American Options

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

Early-Exercise of American Options

American vs. European Option Premium

The main difference of American options compared to their European counterparts is that they can be exercised early.
Knowing when to optimally exercise an American call or put is a challenging problem.
This is the main reason why it is harder to price American options compared to European options.
In the following \Cam or \Pam denote an American call or put respectively, whereas their European counterparts are denoted by C or P.
Because an American option has all the benefits of a European option, but in addition has the possibility of early exercise, it has to be the case that its premium is at least as high as the premium on a European option with the same characteristics, i.e. we must have that \Cam \geq C and \Pam \geq P.

American Call Option on a Non-Dividend Paying Asset

It is never optimal to exercise early an American call option when the asset pays no dividends as long as the risk-free rate is positive.
To see why, notice that if r > 0 and T > 0, then we have that: \Cam \geq C \geq S - K e^{-r T} > S - K.
If it were optimal to exercise early, the price of the American call would be equal to its intrinsic value, violating the strict inequality.
- Since the time-value of the European call is strictly positive, you destroy value if you exercise immediately even if the option is ITM!
This implies that for non-dividend paying stocks the value of an American call option with maturity T and strike price K is the same as the value of a European call option with the same characteristics.

Early-Exercise of American Options

When there are cash dividends, it might be optimal to exercise early an American call option just before the stock goes ex-dividend.
For American puts the situation is similar, even when on assets with no-dividends.
In summary, exercising early is all about opportunity costs. If there is no opportunity cost in waiting, then it is never optimal to exercise early.

Price Bounds on American Options

Put-Call Parity

For American options written on non-dividend paying stocks, the put-call parity holds as two inequalities if r > 0: S - K \leq \Cam - \Pam \leq S - K e^{-r T}
Interestingly, these inequalities are reversed if r < 0: S - K e^{-r T} \leq \Cam - \Pam \leq S - K
Both inequalities imply that put-call parity holds in the same way as for European options if r = 0: \Cam - \Pam = S - K

Example 1

On 2/8/17, Facebook (FB) closing price was $134.20.
As of that date, the stock does not pay dividends.
Options on FB as on other stocks are American.
Let us consider options on FB with maturity date 6/17/17.
This implies that T = 129 / 365 = 0.35.

Example 1 (cont’d)

If we use a continuously compounded interest rate of 1.5% per year, we can compute the no-arbitrage bounds on these options.
As can be seen from the table, the difference between calls and puts is well within the bounds predicted by the theory for all strikes.

Strike	Call Price	Put Price	S - K	\Cam - \Pam	S - K e^{-r T}
120	17.30	2.50	14.20	14.80	14.83
125	13.61	3.75	9.20	9.86	9.86
130	10.25	5.45	4.20	4.80	4.89
135	7.45	7.60	-0.80	-0.15	-0.09
140	5.13	10.35	-5.80	-5.22	-5.06
145	3.47	13.65	-10.80	-10.18	-10.03

Option Bounds

We always have that the American call is greater than its intrinsic value and less than the current spot price, i.e., \max(S - K, 0) \leq \Cam \leq S
For American put options we have a similar result: \max(K - S, 0) \leq \Pam \leq K
Note that these bounds are not necessarily the tightest, depending on whether ot not it might be worthwhile to exercise the option early.

Binomial Pricing of American Options

Early-Exercise and American Option Pricing

In order to accommodate the binomial pricing framework to American options, we need to allow for the possibility of early exercise at any point in time.
This means that we should always compare the intrinsic value of the option, i.e. the value of exercising now, with the value of continuing given by the discounted risk-neutral expected value of future payoffs.

A Two Period Binomial Model

We will start by pricing an American put option on a non-dividend paying asset.
To make the valuation problem interesting, we will use a two-period binomial tree.
As with European options, the spot rate is given by S and each period the spot rate goes up by u or goes down by d with risk-neutral probabilities p and 1 - p, respectively
Therefore:
- S_{u} = S \times u and S_{d} = S \times d
- S_{uu} = S_{u} \times u, S_{ud} = S_{u} \times d = S_{d} \times u = S_{du} and S_{dd} = S_{d} \times d
An American put option expiring at T and strike K trades at \Pam
The time-step is then \Delta T = T / 2

Two Period Tree for the Spot and American Put Option

Pricing the American Put Option

The American put price at expiration is the intrinsic value of the option: \Pam_{ij} = \max(K - S_{ij}, 0) \text{ for } i, j \in \{u, d\}
If the spot rate goes up:
- The intrinsic value of the put is I_{u} = \max(K - S_{u}, 0).
- The continuation value is H_{u} = (p \Pam_{uu} + (1 - p) \Pam_{ud}) e^{-r \Delta t}.
- If I_{u} > H_{u} then the option should be exercised immediately, otherwise we should wait \Rightarrow \Pam_{u} = \max(H_{u}, I_{u}).
Similarly, if the spot rate goes down, we have that \Pam_{d} = \max(H_{d}, I_{d}).
Finally, the value of the American put is given by \Pam = \max(H, I), where I = \max(K - S, 0) and H = \left( p \Pam_{u} + (1 - p) \Pam_{d} \right) e^{-r \Delta t}

Example 2

Let’s price an American put option with maturity 6 months and strike $100 written on a non-dividend paying stock using a two-step binomial model.
The current stock price is $100, and it can go up or down by 5% each period for two periods.
Each period represents 3-months, i.e. \Delta t = 0.25. The risk-free rate is 6% per year (continuously compounded).

Example 2 (cont’d)

Example

The risk-neutral probability of an up-move is: p = \frac{e^{r \Delta t} - d}{u - d} = \frac{e^{0.06 \times 0.25} - 0.95}{1.05 - 0.95} = 0.6511

If the stock price goes up \small \begin{align*} H_{u} & = \left( 0 \times p + 0.25 \times (1 - p) \right) e^{-0.06 \times 0.25} \\ & = 0.09 \\ I_{u} & = \max(100 - 105, 0) \\ & = 0 \\ \Pam_{u} & = \max(0.09, 0) \\ & = 0.09 \end{align*}

If the stock price goes down

\small \begin{align*} H_{d} & = \left( 0.25 \times p + 9.75 \times (1 - p) \right) e^{-0.06 \times 0.25} \\ & = 3.51 \\ I_{d} & = \max(100 - 95, 0) \\ & = 5 \\ \Pam_{d} & = \max(3.51, 5) \\ & = 5 \end{align*}

Example

Finally, \begin{align*} H & = \left( 0.09 \times p + 5 \times (1 - p) \right) e^{-0.06 \times 0.25} = 1.78 \\ I & = \max(100 - 100, 0) = 0 \end{align*} so that \Pam = \max(1.78, 0) = \$1.78.
- Note that the value of a European put with the same characteristics is P = \$1.26, implying that the early-exercise premium is 1.78 - 1.26 = \$0.52.
- Also, you can verify that the value of an American call with the same characteristics is the same as the value of a Euroean call, which is $4.22.

Factors Affecting American Option Prices

Variable	American Call	American Put
Current stock price	+	-
Strike price	-	+
Time-to-expiration	+	+
Volatility	+	+
Risk-free rate	+	-