American Options

Options and Futures
Lorenzo Naranjo

Spring 2024

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

Factors Affecting American Option Prices


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