American Option is a contract that may be exercised at any time between the date of purchase and the expiration date. A call option gives the holder the right to buy the underlying asset at a certain prespecified price, called strike. A put option give the holder the right to sell the underlying asset at the strike. American options are most popular in equity markets. In the FX world, European options are more preferable.
In addition to their function as stand alone products, American options are important as a feature in more complex products. We say that a derivative contract has American style optionality if all / parts of it can be exercised early. American options on simple underlying assets can be priced by backward induction in binomial / trinomial tree models or methods of dynamic programming. American options on complex underlying assets (not freely traded and dependent on many factors) or exotic structures with American features can be priced using the Longstaff-Schwartz procedure or methods of reinforcement learning.
AMERICAN OPTION REFERENCES
Hull, J. (2011). Options, Futures, and Other Derivatives (8th ed). Pearson / Prentice Hall.
Duffie, D. (2001). Dynamic Asset Pricing Theory (3rd ed). Princeton University Press.
Bjork, T. (2009). Arbitrage Theory in Continuous Time (3rd ed). Oxford University Press.
Lipton, A. (2001). Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach. World Scientific.
Longstaff, F. A. & Schwartz, E. S. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. Review of Financial Studies vol. 14.
Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley Finance, New York.
Passarelli, D. (2008). Trading Option Greeks: How Time, Volatility, and Other Pricing Factors Drive Profit. Bloomberg Press, New York.
AMERICAN OPTION RESOURCES