94e^{0.05} = 98.82$ in a year against the 100 the bond pays. Every argument in this lesson uses only this bond, the stock, and options on the stock.