Put-Call Parity for American Options: The Bounds and Where They Come From
For European options, put-call parity is an equation. The call price minus the put price equals the stock price minus the present value of the strike, and there is nothing to argue about. For American options it becomes a pair of inequalities, and the difference between the call and the put can sit anywhere inside a band.
Most treatments state that band and move on. The band is worth deriving, because each side comes from a different argument, and knowing which argument produced which side is what lets you use the result rather than recite it.
Key Takeaways
- For European options put-call parity is an equation. For American options it becomes a band, because an option can be exercised before expiry.
- The band runs from the stock price minus the strike, up to the stock price minus the present value of the strike.
- Its width is the interest on the strike over the option’s life, so a higher rate or a longer maturity widens it.
- The upper bound holds because the American call equals the European call on a non dividend payer, while the American put is worth more than the European put.
- The lower bound comes from a portfolio that can never end negative, so it cannot be set up for a negative cost.
- With dividends, subtract their present value from the stock price at both ends of the band.
Where the European equation comes from
Put-call parity links a call and a put written on the same underlying, with the same strike and the same expiry. For European options on a non dividend paying stock it is exact:
where c and p are the European call and put prices, S0 is the stock price today, X is the strike, r is the continuously compounded risk free rate, and T is the time to expiry.
The reason it is exact is that two different portfolios end up holding the same thing. Buy the call and put aside cash equal to the discounted strike, and at expiry you hold either the stock or the strike, whichever is larger. Buy the put and the stock, and at expiry you again hold the larger of the two. Two portfolios with identical payoffs in every state must cost the same today, or a trader can buy one, sell the other and collect the difference for nothing.
That equivalence is what makes synthetic positions possible. Rearranged, the same equation says a long call plus cash behaves exactly like a long put plus the stock, so any one of the four instruments can be manufactured from the other three. A desk that cannot buy a put directly can build one, and the price it should pay is fixed by the equation rather than negotiated.
Everything after this rests on one fact about that argument. It works because a European option cannot be touched before expiry, so both portfolios are guaranteed to survive intact until then. Remove that guarantee and the equality goes with it.
Why American options give a band and not an equation
An American option can be exercised at any moment, which means the portfolio that contains one can be broken open early. The replication argument no longer holds as an equality, because one leg can disappear before the other.
For American options on a non dividend paying stock, the result becomes:
where C and P are the American call and put prices. Capital letters for American, lower case for European, is the usual convention and this article follows it.
Look at what the two ends are made of. The upper end is the European parity expression exactly. The lower end is the same expression with the strike left undiscounted. The width of the band is therefore X minus its own present value, which is nothing more than the interest on the strike over the life of the option.
The band is not vague. Its width is X − Xe−rT, the interest that could be earned on the strike between now and expiry. A longer life or a higher rate widens it. As either goes to zero, the band closes and the American relationship collapses back onto the European equation.
Deriving the upper bound
The upper bound comes from two facts, one about the call and one about the put.
The call first. On a non dividend paying stock, an American call is never exercised early, so the right to do so carries no value and the American call is worth exactly what the European call is worth:
The put is different. An American put can be worth exercising early, particularly when it is deep in the money, so the extra right does carry value. That gives an inequality rather than an equality:
Now take the European equation and substitute. Because C equals c and P is at least p, the difference C minus P can only be smaller than or equal to c minus p:
The upper bound is simply European parity, reached by noting that the American call gains nothing from its extra right while the American put gains something. All of the slack in the relationship is created by the put.
Deriving the lower bound
The lower bound needs a genuine no arbitrage argument, and it is the half that most treatments skip.
Build a position today: buy the American call, sell the American put, short the stock, and lend an amount X at the risk free rate. The cost of setting this up is C − P − S0 + X. Now check what happens in every case.
Follow the position through the three things that can happen to it. The claim to test is that its value is never negative, whatever the stock does.
- The put is exercised against you at some time t before expiry. You must buy the stock at X, and that share closes your short. Your loan has grown to Xert, so paying X out of it leaves Xert − X, which is positive. You still hold the call, which is never worth less than zero.
- The stock finishes above the strike at expiry. The call is worth ST − X, the put expires worthless, closing the short costs ST, and the loan is worth XerT. Adding those gives XerT − X, positive again.
- The stock finishes below the strike at expiry. The call expires worthless, the put is exercised against you at a cost of X − ST, closing the short costs ST, and the loan is worth XerT. That also comes to XerT − X.
Answer: the position ends with a non negative value in every case. A position that can never lose cannot be set up for a negative cost, so C − P − S0 + X ≥ 0, which rearranges to C − P ≥ S0 − X.
Read the three cases again and notice what they have in common. Each one ends at Xert − X, the interest earned on the loan. That is the same quantity as the width of the band, arriving from the other direction, which is the sign that the two bounds are describing one thing rather than two.
A worked example, and the trade when the band breaks
Numbers make the band concrete, and they also show what to do when a quoted pair of prices sits outside it.
A non dividend paying stock trades at 45. American options on it carry a strike of 42 with nine months to expiry, and the risk free rate is 6% continuously compounded. The call is quoted at 6.20 and the put at 1.00. Is that pair of quotes consistent?
- Compute the lower bound. S0 − X = 45 − 42 = 3.00.
- Discount the strike. 42 x e−0.06 x 0.75 = 42 x 0.95600 = 40.15.
- Compute the upper bound. 45 − 40.15 = 4.85.
- Compute the observed difference. 6.20 − 1.00 = 5.20.
- Compare. 5.20 sits above 4.85, so the pair is outside the band by 0.35 per share.
Answer: the call is expensive relative to the put. Sell the call, buy the put, buy the stock, and borrow the discounted strike of 40.15. That collects 6.20 − 1.00 − 45 + 40.15 = 0.35 today. At expiry the stock and the put together deliver the larger of the stock price and 42, the short call takes away anything above 42, and repaying the loan costs 42, so the position nets to zero. The 0.35 was riskless.
A difference below the lower bound is the mirror image. The put is then expensive relative to the call, and the trade reverses: buy the call, sell the put, short the stock and lend the proceeds, which is exactly the position used to derive the lower bound in the previous section. That is not a coincidence. Each bound is derived from the trade that would be available if it were violated, which is why the derivation and the trade are the same piece of reasoning read in opposite directions.
Dividend paying stocks
A dividend goes to whoever owns the share, and neither the call holder nor the put holder owns it. When a dividend is expected during the life of the options, the stock is worth less to an option holder by exactly the present value of what will be paid out, and both bounds shift by that amount.
Write PV(D) for the present value of the dividends expected before expiry. The relationship becomes:
where PV(D) is the present value of the dividends payable during the option’s life, discounted at the risk free rate.
| Underlying | Lower bound on C − P | Upper bound on C − P |
|---|---|---|
| Non dividend paying stock | S0 − X | S0 − Xe−rT |
| Dividend paying stock | S0 − PV(D) − X | S0 − PV(D) − Xe−rT |
| European options, for contrast | c − p = S0 − Xe−rT, an equation rather than a band | |
Adding the dividend term rather than subtracting it. The dividend reduces what the stock is worth to an option holder, so it comes off the stock price on both ends. A useful check is the direction of travel: a dividend makes the call less valuable and the put more valuable, so C − P must fall, which means the band must move down.
One further caution belongs with dividends. The upper bound was built on the fact that an American call on a non dividend payer is never exercised early. Once a dividend enters, that fact stops holding, early exercise of the call becomes possible immediately before an ex-dividend date, and the argument for the upper bound weakens rather than disappearing. Treat the dividend-adjusted band as a working approximation, not as the same result with an extra term.
What the bounds do not tell you
Because the width is nothing but interest on the strike, the inputs move it in predictable directions, and knowing which way each one pushes is usually enough to catch an error before the arithmetic is finished.
| If this rises | The band | Because |
|---|---|---|
| Risk free rate | Widens | More interest can be earned on the strike before expiry |
| Time to expiry | Widens | The strike is discounted over a longer period |
| Strike price | Widens, and both ends fall | A larger strike carries more interest, and both ends subtract it |
| Stock price | Shifts up, width unchanged | The stock price enters both ends identically |
| Expected dividends | Shifts down, width unchanged | PV(D) is subtracted from both ends |
The band is a consistency check, not a pricing model. It constrains the difference between two prices and says nothing about either price on its own. A call and a put can both be badly mispriced and still sit comfortably inside the band, as long as they are mispriced together.
It also gets looser exactly where options are most interesting. The width is the interest on the strike, so a long dated option in a high rate environment carries a wide band inside which almost any difference is permitted. For an option with a month to run at a low rate the band is narrow and the check is sharp.
When an actual value is needed rather than a bound, the tool is a binomial tree, where exercise is tested against continuation at every node. The band is what you use to sanity check a quote in seconds; the tree is what you use to price.
One habit makes the check reliable under time pressure. Compute the upper bound first, because it is the European parity number and you will need it anyway, then get the lower bound by simply adding back the interest on the strike rather than recomputing from scratch. Two numbers, one subtraction between them, and the width doubles as an arithmetic check: if the gap between your two bounds is not close to the interest on the strike, one of them is wrong.
Questions here usually give you a stock price, a strike, a rate, a maturity and a pair of quotes, and ask whether an arbitrage exists. Work in a fixed order. Compute the lower bound as the stock price minus the strike. Discount the strike and compute the upper bound. Subtract the dividend present value from both if the stock pays one. Then compare the quoted difference against the two ends and, if it falls outside, name the trade: sell the expensive side, buy the cheap side, and use the stock and a loan to close the position.
The short version
- European put-call parity is an equation. For American options it becomes a band.
- The band runs from the stock price minus the strike, up to the stock price minus the discounted strike.
- Its width is the interest on the strike over the option’s life, so a higher rate or a longer life widens it.
- The upper bound follows from the American call being worth the same as the European call while the American put is worth more.
- The lower bound follows from a portfolio that can never end negative, so it cannot cost less than nothing today.
- With dividends, subtract their present value from the stock price at both ends.
- The band checks two prices against each other. It does not price either of them.
Frequently Asked Questions
Why is put-call parity an inequality for American options?
Because an American option can be exercised before expiry, so the replicating portfolio that produces the European equation can be broken open early and the two sides are no longer guaranteed to match. The extra right is worth something for the put and nothing for the call on a non dividend payer, and that asymmetry turns the equation into a band.
What are the bounds for American options on a non dividend paying stock?
The difference between the call and put prices lies between the stock price minus the strike at the lower end, and the stock price minus the present value of the strike at the upper end. The upper end is the European parity expression, and the lower end is the same expression with the strike left undiscounted.
How wide is the band?
Its width is the strike minus the present value of the strike, which is simply the interest that could be earned on the strike between now and expiry. A longer maturity or a higher interest rate makes the band wider and the consistency check weaker. As either approaches zero the band closes and the American relationship collapses back onto the European equation.
How do dividends change the bounds?
Subtract the present value of the dividends expected during the option’s life from the stock price at both ends of the band. A dividend makes the call less valuable and the put more valuable, so the whole band shifts down. Note also that with dividends an American call can be worth exercising early, so the argument behind the upper bound weakens and the adjusted band should be treated as a working approximation.
What do I do if the quoted prices fall outside the band?
Sell the expensive side and buy the cheap one, using the stock and a risk free loan to close the position. If the difference sits above the upper bound the call is expensive, so sell the call, buy the put, buy the stock and borrow the discounted strike. If it sits below the lower bound the put is expensive and the trade reverses.
Can I use the bounds to price an American option?
No. The band constrains the difference between a call and a put, not either price on its own, so both can be mispriced together and still satisfy it. For an actual value, use a binomial tree that tests exercise against continuation at every node. The band is a quick consistency check, not a valuation method.
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