# Binomial Black-Scholes with Richardson Extrapolation (BBSR) Method

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This Demonstration shows the convergence of the binomial Black–Scholes with Richardson extrapolation (BBSR) method [2] compared to the standard binomial Cox-Ross-Rubinstein (CRR) method [1], 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 grid size .

Contributed by: Michail Bozoudis (March 2016)

Suggested by: Michail Boutsikas

Open content licensed under CC BY-NC-SA

## Snapshots

## 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:

,

where denotes the normal cumulative distribution function,

, and

.

Then, by working backward to the present, the process is similar to the binomial method. The Black–Scholes with Richardson extrapolation (BBSR) method [2] adds two-point Richardson extrapolation to the BBS method. For example, the BBSR method computes the BBS price corresponding to a pair of options with and time steps, and respectively, and then sets the approximate price to . Empirical results [2] indicate the improvement of the BBSR over the BBS method.

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.

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