Put-call parity
1. The idea
Compare two portfolios held until the options expire:
- A: a European call, and a bond that pays the strike price \(K\) at expiry.
- B: a European put with the same strike price and expiry, and one share.
If the share ends above \(K\), portfolio A exercises the call and pays the bond’s \(K\) for the share, so it ends with the share; portfolio B lets its put expire and also ends with the share. If the share ends below \(K\), portfolio A lets its call expire and keeps the bond’s \(K\); portfolio B exercises its put and sells the share for \(K\). Either way, A and B end with the same amount: the larger of the share price and \(K\). So they must cost the same today:
\[ c + K e^{-rT} = p + S_0, \]
where \(c\) and \(p\) are the call’s and the put’s prices, \(S_0\) the share price today, \(r\) the risk-free rate with continuous compounding and \(T\) the time to expiry in years. This is put-call parity. An example: a share at $44, a strike price of $45, four months to expiry and a 5% rate. If the call costs $2.41, the put must cost \(p = 2.41 + 45\,e^{-0.05 \times 4/12} - 44 = \$2.67\).
Dividends. A dividend paid before expiry goes to portfolio B only, through its share. So the share counts without it: \(c + Ke^{-rT} = p + S_0 - D\), where \(D\) is the present value of the dividends. For a stock index with a dividend yield \(q\), the index counts as \(S_0 e^{-qT}\).
Arbitrage. When prices break parity, buy the cheaper portfolio and sell the more expensive one. At expiry the two cancel, whatever the share price, so the gap between their prices is a gain made today. Grown at the risk-free rate until expiry, the same gain is larger: a question must say which of the two it asks for. With bid and ask prices, we buy at the ask and sell at the bid, which can leave no gap to use.
Connecting the dots. Rewritten as \(c - p = S_0 - Ke^{-rT}\), parity says that a bought call and a sold put with the same strike price make a forward contract with delivery price \(K\) (Level 1): both have the payoff \(S_T - K\) at expiry, where \(S_T\) is the share price at expiry. Portfolio B, a share with a put that protects it, is called a protective put. Parity holds for European options; for American options it becomes a range, as the lesson on bounds shows.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The two portfolios on a chart
First, portfolio A’s payoff if the share falls.
Now move the share price at expiry. The two lines lie on top of each other: whatever the share does, the call with the bond and the put with the share have the same payoff.
4. One worked example, then your turn
In the worked example, we find a call’s price on a share that pays a dividend.
Now an arbitrage with hints, one step at a time.
And two on your own: an arbitrage with bid and ask prices, and the interest rate that an index’s option prices imply.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a put’s price, a call’s price, an arbitrage and an implied interest rate. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: