Early Exercise Value of American Puts

Why an American Put Can Be Worth Exercising Early

An American put gives its holder the right to sell the underlying asset for the strike price on or before expiration. If the asset price is well below the strike, exercising now delivers cash sooner than waiting. That timing can matter when interest rates are positive. But exercise also gives up the right to benefit from later price changes, so being in the money does not automatically make immediate exercise optimal.

The decision is a comparison at the current date: exercise now, or retain the option and its future choices. To make that comparison precisely, this tutorial uses the risk-neutral binomial model. A model calculation is only as sound as its assumptions and its application of the American exercise rule at every decision point.

Definition: For an American put with strike $K$ and current underlying price $S$, the immediate exercise value is $\max(K-S,0)$. The continuation value is the discounted risk-neutral expected value of the option at the next decision date, when the holder will again choose optimally between exercising and continuing. At a decision node, the American put’s value is the greater of its immediate exercise value and its continuation value.

The One-Period Risk-Neutral Comparison

First consider a one-period binomial model. The underlying pays no dividends during the period. Its current price is $S$, and at the end of the period it will be either $Su$ or $Sd$, where $u>d>0$. The effective risk-free rate for the period is $i$, with $d<1+i<u$. These assumptions give a risk-neutral probability

q=\frac{1+i-d}{u-d}

of an up move; the probability of a down move is $1-q$. Under this pricing model, a one-period contingent payoff has present value equal to its expected payoff using these probabilities, discounted by $1+i$. The probabilities are pricing weights under the model, not necessarily forecasts of real-world frequencies.

At the end of the period, the put’s payoff is $\max(K-S_T,0)$, where $S_T$ is the ending underlying price. Thus the continuation value from holding for the period is

C=\frac{q\max(K-Su,0)+(1-q)\max(K-Sd,0)}{1+i}.

At the current date, immediate exercise is optimal if $K-S\geq C$; holding is optimal if $C>K-S$, provided the put is in the money. If $S\geq K$, immediate exercise pays zero, and the comparison is with the nonnegative continuation value.

Worked Example: An In-the-Money Put Whose Time Value Favors Holding

A one-period American put has $K=\$80$ and $S=\$72$. Assume a no-dividend underlying, effective risk-free rate $i=5\%$ for the period, and binomial factors $u=1.25$ and $d=0.90$. Determine whether to exercise or hold. Use the one-period risk-neutral model and compare immediate exercise with continuation. This example avoids the mistake of assuming that every in-the-money put should be exercised.

Step 1: Check the model and calculate the risk-neutral probability. The factors satisfy $0.90<1.05<1.25$, so the no-arbitrage condition holds. Then

q=\frac{1.05-0.90}{1.25-0.90} =\frac{0.15}{0.35} =\frac{3}{7} \approx 0.4286, \qquad 1-q=\frac{4}{7}.

Step 2: Find the ending prices and put payoffs. The up price is $72(1.25)=\$90.00$, giving payoff $\max(80-90,0)=\$0.00$. The down price is $72(0.90)=\$64.80$, giving payoff $\max(80-64.80,0)=\$15.20$.

Step 3: Calculate continuation value.

C=\frac{(3/7)(0)+(4/7)(15.20)}{1.05} =\frac{8.685714\ldots}{1.05} =\$8.272108\ldots \approx \$8.27.

Step 4: Compare with exercise. Immediate exercise is $80-72=\$8.00$. Since $C=\$8.27>\$8.00$, the optimal decision is to hold. The calculation is checked by substituting the two terminal payoffs and discounting their weighted average once. Dollar amounts are rounded to the nearest cent only at the final step.

Why Positive Interest Rates Can Favor Early Exercise

There is a particularly clear case in which early exercise has an advantage. Suppose the put is in the money now and both possible next-period prices are still below the strike: $Su<K$ and $Sd<K$. In that case, the put payoff at either next-period node is $K-S_T$, without a zero-payoff branch. The risk-neutral pricing condition is $qu+(1-q)d=1+i$. It implies that the risk-neutral expected next-period price is $S(1+i)$.

Using those facts, continuation for one period is

C =\frac{q(K-Su)+(1-q)(K-Sd)}{1+i} =\frac{K-S(1+i)}{1+i} =\frac{K}{1+i}-S.

Immediate exercise is $K-S$. Subtracting continuation from immediate exercise gives

(K-S)-C =(K-S)-\left(\frac{K}{1+i}-S\right) =\frac{Ki}{1+i}.

For $K>0$ and $i>0$, this difference is positive. In this one-period comparison, when both next-period prices remain below the strike, immediate exercise is worth more than holding for the period. The reason is that exercise receives the strike sooner; waiting discounts its receipt. This is a local result under the stated assumptions, not a rule that all in-the-money puts should be exercised.

Worked Example: A Deeply In-the-Money Put and the Value of Receiving the Strike Sooner

A one-period American put has strike $K=\$100$ and current stock price $S=\$60$. Assume the stock pays no dividends, the effective risk-free rate for the period is $i=5\%$, and the binomial factors are $u=1.20$ and $d=0.80$. Compare exercise with holding. The method is to price the two terminal payoffs under the risk-neutral model; the key check is whether both terminal prices are below the strike.

Step 1: Calculate and check the risk-neutral probability.

q=\frac{1.05-0.80}{1.20-0.80} =\frac{0.25}{0.40} =0.625, \qquad 1-q=0.375.

The condition $0.80<1.05<1.20$ holds. The terminal stock prices are $60(1.20)=\$72.00$ and $60(0.80)=\$48.00$, both below $K=\$100$, as required for the result above to apply.

Step 2: Calculate terminal put payoffs. The up-state payoff is $100-72=\$28.00$; the down-state payoff is $100-48=\$52.00$.

Step 3: Calculate continuation and compare.

C=\frac{(0.625)(28)+(0.375)(52)}{1.05} =\frac{17.50+19.50}{1.05} =\frac{37.00}{1.05} =\$35.238095\ldots \approx \$35.24.

Immediate exercise is $100-60=\$40.00$, which exceeds continuation by $\$40.00-\$35.238095\ldots=\$4.761904\ldots$, or $\$4.76$ to the nearest cent. Exercise is optimal in this one-period model. The result also agrees with the derived difference $Ki/(1+i)=100(0.05)/1.05=\$4.761904\ldots$. Both calculations round to $\$4.76$ at the final step.

Applying the American Exercise Rule in a Multi-Period Tree

With more than one period remaining, a European-style calculation to expiration is not enough. At an intermediate node, the put holder may exercise there, so the value at the next nodes must already reflect their optimal exercise decisions. The backward-induction rule is

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

for an in-the-money node, where $V_t$ is the American put value at that node, and $V_{t+1}^{u}$ and $V_{t+1}^{d}$ are the American values at the next up and down nodes. More generally, the first term is $\max(K-S_t,0)$, which also covers an out-of-the-money node. At expiration, $V_T=\max(K-S_T,0)$. Work backward from expiration, taking the larger of exercise and continuation at each node.

A continuation calculation that uses only terminal payoffs can understate the value of waiting if it fails to include optimal early exercise at a later node. The American rule must be applied there first, and that node’s resulting value must then be used in the preceding node’s continuation calculation.

Worked Example: Backward Induction with Exercise at a Down Node

A two-period American put has $K=\$100$, $S_0=\$85$, binomial factors $u=1.20$ and $d=0.80$ in each period, and a no-dividend underlying. The effective risk-free rate is $i=5\%$ per period. Use the risk-neutral model and backward induction to decide whether to exercise at the root and at each period-one node. The method explicitly applies the American exercise rule at intermediate nodes; this avoids treating the contract as European.

Step 1: Find the risk-neutral probabilities and stock-price tree.

q=\frac{1.05-0.80}{1.20-0.80}=0.625, \qquad 1-q=0.375.

At period one, the up and down prices are $85(1.20)=\$102.00$ and $85(0.80)=\$68.00$. At period two, the prices reached from the up node are $\$122.40$ and $\$81.60$; those reached from the down node are $\$81.60$ and $\$54.40$.

Step 2: Calculate expiration payoffs. At prices $\$122.40,\ \$81.60,\ \$81.60,\ \$54.40$, the respective put payoffs are $\$0.00,\ \$18.40,\ \$18.40,\ \$45.60$.

Step 3: Compare exercise and continuation at the period-one up node. At $S=\$102.00$, immediate exercise is $\max(100-102,0)=\$0.00$. Continuation is

C_u=\frac{(0.625)(0)+(0.375)(18.40)}{1.05} =\frac{6.90}{1.05} =\$6.571428\ldots \approx \$6.57.

Holding is optimal at this node, so its American value is $\$6.57$ before rounding, namely $\$6.571428\ldots$.

Step 4: Compare exercise and continuation at the period-one down node. At $S=\$68.00$, immediate exercise is $100-68=\$32.00$. Continuation, using the period-two payoffs, is

C_d=\frac{(0.625)(18.40)+(0.375)(45.60)}{1.05} =\frac{11.50+17.10}{1.05} =\frac{28.60}{1.05} =\$27.238095\ldots \approx \$27.24.

Since $\$32.00>\$27.238095\ldots$, exercise is optimal at the down node; its American value is $\$32.00$. Using only the terminal continuation calculation at this node would understate the optimal value at the down node and the resulting root continuation value, because it would fail to account for optimal early exercise there.

Step 5: Compare exercise and continuation at the root. The root’s immediate exercise value is $100-85=\$15.00$. Its continuation value uses the American values just calculated:

C_0 =\frac{(0.625)(6.571428\ldots)+(0.375)(32)}{1.05} =\frac{4.107142\ldots+12}{1.05} =\$15.340136\ldots \approx \$15.34.

Because $\$15.340136\ldots>\$15.00$, holding is optimal at the root. The backward-induction checks are consistent: exercise at the period-one down node, hold at the period-one up node, and hold at time zero. All displayed dollar results are rounded to the nearest cent only after the comparisons are made using unrounded values.

What the Exercise Comparison Does—and Does Not—Say

Deep in-the-money status makes it more likely that the next-period put payoff will be close to $K-S_T$ in every state. In that situation, the strike-receipt timing effect can be important. But when some next-period prices exceed the strike, the payoff has a zero floor. The payoff is then not simply $K-S_T$ in all states, and the preceding algebra does not apply. The value of retaining the option’s ability to benefit from a later price decline may make continuation preferable.

In a multi-period tree, a put’s value is therefore not determined by comparing intrinsic value with a European value calculated directly to expiration. The correct comparison is with continuation based on the next nodes’ American values. Exercise can be optimal in a down node while holding remains optimal at an earlier node, as the example showed.

At the current node	Decision rule	Interpretation
Immediate exercise value exceeds continuation	Exercise	Receiving the strike now is worth more than retaining the option.
Continuation exceeds immediate exercise value	Hold	The value of future choices outweighs exercising now.
The values are equal	Either action is optimal in the model	The holder is indifferent at that node.

A common exam trap is to treat intrinsic value as the American put’s full value or to exercise simply because $S<K$. The correct method is to calculate the node’s continuation value under the stated model and compare it with $\max(K-S,0)$. Also check the period attached to the interest rate: if $i$ is the effective rate per period, discount one period by $1+i$, not by a rate stated for a different time unit.

Key takeaway: An American put’s value at each node is the greater of immediate exercise and continuation, with continuation calculated from optimally valued next nodes. Positive rates can favor early exercise when all next-period outcomes remain below the strike, but the exercise decision always depends on the actual exercise-versus-hold comparison.

Check Your Understanding

For each problem, assume the stated binomial model and compare exercise with continuation. Dollar answers are rounded to the nearest cent.

A one-period American put has $K=\$90$, $S=\$84$, $u=1.20$, $d=0.90$, and an effective risk-free rate of $5\%$ per period. The underlying pays no dividends. Find $q$, continuation value, immediate exercise value, and the decision. (Answer: $q=0.5000$; continuation $\$6.86$; exercise $\$6.00$; hold.)
A one-period American put has $K=\$120$, $S=\$72$, $u=1.25$, $d=0.75$, and an effective risk-free rate of $4\%$ per period, with no dividends. Find the risk-neutral continuation value and decision. (Answer: $q=0.5800$; continuation $\$43.38$; exercise $\$48.00$; exercise.)
A one-period American put has $K=\$150$, $S=\$100$, $u=1.10$, $d=0.90$, and an effective risk-free rate of $5\%$ per period. There are no dividends. Calculate the exercise advantage, immediate exercise value minus continuation value. (Answer: exercise $\$50.00$; continuation $\$42.86$; exercise advantage $\$7.14$.)
A two-period American put has $K=\$110$, $S_0=\$90$, $u=1.20$, $d=0.80$, and an effective risk-free rate of $5\%$ per period. The underlying pays no dividends. Apply the American rule at the period-one nodes and then determine the root decision. (Answer: $q=0.6250$; at the up node, exercise $\$2.00$, continuation $\$8.43$, hold; at the down node, exercise $\$38.00$, continuation $\$32.76$, exercise; root continuation $\$18.59$ versus exercise $\$20.00$, exercise.)
A one-period American put has $K=\$200$, $S=\$160$, $u=1.10$, $d=0.90$, and an effective risk-free rate of $2\%$ per period, with no dividends. Both next-period prices are below the strike. Calculate $q$, continuation, and the exercise advantage. (Answer: $q=0.6000$; continuation $\$36.08$; exercise $\$40.00$; exercise advantage $\$3.92$, so exercise.)

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