Options & Put–Call Parity

Calls and puts are rights, and one identity ties their prices together with no model of the stock.

The idea

An option is the right, never the obligation, to trade a share at a fixed price. A call with strike $K$ is the right to buy one share at $K$ when the option expires; a put is the right to sell one share at $K$.

The holder exercises a right only when doing so pays, and that fixes the option's value at expiry. With the share at $S_{T}$, exercising a call trades $K$ for something worth $S_{T}$, worthwhile exactly when $S_{T} > K$; so the call pays $\max(S_{T} - K, 0)$, and likewise the put pays $\max(K - S_{T}, 0)$.

Put–call parity ties the two prices together.

Theorem (Put–call parity).

Let $C$ and $P$ be today's prices of a call and a put on the same share, with the same strike $K$ and the same expiry $T$ years away; let $S$ be today's share price and $r$ the risk-free rate. Then $C + \frac{K}{(1+r)^{T}} = P + S.$

The term $K/(1+r)^{T}$ is the cost today of a risk-free bond maturing to $K$ at expiry. The identity holds in any market free of arbitrage (Arbitrage & the Law of One Price) and assumes nothing about how the stock moves, so any three of the four prices pin down the fourth.

Ways to work on it

• Walkthrough. Call and put payoffs, and the portfolio comparison behind put–call parity.
• Practice. Compute payoffs at expiry and use parity to find a missing price.
• Hardest. Spot the mispricing when put–call parity fails and construct the arbitrage.

Not sure where to start? Take the ten-question placement test.