Convergence of Binomial, Binomial Black-Scholes, and Trinomial Option Pricing Methods

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This Demonstration shows the convergence of the binomial [1], binomial Black–Scholes (BBS) [2], and trinomial [3] methods, depending on the American put option's maturity time discretization. Use the controls to set the option's parameters and time discretization (up to 100 uniform steps); the table shows the American put value approximations at the selected number of time steps. The horizontal black dashed line represents the option's value according to Mathematica's built-in function FinancialDerivative with a 1,000×10,000 grid size.

Contributed by: Michail Bozoudis (November 2014)
Suggested by: Michail Boutsikas
Open content licensed under CC BY-NC-SA


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Details

Under the binomial method (CRR) [1], the underlying asset price is modeled as a recombining tree, where at each node the price can go up and down. 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 binomial tree (equal to the option's maturity divided by the number of time steps).

To make sure that these probabilities are in the interval , the condition should be satisfied.

Once the binomial lattice of all possible asset prices up to maturity has been calculated, the option value is found at each node by working backward from the final nodes to the present.

Under the binomial Black–Scholes (BBS) method [2], which is a variation of the binomial method, the Black–Scholes analytic formula is applied to estimate the values at those nodes one time step before expiration. Then, by working backward to the present, the process is similar to the binomial method. As seen in the graph, the BBS method converges more smoothly than the binomial method.

Under the trinomial method [3], the underlying asset price is modeled as a recombining tree, where at each node the price has three possible paths: up, down, or stable (middle). These values are found by multiplying the value at the current node by the appropriate factor , , or :

,

,

,

with corresponding probabilities

,

,

.

To make sure that all probabilities are in the interval , 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 later nodes (as opposed to two) and their corresponding probabilities.

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] M. Broadie and J. Detemple, "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," The Review of Financial Studies, 9(4), 1996 pp. 1211–1250. doi:10.1093/rfs/9.4.1211.

[3] P. Boyle, "Option Valuation Using a Three Jump Process," International Options Journal, 3, 1986 pp. 7–12.



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