Details

Tilley's bundling algorithm [1] is a Monte Carlo simulation approach [2] for pricing American options. It generates pseudorandom paths that follow GBM and estimates the option’s continuation value by using backwards induction and a bundling technique. At one time step prior to maturity, paths whose stock prices are similar are grouped together to obtain an estimate of the one-period-ahead option value. Proceeding recursively from the option’s expiration date, an optimal exercise policy is obtained. The red dots represent Tilley's "sharp boundary", which is the algorithm's approximation of . The (actual) early-exercise boundary derives from Mathematica's built-in function FinancialDerivative. While the stock price moves above , the early-exercise is not optimal because the option's holding value is greater than its intrinsic value.

Tilley's "sharp boundary" lies within a "transition zone", which is determined by the dashed lines in the graph: • between the zone's upper limit and the "sharp boundary", the possibility of early-exercise is significantly low, • between the "sharp boundary" and the zone's lower limit, the possibility of early-exercise is significantly high, • outside the "transition zone", the early-exercise possibilities are considered 0% and 100%, respectively.

The "alpha" coefficient derives from , where is the number of paths per bundle, and is the total number of paths. • indicates that the simulation algorithm uses a single bundle that includes all paths, • indicates that every single path is treated as a separate bundle, • indicates that the number of bundles equals the number of paths per bundle. Tilley demonstrated that if , the algorithm still provides good approximations for American options, even if the simulation algorithm uses the upper limit of the "transition zone" as the early-exercise boundary.

References

[1] J. Tilley, "Valuing American Options in a Path Simulation Model," Transactions of the Society of Actuaries, 45, 1993 pp. 83–104.

[2] M. Fu, S. Laprise, D. Madan, Y. Su, and R. Wu, "Pricing American Options: A Comparison of Monte Carlo Simulation Approaches," Journal of Computational Finance, 4, 2001 pp. 39–88.