American and European Option Values

Exercise Rights and Option Value

An option’s payoff describes what it is worth if exercised at a particular time; its value also reflects the choices the holder can make before then. A European option can be exercised only at expiration. An American option can be exercised at any permitted time up to and including expiration. The word “American” describes the exercise right, not the location of the underlying asset or the option holder.

The comparison is meaningful when the options have the same underlying, strike price, expiration date, and other contract terms. With those terms held fixed, the American holder has every exercise choice available to the European holder, plus the possibility of exercising earlier. That additional flexibility cannot make the option less valuable. It can, however, be worth nothing in some situations or add value in others.

Definition: Let $c$ and $p$ denote the values of European calls and puts, and let $C$ and $P$ denote the values of otherwise identical American calls and puts. For the same underlying, strike, expiration, and contract terms, $C \geq c$ and $P \geq p$. The differences $C-c$ and $P-p$ are the respective exercise premiums. An exercise premium is nonnegative; it need not be positive.

For a stock price $S$ and strike price $K$, the immediate exercise payoffs are $\max(S-K,0)$ for a call and $\max(K-S,0)$ for a put. These are often called intrinsic values. An American option’s value must be at least its immediate exercise payoff, because the holder can choose to exercise now. But the American option’s value is not generally equal to its intrinsic value: waiting may be more valuable.

Why the American Value Cannot Be Lower

The key argument is about available strategies. Any strategy that exercises a European option at expiration is also available to the American holder: the American holder can simply choose not to exercise early and wait until expiration. The American holder therefore has at least all the European holder’s feasible choices.

Under the standard no-arbitrage pricing framework, an option’s value reflects the best value obtainable from its permitted exercise choices. Since the American holder can follow the European strategy, the American value cannot be smaller. If early exercise is optimal in at least one situation that affects value, the American option may be worth more; if not, the extra right may add no value.

\text{American value} \geq \text{European value}

In a discrete-time binomial model, the same reasoning can be expressed as a backward-induction calculation. At a node, the holder chooses between exercising immediately and keeping the option for the next time step. The continuation value is the discounted risk-neutral expected value at the next nodes.

V_t=\max\left(g_t,\frac{qV_{t+1}^{u}+(1-q)V_{t+1}^{d}}{1+i}\right)

Here $V_t$ is the American option value at the current node, $g_t$ is the immediate exercise payoff, $V_{t+1}^{u}$ and $V_{t+1}^{d}$ are the values at the next up and down nodes, $i$ is the effective interest rate per period, and $q$ is the risk-neutral probability of an up move. This formula assumes a one-period binomial step with no arbitrage: if the stock’s up and down factors are $u$ and $d$, then $d<1+i<u$, and $q=((1+i)-d)/(u-d)$. At expiration, the option value is its payoff, because no further exercise or continuation choice remains.

A One-Period Comparison

Before considering early exercise at intermediate dates, a one-period tree illustrates how the comparison works when the only choices are to exercise now or at expiration. In a one-period model, the European value is the discounted risk-neutral expected expiration payoff. The American value is the larger of that continuation value and the immediate exercise payoff.

Worked Example: Calls and Puts on a One-Period Tree

A hypothetical stock is priced at $S_0=\$100$. In one year it will be either $\$130$ or $\$80$. The effective annual interest rate is $i=0\%$, and the strike is $K=\$100$ for both a call and a put. Find the European and American values under the one-period binomial model.

Method and model. The stock’s up and down factors are $u=1.30$ and $d=0.80$, and the gross risk-free return is $1+i=1.00$. The no-arbitrage condition holds because $0.80<1.00<1.30$. The risk-neutral up probability is $q=(1.00-0.80)/(1.30-0.80)=0.20/0.50=0.40$; the down probability is $1-q=0.60$.

Call. The expiration payoffs are $\max(130-100,0)=\$30$ in the up state and $\max(80-100,0)=\$0$ in the down state. Therefore, the European call value is $0.40(\$30)+0.60(\$0)=\$12$, with no discounting because $i=0\%$. Immediate exercise at time zero pays $\max(100-100,0)=\$0$. The American call value is the greater of $\$0$ and $\$12$, so it is $\$12$.

Put. The expiration payoffs are $\max(100-130,0)=\$0$ in the up state and $\max(100-80,0)=\$20$ in the down state. The European put value is $0.40(\$0)+0.60(\$20)=\$12$. Immediate exercise also pays $\max(100-100,0)=\$0$, so the American put value is $\$12$.

Check and common mistake. Both American values equal their European counterparts, so both exercise premiums are $\$0$. A mistake would be to assign either American option a value of zero because its immediate exercise payoff is zero. That ignores the positive value of waiting for the expiration payoffs.

Backward Induction and a Positive Exercise Premium

With multiple exercise dates, an American option can be worth more than its European counterpart. At every nonterminal node, compare the immediate exercise payoff with the continuation value. The continuation value is not the undiscounted average of the next values: it is the risk-neutral expected value discounted by the period’s gross risk-free return.

Set terminal values.
At expiration, calculate the call payoff $\max(S-K,0)$ or put payoff $\max(K-S,0)$ at each terminal stock price.

Move back one period.
At each earlier node, calculate the immediate exercise payoff and the discounted risk-neutral expected value of continuing.

Choose the larger value.
The American option value at that node is the maximum of immediate exercise and continuation. Repeat until the initial node is reached.

Worked Example: An American Put with an Exercise Premium

A hypothetical stock is priced at $\$100$, and the put strike is $K=\$105$. There are two one-year periods to expiration. In each period the stock either rises by a factor $u=1.20$ or falls by a factor $d=0.80$. The effective annual interest rate is $i=10\%$. Find the European put value, the American put value, and the exercise premium.

Method and risk-neutral probability. The no-arbitrage condition holds because $0.80<1.10<1.20$. Thus $q=(1.10-0.80)/(1.20-0.80)=0.30/0.40=0.75$, and the down probability is $0.25$. The discount factor per year is $1/1.10$.

Stock prices and terminal payoffs. After two periods, the stock prices are $100(1.20)^2=\$144$ after two up moves, $100(1.20)(0.80)=\$96$ after one up and one down move, and $100(0.80)^2=\$64$ after two down moves. The corresponding put payoffs are $\$0$, $\$9$, and $\$41$.

European value. The risk-neutral probabilities of the three terminal stock-price outcomes are $0.75^2=0.5625$, $2(0.75)(0.25)=0.3750$, and $0.25^2=0.0625$. The expected expiration payoff is $0.5625(\$0)+0.3750(\$9)+0.0625(\$41)=\$3.375+\$2.5625=\$5.9375$. Discounting for two years gives $\$5.9375/(1.10)^2=\$5.9375/1.21=\$4.9070$, or **\$4.91** to the nearest cent.

American values at the one-year nodes. At the up node, the stock price is $\$120$, so immediate exercise pays $\max(105-120,0)=\$0$. Continuing leads to terminal put values $\$0$ or $\$9$. Its value is $(0.75(\$0)+0.25(\$9))/1.10=\$2.25/1.10=\$2.0455$. The node value is therefore $\max(\$0,\$2.0455)=\$2.0455$.

At the down node, the stock price is $\$80$, so immediate exercise pays $\$25$. Continuing leads to put values $\$9$ or $\$41$. Its value is $(0.75(\$9)+0.25(\$41))/1.10=(\$6.75+\$10.25)/1.10=\$17/\!1.10=\$15.4545$. The node value is $\max(\$25,\$15.4545)=\$25$, so exercise is optimal there.

Initial American value and check. At time zero, immediate exercise pays $\$5$. The continuation value is $(0.75(\$2.0455)+0.25(\$25))/1.10=(\$1.5341+\$6.25)/1.10=\$7.0764$. Thus the American value is $\max(\$5,\$7.0764)=\$7.0764$, or **\$7.08** to the nearest cent. Its exercise premium is $\$7.0764-\$4.9070=\$2.1694$, or **\$2.17** to the nearest cent. The American value exceeds the European value, as required.

Common mistake avoided. At the down node, using only the continuation value of $\$15.4545$ would undervalue the American put. The holder can instead exercise for $\$25$, so the node value must be the larger amount.

When the Exercise Premium Is Zero

An American option may have the same value as its European counterpart even though its holder has more choices. The extra exercise choices add no value when waiting is at least as good as exercising at every relevant node. This is a comparison of values under the specified model, not a claim that exercise rights are always worthless.

The next calculation uses the same stock tree and interest rate as the put example, but a call with the same $\$105$ strike. It evaluates both exercise choices at each node without relying on a general rule about when calls should be exercised.

Worked Example: A Call with No Exercise Premium in the Tree

Use the two-period stock model from the previous calculation: $S_0=\$100$, $u=1.20$, $d=0.80$, $i=10\%$ per year, and $q=0.75$. A call has strike $K=\$105$. Find its European and American values in this model.

European value. The terminal stock prices are $\$144$, $\$96$, and $\$64$, so the call payoffs are $\$39$, $\$0$, and $\$0$. Using the terminal probabilities $0.5625$, $0.3750$, and $0.0625$, the expected payoff is $0.5625(\$39)+0.3750(\$0)+0.0625(\$0)=\$21.9375$. The European value is $\$21.9375/1.21=\$18.1302$, or **\$18.13** to the nearest cent.

American values at the one-year nodes. At the up node, $S=\$120$, so immediate exercise pays $\$15$. The continuation value is $(0.75(\$39)+0.25(\$0))/1.10=\$29.25/1.10=\$26.5909$. The node value is $\$26.5909$, since continuation is larger. At the down node, $S=\$80$; immediate exercise pays zero, and both following stock prices, $\$96$ and $\$64$, are below the strike. Continuation also pays zero, so the node value is zero.

Initial value and check. Immediate exercise at time zero pays $\max(100-105,0)=\$0$. The continuation value is $(0.75(\$26.5909)+0.25(\$0))/1.10=\$19.9432/1.10=\$18.1302$. The American value is therefore $\$18.13$, equal to the European value to the nearest cent. The exercise premium in this tree is zero. At each node, the chosen value is at least the immediate exercise payoff, and the initial American value equals the European value.

Common mistake avoided. Do not add the call’s immediate exercise payoff to its continuation value. The holder chooses between those alternatives, so the node value is their maximum, not their sum.

Interpreting the Comparison

The inequality between American and European values is a general consequence of the exercise rights, but a numerical exercise premium depends on the contract and the circumstances. A positive premium means the early-exercise choice affects value under those circumstances. A zero premium means that having the additional choice does not improve value there. In a binomial calculation, the maximum operation at each node captures both possibilities.

A reliable comparison keeps three quantities distinct:

Expiration payoff: what the option pays if exercised at expiration.
Immediate exercise payoff: what the holder receives by exercising at the current stock price.
Continuation value: the discounted risk-neutral expected value of retaining the option.

Confusing these quantities is a frequent source of errors. For example, immediate exercise payoff is a lower bound for an American option’s value, not necessarily its value. Likewise, the European value is a lower bound for the otherwise identical American option, but the American value is not found by adding the European value to the exercise payoff. At each exercise date, the holder selects the more valuable alternative.

Key takeaway: For otherwise identical contracts, American value is at least European value because the American holder can always wait until expiration. In a binomial model, calculate the American value by comparing immediate exercise with discounted continuation at every earlier node; the exercise premium may be positive or zero.

Check Your Understanding

For each problem, identify the relevant exercise comparison and check the final value against the American-versus-European ordering.

A one-year binomial model has $S_0=\$100$, $u=1.20$, $d=0.90$, and effective annual interest rate $i=4\%$. A call has $K=\$100$, and exercise is allowed only at time zero or expiration. Find its American value. (Answer: $\$8.97$)
An American put and an otherwise identical European put have values of $\$12.60$ and $\$11.85$, respectively. What is the exercise premium, and does the comparison satisfy the required value ordering? (Answer: $\$0.75$; yes, the American value is higher.)
A one-period put has $S_0=\$70$, $K=\$75$, $u=1.20$, $d=0.80$, and effective annual interest rate $i=5\%$. Find its American value when exercise is allowed at time zero or expiration. (Answer: $\$6.79$)
At an intermediate node of an American option tree, immediate exercise pays $\$18.00$, while discounted continuation is worth $\$20.50$. What is the node value and which choice is optimal? (Answer: $\$20.50$; continue.)
In a two-period put tree, at a down node the immediate exercise payoff is $\$25.00$ and the discounted continuation value is $\$15.00$. What value should backward induction assign to this node, and what action is optimal? (Answer: $\$25.00$; exercise immediately.)

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