9807

Pricing Put Options with the Trinomial Method

This Demonstration applies the trinomial method (also known as the "three-jump process" [1]) to estimate the value of a put option. Use the controllers to set the option's parameters and the time discretization, in order to approximate the American and the European puts. The European put can be exercised only at its maturity, while the American put can be exercised at any time up to maturity. The black line represents the early exercise boundary for the American put. Whenever the asset price drops below this boundary, the American put's intrinsic value becomes greater than its holding value, and it is optimal for the holder to exercise the option. Random trinomial paths show where the asset price is more likely to move.

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Under the trinomial method, the underlying asset price is modeled as a recombining tree, where at each node the price has three possible paths: an up, down, and stable, or middle, path. These values are found by multiplying the value at the current node by the appropriate factor , , or :
with corresponding probabilities:
,
where is the asset price volatility, is the continuous dividend yield, is the risk-free rate, and is the length of each time step in the trinomial tree (equal to the option's maturity divided by the number of time steps).
The above probabilities derive from the application of the explicit finite-difference method [2] to solve the Black–Scholes partial differential equation. To make sure that all probabilities are in the interval (0,1), the condition should be satisfied.
Once the trinomial lattice of all possible asset prices up to maturity has been calculated, the option value is found at each node largely as for the binomial model, by working backward from the final nodes to the present. The difference between the trinomial and binomial models is that the option value at each non-final node is determined based on the three, as opposed to two, later nodes and their corresponding probabilities.
The trinomial model is considered to produce more accurate results than the binomial model when fewer time steps are modeled, and is therefore used when computational speed or resources may be an issue.
References
[1] P. Boyle, "Option Valuation Using a Three Jump Process," International Options Journal, 3, 1986 pp. 7–12.
[2] M. Brennan and E. Schwartz, "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis," The Journal of Financial and Quantitative Analysis, 13(3), 1978 pp. 461–474. doi:10.2307/2330152.
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