Option payoffs, profit, and put-call parity
If two portfolios pay the same amount at expiration no matter where the stock ends, must they cost the same today?
Yes, in a market with no arbitrage. Some mixes of options and bonds pay exactly what a share pays at expiration, so their prices are tied to the share price. Stage 1 asks you to find such a mix; stage 3 proves the price link, called put-call parity.
Before you start · 3 quick checks on continuous compounding and discounting, about 2 min
Every block is held until a fixed date \(T\), measured in years. \(S_T\) is the stock price at \(T\), and \(x^+ = \max(x, 0)\).
- European call with strike \(K\): the right to buy one share for \(K\) at \(T\). Payoff \((S_T - K)^+\). Its price today is \(C\).
- European put with strike \(K\): the right to sell one share for \(K\) at \(T\). Payoff \((K - S_T)^+\). Its price today is \(P\).
- Share of a stock that pays no dividends before \(T\): payoff \(S_T\). Its price today is \(S_0\).
- Zero-coupon bond paying \(F\) at \(T\): payoff \(F\) in every case. Its price today is \(Fe^{-rT}\), where \(r\) is the continuously compounded risk-free rate.
Long means you bought the block. Short means you sold it, and your payoff is the negative of the buyer's. A short bond is a loan you must repay.
Trading conditions, used on the whole page:
- Short sales allowed: you can sell a share you do not own and buy it back later.
- Each option can be bought or written (sold) at one price, its quoted price.
- Unlimited borrowing and lending at the risk-free rate \(r\).
- No transaction costs or taxes, and one price for buying and selling each block.
- The options are European: they can be exercised only at \(T\), and no block has a cash flow before expiration. The stock pays no dividends before \(T\).
These conditions describe how trading works. They do not say that prices are free of arbitrage; that is a separate assumption, stated with the theorem in stage 3. The dealer question at the end of stage 1 and problem P4 use only the trading conditions. The bull spread question in stage 2 also assumes no arbitrage, and says so.
Profit at \(T\) is the payoff minus the cost of the position carried to \(T\) at the rate \(r\):
\[ \text{profit} = \text{payoff} - (\text{cost today})\,e^{rT}. \]Why carry the cost to expiration
The premium is paid at time 0 and the payoff arrives at \(T\). To compare them, both must be measured at the same date. Money spent on the premium could have earned \(r\) instead, so its cost at \(T\) is \(e^{rT}\) times the premium.
With this convention a bond has zero profit: it costs \(Fe^{-rT}\), and \(F - Fe^{-rT}e^{rT} = 0\). Some textbooks subtract the premium without interest. For a few months to expiration and a small \(r\), the two versions differ by little.
Build the target payoff
Each goal below draws a target payoff as a wide gray band. Pick a block, a strike, and buy or sell it. Your position's payoff is drawn as a line on top of the target, and the page tells you where the two differ. Strikes run from 80 to 120 in steps of 10. Buying or selling the same block again changes its quantity.
How the match is judged
Every block's payoff is a straight line in \(S_T\) except for one bend at its strike. A sum of blocks is therefore made of straight pieces that can bend only at the strikes 80, 90, 100, 110 and 120.
Two such payoffs agree everywhere on \(S_T \ge 0\) exactly when they agree at \(S_T = 0\), at each strike, and at two prices above 120 (the page uses 200 and 300). Between consecutive checkpoints both are straight lines, and two lines that agree at two points are the same line.
Same payoff, same price?
From payoff to profit
Where the quotes come from
The call prices are values from a standard pricing model with 25% volatility. You do not need that model here. Each put price is then set by put-call parity, proved in stage 3. The page computes every cost and profit with these unrounded prices, so parity holds in them exactly. The table shows them rounded to the cent, so its rounded numbers satisfy parity only to within about a cent.
Profit goals
The builder now shows profit at expiration as a solid line and the payoff as a dashed line. The tiles show the cost, the worst and best profit, and the break-even final prices.
Put-call parity
Assumptions.
- A European call and a European put on the same stock have the same strike \(K \gt 0\) and the same expiration \(T \gt 0\). Their prices today are \(C\) and \(P\).
- The stock pays no dividends before \(T\). Its price today is \(S_0 \gt 0\), and its price at \(T\) is \(S_T \ge 0\).
- The continuously compounded risk-free rate \(r \gt 0\) is constant, so a bond paying \(K\) at \(T\) costs \(Ke^{-rT}\).
- Frictionless market: the trading conditions of the Setup hold. There are no transaction costs or taxes, options are bought and written at one price, borrowing and lending at \(r\) are unlimited, short sales are allowed, and no block has a cash flow before expiration.
- No arbitrage: no trade costs nothing today and has a payoff at \(T\) that is never negative and is positive with positive probability. In particular, no trade brings in positive cash today and has a payoff at \(T\) that is never negative.
Conclusion.
\[ C - P = S_0 - Ke^{-rT}. \]The argument works for any constant rate, including \(r \le 0\). The assumption \(r \gt 0\) matches the examples on this page, where a bond costs less than it pays.
| Trade | Cash today | At \(T\), \(S_T \lt K\) | At \(T\), \(S_T \ge K\) |
|---|---|---|---|
| Write the call | \(+C\) | \(0\) | \(-(S_T - K)\) |
| Buy the put | \(-P\) | \(K - S_T\) | \(0\) |
| Borrow \(Ke^{-rT}\) | \(+Ke^{-rT}\) | \(-K\) | \(-K\) |
| Buy one share | \(-S_0\) | \(S_T\) | \(S_T\) |
| Total | \(C - P + Ke^{-rT} - S_0 \gt 0\) | \(0\) | \(0\) |
| Trade | Cash today | At \(T\), \(S_T \lt K\) | At \(T\), \(S_T \ge K\) |
|---|---|---|---|
| Buy the call | \(-C\) | \(0\) | \(S_T - K\) |
| Write the put | \(+P\) | \(-(K - S_T)\) | \(0\) |
| Lend \(Ke^{-rT}\) | \(-Ke^{-rT}\) | \(K\) | \(K\) |
| Short one share | \(+S_0\) | \(-S_T\) | \(-S_T\) |
| Total | \(S_0 - (C - P + Ke^{-rT}) \gt 0\) | \(0\) | \(0\) |
Problems
All prices are hypothetical. All options are European, the stocks pay no dividends before expiration, and the trading conditions of the Setup hold. P1 and P6 use parity, so they also assume no arbitrage. P4 and P5 do not: they ask what the trading conditions allow when quoted prices break parity.
More problems
Further reading
- J. C. Hull, Options, Futures, and Other Derivatives, 10th ed. (Pearson, 2018), ch. 11, "Properties of Stock Options," and ch. 12, "Trading Strategies Involving Options."
- R. L. McDonald, Derivatives Markets, 3rd ed. (Pearson, 2013), ch. 2, "An Introduction to Forwards and Options"; ch. 3, "Insurance, Collars, and Other Strategies"; ch. 9, "Parity and Other Option Relationships."
- H. R. Stoll, "The Relationship Between Put and Call Option Prices," Journal of Finance 24(5) (1969), 801–824, doi:10.1111/j.1540-6261.1969.tb01694.x.