Why Non-Dividend American Calls Are Not Exercised Early

Why Waiting Can Be More Valuable Than Exercising

An American call gives its owner the right to buy a share for the exercise price at any time up to expiration. When the stock pays no dividends, exercising early gives up that right in exchange for owning the share sooner. The central comparison is between the call’s exercise value, which is $S-K$ when the call is in the money, and its continuation value, the value of retaining the option.

With a positive interest rate, waiting also postpones paying the exercise price. A precise comparison shows that, under the assumptions below, an in-the-money American call is worth more held than exercised before expiration. If the rate is zero, early exercise is still never better, but the comparison need not be strict.

Definition: Consider a call on a non-dividend-paying stock with current stock price $S$, exercise price $K>0$, and time $\tau>0$ remaining until expiration. Its immediate exercise value is $\max(S-K,0)$. Its continuation value is the value of keeping the option rather than exercising now. An American call permits exercise at any time through expiration; a European call permits exercise only at expiration.

The no-early-exercise conclusion uses a frictionless market: the stock and risk-free borrowing and lending are tradable, there are no transaction costs or taxes, and the stock pays no cash distributions before expiration. Let $i$ be a nonnegative effective annual risk-free rate, with deterministic discounting over $\tau$ years. The present value of $K$ payable at expiration is $K/(1+i)^\tau$. The result does not require a particular probability distribution for future stock prices.

The Call Value Bound That Drives the Result

At any time before expiration, a European call is worth at least the current stock price minus the present value of its exercise price:

C_E \geq S-\frac{K}{(1+i)^\tau}.

Here $C_E$ denotes the European call value for the same stock, strike, and expiration. To see why the bound holds, compare two positions that pay at expiration: a call plus a risk-free investment that grows to $K$, and one share of stock. At expiration, the first position is worth $\max(S_T-K,0)+K$, where $S_T$ is the stock price then. This amount is at least $S_T$ in every state. If the call plus the bond cost less than the share today, an investor could buy both, short the share, and receive a positive amount initially while having a nonnegative payoff at expiration. That would be an arbitrage. Therefore $C_E+K/(1+i)^\tau\geq S$, which gives the stated bound.

An American call is worth at least as much as the corresponding European call, since its owner can choose to wait until expiration and exercise then. Consequently, its value $C_A$ satisfies the same lower bound. For an in-the-money call, subtract the exercise value $S-K$:

C_A-(S-K) \geq \left(S-\frac{K}{(1+i)^\tau}\right)-(S-K) = K\left(1-\frac{1}{(1+i)^\tau}\right) \geq 0.

If $i>0$, $K>0$, and $\tau>0$, then $1/(1+i)^\tau<1$. The right-hand side is strictly positive, so exercising an in-the-money call early gives up value. If $i=0$, the bound guarantees only that continuing is at least as valuable as exercising. It may be strictly better because of uncertainty, but equality is possible in special cases.

Theorem: In a frictionless market with no dividends on the underlying stock and a nonnegative risk-free rate, early exercise of an American call cannot be better than holding the call. For a positive rate, positive strike, positive time remaining, and an in-the-money call, holding is strictly better under the stated bound. At a zero rate, early exercise can at most tie; it cannot improve on holding.

Worked Comparisons

Worked Example: Applying the Bound to an In-the-Money Call

Method: compare the present-value bound with intrinsic value. In a hypothetical frictionless market, a non-dividend-paying stock is priced at $S=\$100$. A call has strike $K=\$80$, two years to expiration, and an effective annual risk-free rate $i=6\%$. No probability distribution for the stock price is specified or needed for this bound. Mortality is not applicable.

The immediate exercise value is:

S-K=\$100-\$80=\$20.

The present value of the exercise price is:

\frac{K}{(1+i)^\tau} = \frac{\$80}{(1.06)^2} = \frac{\$80}{1.1236} \approx \$71.20.

The call value is therefore at least:

C_A \geq \$100-\$71.20=\$28.80.

This lower bound exceeds the $\$20$ exercise value by approximately $\$8.80$. Holding the call is thus worth at least approximately $\$28.80$, while exercising produces a share worth $\$100$ after paying $\$80$. The comparison uses the value of the share net of the exercise payment; it does not claim that the two actions leave the investor with identical positions.

Check: The exact discounted strike is $\$80/1.1236$, approximately $\$71.1997$; the bound is approximately $\$28.8003$, and the excess over intrinsic value is approximately $\$8.8003$. Amounts are rounded to the nearest cent only at the end. The common mistake avoided is discounting the stock price instead of the future exercise payment.

Worked Example: One-Period Binomial Call Value

Method: value the call by risk-neutral expectation and compare it with exercise. In a hypothetical one-period model, a non-dividend-paying stock is currently $\$52$. In one year it will be either $\$65$ or $\$44.20$. The strike is $\$48$, the effective annual risk-free rate is $i=4\%$, and there are no intermediate cash distributions. The model is frictionless; its risk-neutral up and down probabilities are determined by no-arbitrage. Mortality is not applicable.

The up and down factors are $u=65/52=1.25$ and $d=44.20/52=0.85$. Since $d<1.04<u$, the risk-neutral up probability is:

p=\frac{(1+i)-d}{u-d} =\frac{1.04-0.85}{1.25-0.85} =\frac{0.19}{0.40} =0.475.

The call’s expiration payoffs are $\max(65-48,0)=\$17$ in the up state and $\max(44.20-48,0)=\$0$ in the down state. Its continuation value is:

C_E=\frac{0.475(\$17)+0.525(\$0)}{1.04} =\frac{\$8.075}{1.04} \approx \$7.76.

The immediate exercise value is $52-48=\$4$. The one-period American call is worth the greater of exercise and continuation values, so its value is approximately $\$7.76$, not $\$4$. Waiting is better by approximately $\$3.76$.

Check: The risk-neutral probabilities sum to $0.475+0.525=1$, and the expected stock price is $0.475(65)+0.525(44.20)=\$54.08=\$52(1.04)$. The option value is rounded to the nearest cent. The common mistake avoided is discounting the expected payoff at an unrelated probability instead of the risk-neutral probability consistent with the risk-free growth rate.

Worked Example: Checking Exercise at Two Intermediate Nodes

Method: use backward induction at each possible exercise date. Consider a hypothetical two-period binomial model for a non-dividend-paying stock. The initial stock price is $\$100$, the strike is $\$90$, and the effective annual risk-free rate is $i=10\%$. In each year the stock either rises by a factor $u=1.2$ or falls by a factor $d=0.8$. The market is frictionless, with no distributions. Mortality is not applicable.

The risk-neutral up probability in each period is:

p=\frac{1.10-0.80}{1.20-0.80}=0.75, \qquad 1-p=0.25.

After the first up move, the stock is $\$120$, so immediate exercise is worth $\$30$. In the next period the stock will be $\$144$ or $\$96$, giving call payoffs of $\$54$ or $\$6$. The continuation value at the up node is:

\frac{0.75(\$54)+0.25(\$6)}{1.10} = \frac{\$42}{1.10} \approx \$38.18.

Because $\$38.18>\$30$, exercising at the up node is inferior to continuing.

After the first down move, the stock is $\$80$; immediate exercise is worth zero. In the next period the stock will be $\$96$ or $\$64$, giving call payoffs of $\$6$ or $\$0$. The continuation value is:

\frac{0.75(\$6)+0.25(\$0)}{1.10} = \frac{\$4.50}{1.10} \approx \$4.09.

Continuing is also better at the down node. At the initial date, the stock is $\$100$ and immediate exercise is worth $\$10$. Using the two continuation values just calculated, the value of waiting is:

\frac{0.75(\$38.1818\ldots)+0.25(\$4.0909\ldots)}{1.10} \approx \$26.96.

Thus the initial call value is approximately $\$26.96$, above the $\$10$ exercise value. Intermediate calculations retain unrounded values; displayed dollar results are rounded to the nearest cent. The common mistake avoided is checking only the initial date and overlooking that an American option must be checked at every possible early exercise node.

Check: At the up node the next-period expected payoff is $\$42$; at the down node it is $\$4.50$. Discounting each by $1.10$ gives the continuation values used above. The initial continuation calculation also discounts the probability-weighted node values by $1.10$ for the first period.

Holding, Selling, and Exercising Are Different Choices

If the owner can sell the call at its market value, selling realizes the option’s value without paying the strike. Under the positive-rate assumptions, an in-the-money call’s value exceeds its exercise value, so exercising and immediately selling the resulting share would yield less than selling the call. Selling the call does, however, end the owner’s option position. It is not the same choice as holding the option, and it does not preserve the same exposure to future stock-price changes.

An owner who wants to retain the option’s exposure can simply hold it. An owner who specifically wants to own a share may still have a reason to exercise for purposes outside the model, but that does not make early exercise the value-maximizing action in the frictionless setting. Taxes, transaction costs, liquidity constraints, or restrictions on trading can alter an investor’s practical decision; they are excluded from the theorem.

Worked Example: Comparing a Sale with Immediate Exercise

Method: compare net liquidation amounts, while keeping the positions distinct. Suppose a hypothetical non-dividend-paying stock is priced at $\$70$, and an American call with strike $\$60$ has a market bid of $\$13$. There is one year remaining and the effective annual risk-free rate is $i=5\%$. Assume a frictionless market apart from the stated bid, and no distributions or mortality.

Exercise requires paying $\$60$ for a share worth $\$70$, so the net value of exercising and immediately selling the share is $\$70-\$60=\$10$. Selling the call at its stated bid produces $\$13$. Thus, if the owner wants to exit rather than retain stock exposure, selling the option yields $\$3$ more than exercising and immediately liquidating the share.

The no-arbitrage lower bound is also consistent with the quote:

C_A \geq \$70-\frac{\$60}{1.05} \approx \$12.86.

The $\$13$ bid is above this bound, and the $\$10$ exercise value is below it. The comparison does not say that holding cash after selling is equivalent to owning the stock after exercise; those are different positions. Amounts are rounded to the nearest cent. The common mistake avoided is treating sale proceeds and the stock received on exercise as identical assets without first stating the investor’s objective.

Check: $\$60/1.05\approx \$57.14$, so the lower bound is approximately $\$12.86$. The stated sale proceeds exceed the $\$10$ net liquidation value by $\$3$.

Assumptions and Common Exam Traps

The result depends on the assumptions. If the stock pays dividends, holding the call does not provide the dividends that a shareholder would receive, so the comparison changes. A negative interest rate also invalidates the nonnegative-rate inequality used here: discounting the strike can then make its present value exceed its face amount. These cases are outside the theorem proved above.

A frequent error is to compare the call’s value with $S-K/(1+i)^\tau$ but then describe that number as the exercise payoff. It is a lower bound on the value of holding the call; the actual exercise payoff is $S-K$. Another error is to claim strict superiority at a zero rate based only on the bound. At $i=0$, the bound reduces to $C_A\geq S-K$, which allows equality. The conclusion is therefore “early exercise cannot be better,” with strict preference established by this argument when the rate is positive.

State the setting.
Confirm that the call is American, the stock pays no dividends, the rate is nonnegative, and time remains before expiration.

Calculate exercise value.
For an in-the-money call this is $S-K$; otherwise it is zero.

Compare with continuation.
Use $C_A\geq S-K/(1+i)^\tau$ and compare that bound with $S-K$.

Interpret the result carefully.
With a positive rate, holding is strictly better for an in-the-money call before expiration. With a zero rate, the bound proves only that holding is no worse.

Key takeaway: A non-dividend-paying American call does not benefit from early exercise in a frictionless market with nonnegative rates. Holding preserves the option and delays payment of the strike; selling realizes the option’s market value, while exercising exchanges the option for stock and requires paying the strike.

Check Your Understanding

For each problem, identify the relevant comparison and check the assumptions before choosing an action.

A non-dividend-paying stock is worth $\$72$. A call has strike $\$60$, one year remaining, and the effective annual rate is $5\%$. Find the exercise value and the lower bound on holding value, then state which is larger. (Answer: Exercise value $=\$12.00$; lower bound $=\$72-\$60/1.05\approx \$14.86$; holding is better by at least approximately $\$2.86$.)
In a one-period binomial model, a non-dividend-paying stock is $\$40$, with next-period values $\$50$ and $\$32$. A call’s strike is $\$36$, the effective annual rate is $5\%$, and expiration is one year away. Find the risk-neutral up probability, the call value, and the exercise value. (Answer: $p=(1.05-0.80)/(1.25-0.80)=0.5556$; call value $=[0.5556(\$14)+0.4444(\$0)]/1.05\approx \$7.41$; exercise value $=\$4.00$, so continuation is better.)
A non-dividend-paying stock is known with certainty to remain at $\$80$ for one year. The strike is $\$65$, and the effective annual rate is zero. Under a frictionless market, compare exercising now with holding to expiration. (Answer: Both values are $\$15.00$; early exercise ties holding, so it is not better but need not be strictly worse.)
In a two-period binomial model, $S_0=\$80$, $K=\$75$, $u=1.25$, $d=0.80$, and $i=5\%$ per period. The stock pays no dividends. At the first up node, calculate immediate exercise and one-period continuation values using risk-neutral probabilities. (Answer: $p=(1.05-0.80)/(1.25-0.80)=5/9$; at $S=\$100$, next stock values are $\$125$ and $\$80$, with call payoffs $\$50$ and $\$5$; continuation $=[(5/9)(\$50)+(4/9)(\$5)]/1.05\approx \$28.57$, versus exercise value $\$25.00$.)
An owner holds an American call on a non-dividend-paying stock priced at $\$95$. The strike is $\$90$, one year remains, the effective annual rate is $8\%$, and the call can be sold at a $\$14$ bid. Compare the bid with the call’s exercise value and the no-arbitrage lower bound. If the owner wants to exit and immediately liquidate any share received on exercise, which action gives more cash? (Answer: Exercise-and-liquidate value $=\$5.00$; lower bound $=\$95-\$90/1.08\approx \$11.67$; selling yields $\$14.00$, which is $\$9.00$ more than exercising and liquidating.)

What You'll Learn