Put-call parity, and what it forces to be true
Put-call parity says the price of a call minus the price of a put at the same strike and expiry equals the stock price minus the strike discounted at the risk-free rate. It holds by arbitrage, not by assumption, which makes it the only thing on this page that does not depend on Black-Scholes being right.
Everything else in this cluster comes out of a model with six assumptions, four of which are false. This one comes out of the fact that if it were violated, somebody would take the free money.
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
The put and the call at the same strike must carry the same implied volatility.
Parity is a relationship between prices, and prices translate into implied volatilities. If the $72.50 call implied 32 percent and the $72.50 put implied 36 percent, the two sides of the equation would not balance, and the arbitrage would exist.
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
Rearrange the equation and each rearrangement is a position you can actually build.
| You want | Build it from |
|---|---|
| Long stock | long call, short put |
| Short stock | short call, long put |
| Long call | long stock, long put |
| Short put | long stock, short call |
Look at the last row. A covered call is a synthetic short put. Same payoff, same risk, same shape, different name and different margin treatment.
That is not trivia. It means the argument about whether covered calls are safer than cash-secured puts is, at the same strike and expiry, an argument about nothing. They are the same trade. The real differences are capital, tax treatment and what happens on assignment, not risk profile.
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.
In practice, parity is close enough that a persistent violation you can see on a retail screen is almost certainly a stale quote rather than an opportunity. If you think you have found one, check the timestamps and check the bid-ask on both legs before you check your brokerage balance.
OptionsKing scores every candidate strike on a deterministic 0 to 100 scale, blends that score with the return on the capital the trade ties up, and shows you the highest-ranked handful. There is no minimum score. How it works covers what the ranking does and does not tell you.
Questions people actually ask
What is put-call parity?
The relationship that a call minus a put at the same strike and expiry equals the stock price minus the strike discounted at the risk-free rate. It holds by arbitrage rather than by modelling assumption.
Do puts and calls at the same strike have the same implied volatility?
They must, or parity would be violated and the arbitrage would exist. Skew is a difference between strikes, not between puts and calls at one strike, which is a distinction most explanations of skew get wrong.
Is a covered call the same as a short put?
Synthetically, yes. Long stock plus a short call has the same payoff as a short put at that strike, so at the same strike and expiry they are the same trade. The differences are capital required, tax treatment and what assignment does.
Does put-call parity hold for American options?
Approximately. Early exercise rights and dividends both break the exact equality, which is why it is stated as an inequality for American options. Any large violation visible on a retail screen is almost always a stale quote.
Sources
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
Do the math on your own trade
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.