Put-call parity: equivalent payoffs and financing

Put-call parity links matching European call and put prices with stock and financing. Understand the payoff identity before applying it to quotes.

By OptionsKing. Publisher and editorial methods.

Build two portfolios with the same terminal payoff, then compare their initial cost. That establishes the European no-arbitrage identity without requiring a particular volatility forecast. The UBER example illustrates the equality; dividends, exercise rights and execution explain why retail quotes may not match it exactly.

The equation, on real numbers

UBER at $68.40. The October 17 $72.50 strike, 42 days out, risk-free rate 4.2 percent, and no dividend before expiry.

• Call: $1.53
• Put: $5.28
• C minus P = -$3.75
• The strike discounted: 72.50 times e to the minus 0.042 times 42/365 = $72.15
• S minus that = 68.40 minus 72.15 = -$3.75

Both sides are $3.75. Not approximately. The relationship is an identity, and any option pricing model that violated it would be arbitraged out of existence before lunch.

The $0.35 of discount on the strike is the entire interest rate content of an option at this maturity, and it is where rho comes from.

Why it has to hold

Build two portfolios.

Portfolio A: buy the call, sell the put, both at $72.50. Portfolio B: buy the stock, borrow the discounted strike.

At expiry, whatever UBER does, both are worth exactly the same. Above $72.50 your call is exercised and you own the stock at $72.50; below it, your short put is assigned and you own the stock at $72.50. Either way you end up long shares at the strike, which is what portfolio B also gives you.

Two things that pay identically must cost identically. If they did not, you would buy the cheap one, sell the dear one, and collect the difference with no risk and no view. That trade is called a conversion or a reversal and it is the bread and butter of the desks who keep parity intact.

The consequence that matters most

Matching European put and call prices imply the same volatility in a consistent pricing model. The comparison requires the same underlying, strike, expiration, rates and dividend assumptions.

If those assumptions match but inferred IVs differ, investigate quote timestamps, executable prices and model inputs. American exercise rights, funding and transaction costs complicate the comparison; a displayed discrepancy is not automatically an executable arbitrage.

So when people describe the volatility skew as "puts are more expensive than calls", the precise version is different and worth getting right. The put and the call at the same strike agree exactly. What differs is one strike against another. The $65 put implies more volatility than the $84 call because they are different strikes, not because puts are inherently richer.

Once you have this, a lot of confused talk about skew resolves itself.

Synthetics

Include the cash or financing leg when rearranging parity. Let PV(K) mean cash today that grows to the strike at expiration. Holding that cash adds K to terminal value; borrowing it creates a K repayment obligation. For one standard equity contract, scale the shares and cash by 100.

Rearrange the equation and each rearrangement is a position you can actually build.

For example, stock plus a put equals a call plus PV(K) cash. Omitting the cash makes a protective stock-and-put position look identical in value to a call when it is not. In this example, $68.40 + $5.28 = $1.53 + $72.15, using rounded per-share values.

A covered call equals a short put plus strike-funding cash under these matching European assumptions. Both terminal values are min(stock price, strike), but brokerage collateral, taxes, dividends and American assignment paths can differ. Compare the complete funded positions, not just the option labels.

Where parity bends

It holds exactly for European options with no dividends. Real American equity options introduce two wrinkles.

Early exercise. American options can be exercised any day, which gives them a small extra value the equation does not carry. It matters mostly for in-the-money puts and for calls around an ex-dividend date.

Dividends. A dividend before expiry lowers the forward price of the stock and shifts the relationship. The example above assumes none, which is why the arithmetic is clean. Merton extended the theory to cover dividends the same year Black-Scholes was published.

Treat an apparent parity mismatch as a reason to inspect the inputs. Check matching terms, exercise style, dividends, financing, quote timestamps and executable bid-ask prices. A discrepancy at displayed midpoints may disappear after costs; it does not by itself establish a tradable profit.

Common questions

Sources

• Merton (1973), Bell Journal of Economics 4(1), 141 to 183
• Black and Scholes (1973), Journal of Political Economy 81(3), 637 to 654
• OCC / Options Industry Council, Options Exercise FAQ
• OCC, Characteristics and Risks of Standardized Options

Rules and thresholds above were checked against these documents on August 4, 2026. Exchange and broker rules change. Confirm anything you are about to act on with your own broker.

Keep reading

• The discount term, as a Greek
• What parity says skew actually is
• The synthetic pair, compared in practice

Do the math on your own trade

• Black-Scholes calculator price and Greeks from five inputs.
• Payoff diagram builder draw the position and see where the line goes flat.
• Cash-secured put calculator credit, break-even, return on secured capital, annualized.

Every price in this article is an illustrative worked example, not a quote. Read The Greeks for the rest of the series, and the disclaimer before you act on any of it. Selling options carries real risk of loss, and the loss can be far larger than the premium you collected.