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.