American Option Bounds And The Limits Of Equality

Why American Options Do Not Have European Put-Call Equality

For European options on a non-dividend-paying asset, put-call parity gives an equality connecting the call price, put price, asset price, and discounted strike. American options differ because their holders may exercise before expiration. That additional choice changes the relationship: under the assumptions in this tutorial, American call and put prices satisfy bounds, not generally the European equality.

The distinction matters when checking option quotes. A difference between the quoted call and put prices that falls outside the American bounds is inconsistent with the stated assumptions and no-arbitrage. A difference inside the bounds passes this particular test, but the bounds alone do not prove that every quote is correct or free of other arbitrage.

Definition: An American call or American put with strike $K$ and expiration $T$ may be exercised at any permitted time up to and including expiration. Write their time-zero prices as $C_A$ and $P_A$. The corresponding European options, exercisable only at expiration, have prices $C_E$ and $P_E$.

Assume a frictionless market with no arbitrage, a non-dividend-paying underlying asset with current price $S_0$, and a risk-free rate that is constant and nonnegative over the option term. Let $T$ be measured in years and let $i$ be the effective annual rate. The discount factor for the term is $v^T=(1+i)^{-T}$. The options have the same underlying, strike, and expiration, and exercise is allowed under the usual American contract terms. These bounds do not apply unchanged when the asset pays dividends, rates are negative, or the contracts have different terms.

Theorem: Under these assumptions, the American call and put prices satisfy $$ S_0-K \leq C_A-P_A \leq S_0-Kv^T. $$ The difference $C_A-P_A$ is bounded below by the asset price minus the undiscounted strike and above by the asset price minus the present value of the strike.

Why the European Equality Becomes a Pair of Bounds

A European call on a non-dividend-paying asset is not made more valuable by exercising it early when interest rates are nonnegative. Exercising early gives up the remaining time value and requires paying $K$ sooner than necessary; it does not capture a dividend, because there is none. Thus the American call has the same value as the European call: $C_A=C_E$. An American put, by contrast, can be worth more than its European counterpart because early exercise can give the holder the strike sooner.

This also gives a short derivation of the upper bound. European put-call parity for the matching European options is $C_E-P_E=S_0-Kv^T$. Since the American put includes all the exercise opportunities of the European put, $P_A\geq P_E$. Combining those facts,

C_A-P_A=C_E-P_A\leq C_E-P_E=S_0-Kv^T.

The lower bound is the other side of the standard American put-call comparison. Its no-arbitrage argument accounts for the different exercise rights: the American call-minus-put position cannot be worth less than $S_0-K$. If it were, the position could be compared with holding the asset and borrowing the strike; the exercise rights would permit a nonnegative payoff comparison while the initial prices would favor an arbitrage. Together with the upper comparison above, this yields the stated interval. In contrast to the upper-bound derivation, the lower comparison does not follow just by subtracting $P_A\geq P_E$; it uses the American exercise opportunities.

The interval has a simple interpretation. The lower endpoint treats the strike as if it were paid now; the upper endpoint treats it as a payment at expiration and discounts it. Because $i\geq0$, $v^T\leq1$, so $S_0-K\leq S_0-Kv^T$. If $i=0$, the endpoints coincide, and the difference is pinned down at $S_0-K$. If rates are positive, the interval has width $K(1-v^T)$.

Worked Example: Testing a Pair of American Quotes

A hypothetical non-dividend-paying asset is priced at $S_0=\$120$. Matching American options have strike $K=\$110$, expiration $T=2$ years, and hypothetical quotes $C_A=\$23.00$ and $P_A=\$7.00$. The effective annual risk-free rate is $i=5\%$, constant over the term. Do the quotes pass the American put-call difference bounds?

Method. Calculate both endpoints using the same two-year discount factor, then compare their interval with the quoted difference.

First, the discount factor is $$ v^T=(1.05)^{-2}=\frac{1}{1.1025}=0.907029\ldots. $$

The lower endpoint is $$ S_0-K=120-110=\$10.00. $$

The upper endpoint is $$ S_0-Kv^T=120-(110)(0.907029\ldots) =120-99.7732\ldots =\$20.2268\ldots. $$

The quoted difference is $$ C_A-P_A=23-7=\$16.00. $$

Since $\$10.00\leq\$16.00\leq\$20.2268\ldots$, the quotes pass this necessary American difference-bound test. The conclusion is limited: satisfying the interval does not establish that the quotes are uniquely determined or rule out every other pricing inconsistency. Common mistake avoided: using the European equality as an exact requirement for American prices rather than checking whether their difference lies in the allowed interval.

Early Exercise and the Limits of Equality

For European options, the difference is exactly $S_0-Kv^T$. With no dividends and nonnegative rates, the American call still equals the European call, but the American put can have an early-exercise premium. In symbols, write that nonnegative premium as $\pi=P_A-P_E$. The call-put difference then has the form

C_A-P_A=C_E-P_E-\pi=S_0-Kv^T-\pi.

When $\pi>0$, this difference is below the European parity value. It need not equal either endpoint of the American interval. The bounds describe the range allowed by the exercise rights; they do not assert that every value in the range is attainable for every specified set of option parameters.

The American call's no-early-exercise result is specific to the no-dividend assumption and nonnegative rates. If the asset pays dividends, exercising a call before an ex-dividend date can have a benefit that must be weighed against paying the strike early. That changes the analysis, so the equality $C_A=C_E$ and the resulting bounds should not be carried over without adjustment.

Worked Example: Finding a Minimum Put Quote

A hypothetical asset has $S_0=\$150$, with matching American options at $K=\$140$ and $T=1$ year. The effective annual risk-free rate is $i=4\%$, constant over the term. A hypothetical American call quote is $C_A=\$24.00$. What is the smallest put price consistent with the upper difference bound?

Method. Rearrange $C_A-P_A\leq S_0-Kv^T$ to obtain $P_A\geq C_A-S_0+Kv^T$. Also require the put price to be nonnegative.

The one-year discount factor is $$ v^T=(1.04)^{-1}=\frac{1}{1.04}=0.961538\ldots. $$

The upper bound on the call-put difference is $$ S_0-Kv^T=150-\frac{140}{1.04} =150-134.61538\ldots =\$15.38462\ldots. $$

Rearranging the upper bound gives $$ P_A\geq C_A-S_0+Kv^T =24-150+134.61538\ldots =\$8.61538\ldots. $$

This is positive, so the nonnegative-price condition does not change the result. Rounded to the nearest cent, the smallest put price consistent with this bound is \$8.62. This is a bound-derived minimum, not a claim that the put must be worth exactly that amount. Common mistake avoided: misadding the three terms in the rearranged expression; $24-150+134.61538\ldots$ is about $8.62$, not $19.38$.

Worked Example: Detecting a Sign Error in the Upper Endpoint

A hypothetical non-dividend-paying asset has $S_0=\$95$. Its matching American options have $K=\$100$, $T=1$ year, and hypothetical quotes $C_A=\$7.00$, $P_A=\$17.00$. The constant effective annual rate is $i=10\%$. Do the quotes pass the bounds?

Method. Calculate the lower and upper endpoints directly from $S_0-K$ and $S_0-Kv^T$; do not attach a minus sign to the whole upper expression.

Here $$ v^T=\frac{1}{1.10}=0.909090\ldots. $$

The lower endpoint is $$ S_0-K=95-100=-\$5.00, $$ and the upper endpoint is $$ S_0-Kv^T=95-\frac{100}{1.10} =95-90.90909\ldots =\$4.09091\ldots. $$

The quoted difference is $$ C_A-P_A=7-17=-\$10.00. $$

The required interval is approximately $[-\$5.00,\$4.09]$, and $-\$10.00<-\$5.00$. Therefore, the quotes fail the American difference bounds. Common mistake avoided: changing $S_0-Kv^T=\$4.09$ to $-\$4.09$; only the subtraction in the expression determines its sign.

Using the Bounds Carefully

To apply the result, first confirm that the call and put have the same underlying, strike, and expiration. Then identify the appropriate rate and term, calculate the discount factor, and form the call-minus-put difference in that order. The bounds are for $C_A-P_A$; reversing the subtraction reverses the signs and requires reversing the endpoints as well.

Check the contract and model.
Confirm American exercise, no dividends, a nonnegative rate, and matching strike and expiration.

Discount the strike.
For effective annual rate $i$ and term $T$ in years, calculate $v^T=(1+i)^{-T}$.

Calculate the interval.
Use $S_0-K$ as the lower endpoint and $S_0-Kv^T$ as the upper endpoint.

Compare the quote difference.
Compute $C_A-P_A$ and check whether it is at least the lower endpoint and at most the upper endpoint.

A failure indicates that at least one premise, input, or quote is inconsistent with the bound. It is not a reason to “fix” a negative endpoint by taking its absolute value. Nor does a pass certify that the options satisfy European parity: American exercise means equality is generally unavailable. These bounds are aligned with the SOA Exam FM (Financial Mathematics) syllabus, where recognizing the correct inequality and discounting convention is more useful than forcing an equality that the contracts do not support.

Key takeaway: For matching American options on a non-dividend-paying asset with nonnegative rates, compare $C_A-P_A$ with the interval from $S_0-K$ to $S_0-Kv^T$. European parity remains an equality for European options; early exercise prevents assuming that same equality for American puts.

Check Your Understanding

Use the American difference bounds under the assumptions stated in each question. Round dollar answers to the nearest cent where needed.

A hypothetical asset has $S_0=\$80$, $K=\$75$, $T=1$, and constant effective annual rate $i=4\%$. Find the lower and upper bounds for $C_A-P_A$. (Answer: $\$5.00\leq C_A-P_A\leq\$7.88$)
A hypothetical asset has $S_0=\$132$, $K=\$140$, $T=2$, and constant effective annual rate $i=5\%$. Quotes are $C_A=\$13.00$ and $P_A=\$8.00$. Do the quotes pass the difference bounds? (Answer: Yes; approximately $-\$8.00\leq\$5.00\leq\$5.02$)
A hypothetical one-year American call is quoted at $\$22.00$ on an asset with $S_0=\$210$ and $K=\$200$. The constant effective annual rate is $i=6\%$. What is the smallest put price consistent with the upper difference bound? (Answer: $\$0.68$)
A hypothetical asset has $S_0=\$60$, $K=\$70$, $T=2$, and constant effective annual rate $i=8\%$. Matching American quotes are $C_A=\$5.00$ and $P_A=\$14.00$. Do they pass the bounds? (Answer: Yes; approximately $-\$10.00\leq-\$9.00\leq-\$0.01$)
For matching American options on a hypothetical non-dividend-paying asset, suppose $i=0$, with $S_0=\$104$, $K=\$100$, and any positive expiration term. What must $C_A-P_A$ equal under the bounds? (Answer: $\$4.00$)

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