# Trinomial Tree Option Pricing Method

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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.

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Contributed by: Darius Kirevicius (December 2014)

Open content licensed under CC BY-NC-SA

## Details

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: .

References

[1] 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.

[2] J. C. Hull, *Options, Futures, and Other Derivatives*, 8th ed., Boston: Prentice Hall, 2012.

[3] N. N. Taleb, *Dynamic Hedging: Managing Vanilla and Exotic Options*, New York: John Wiley & Sons, Inc., 1997.

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