Trinomial Tree Option Pricing Method
This Demonstration illustrates the application of the recombining trinomial tree method to approximate the value of the European- or American-type call/put option, assuming constant volatility and risk-free interest rate. A call/put option gives its owner the right but not the obligation to purchase/sell the underlying asset (e.g., equity share) at a price agreed in advance—the strike price. The European-type call/put option may be exercised only on the expiration date of the option. In contrast, the American-type call/put option may be exercised at any time between its inception and expiration. Therefore, the American-type call/put option offers its owner more optionality relative to the European-type option. Hence, the price of the American-type call/put option is at least the same as the European-type call/put option, or more.[more]
Each tree node contains the price of the underlying security (top) and value of the option (bottom). Optimal option exercise is indicated by red bold font for relevant tree nodes. In the case of the European-type call/put options, the optimal exercise refers to instances when the European call/put option is in the money at expiration—the underlying asset price is greater than the strike price. In the case of American-type call/put options, the optimal exercise refers to instances when the American-type call/put option is in the money at expiration as well as when the value of immediate exercise (the underlying asset price minus the strike price) before expiration is greater than the value of continuing with the option. Further, instances when the immediate exercise value is greater than the value of continuing with the option also constitute the optimal exercise price of the American-type call/put option.[less]
The Demonstration illustrates application of the recombining trinomial tree model to approximate the value of the European- and American-type call/put options. The recombining trinomial tree is generated by allowing only three things to happen to the price of the underlying asset: increase, decrease, or remain unchained, one unit of time later (e.g., one tick, day, week, etc.). Specifically, the underlying asset price is allowed to: (1) increase by the factor ; (2) decrease by the factor ; or (3) remain unchanged, hence scaled by the factor , where is the annualized volatility of the underlying asset and is the unit of time between successive tree nodes. Each of the three possible outcomes , , and is assigned its risk-neutral probabilities , , and :
where , , is the risk-free rate (annualized), is the dividend yield of the underlying asset (annualized), and and are as defined above. Further, to ensure that probabilities , , and are in the interval of (0, 1) and their sum is equal to 1, the following condition must hold: .
 J. Cox, S. Ross, and M. Rubinstein, "Option Pricing: A Simplified Approach," Journal of Financial Economics, 7(3), 1979 pp. 229–263. doi:10.1016/0304-405X(79)90015-1.
 J. C. Hull, Options, Futures, and Other Derivatives, 8th ed., Boston: Prentice Hall, 2012.
 N. N. Taleb, Dynamic Hedging: Managing Vanilla and Exotic Options, New York: John Wiley & Sons, Inc., 1997.