American Option Pricing

An American option can be exercised at any time before expiration. That right makes it worth at least as much as an otherwise identical European option, and it makes pricing harder: at every moment, the holder must decide whether exercising now is better than holding. There is no simple closed form formula like Black Scholes for most American options. Instead, practitioners use trees, finite difference methods, analytical approximations or simulation with regression. This lesson explains the problem and the main solutions.

The early exercise problem#

At each point in time, an American option's value is the larger of:

value = max(exercise value now, expected value of continuing)

The price at which exercising becomes optimal is called the early exercise boundary. For an American put, the boundary is a stock price below which you should exercise; it rises towards the strike as expiration approaches. Finding this boundary is the core of American pricing.

When is early exercise valuable? See Early Exercise for the intuition. In short:

Method 1: binomial and trinomial trees#

Trees handle early exercise naturally: at each node, compare continuation value with exercise value and keep the larger. With a few hundred steps, trees give accurate prices for single asset American options. They are the workhorse for listed equity options. See Binomial and Trinomial Trees.

Method 2: finite difference methods#

The Black Scholes model can be written as a partial differential equation describing how the option value changes with price and time. Finite difference methods solve it on a grid of prices and times, working backwards from expiry and applying the early exercise condition at each grid point. Variants include explicit, implicit and Crank Nicolson schemes. They are fast and accurate for one or two underlyings and are widely used in bank pricing libraries.

Method 3: analytical approximations#

Several formulas approximate American prices quickly:

• Barone Adesi and Whaley (1987): a quadratic approximation, fast and usually accurate to a few cents.
• Bjerksund and Stensland (1993, 2002): approximations based on a flat exercise boundary, popular in trading systems.
• Black's approximation for calls with dividends: compare the European value with exercise just before each dividend and take the larger.

These are useful when thousands of options must be priced quickly, for example in real time option chains.

Method 4: Longstaff Schwartz Monte Carlo#

Monte Carlo simulation moves forward in time, but early exercise decisions need to know future values. Francis Longstaff and Eduardo Schwartz solved this in 2001:

1. Simulate many price paths.
2. At expiration, record payoffs.
3. Step backwards: at each date, for paths where the option is in the money, regress the discounted future cash flows on functions of the current price (such as price and price squared).
4. Use the fitted regression as the estimated continuation value; exercise where the immediate payoff is larger.
5. Average the resulting discounted cash flows.

This makes simulation practical for American and Bermudan options with several underlyings or complex models. See Monte Carlo Option Pricing and Regression Analysis.

Comparing methods#

Dividends#

Discrete dividends complicate American pricing because the stock drops on ex dividend dates. Trees and finite differences can model the drop explicitly. Accurate dividend forecasts matter, since early exercise of calls depends on them. See Dividends.

Implied volatility from American prices#

Because listed equity options are American, implied volatilities quoted by exchanges and brokers usually come from American pricing models (often trees), not the basic Black Scholes formula. Using the wrong model can distort implied volatility for deep in the money puts and dividend paying stocks. See Implied Volatility (IV).

Frequently asked questions#

Why can't Black Scholes price American options?#

Because Black Scholes assumes exercise only at expiration. American options can be exercised early, which requires finding the optimal exercise boundary.

What is the best method for pricing American options?#

For single stocks, binomial trees or finite difference methods are standard. For many underlyings, the Longstaff Schwartz simulation method is common.

Are American calls worth more than European calls?#

Only when the stock pays dividends. Without dividends, early exercise of a call is never optimal, so the prices are equal.

Next, learn how models account for the volatility smile in Local Volatility.