This Demonstration shows the "hold-or-exercise" process during the life of an American put option. The early exercise boundary is constructed according to the quadratic approximation analytical method . Pseudorandom geometric Brownian motion (GBM) paths simulate the asset price through time; whenever a path touches the boundary (dashed red line in the first graph), the option is instantly exercised. For each path, the intrinsic value of the option at the time of exercise is discounted, and Monte Carlo integration helps to estimate the American put at . The more GBM paths, the better the approximation.
There are 30 uniform time steps and you can set up to a maximum of 1000 GBM paths. (the source code is provided so you can change these parameters). The "seed randomize" controller creates different pseudo-random GBM paths.
In the first graph, the dashed red line represents the approximation of the early exercise boundary. It is constructed according to the quadratic approximation analytical method . The light blue area shows where the asset price is expected to be in the future according to the GBM model, at the selected "GBM confidence level". Only one for every 20 GBM paths is shown in the graph, as you increase the number of paths from 100 to 1000.
The second graph shows the events of early exercise per time step (before the expiry of the option) as you increase the number of GBM paths from 100 to 1000.
The third graph shows the American put convergence as you increase the number of GBM paths from 100 to 1000. The horizontal dashed line represents the American put approximation with a 2000-step binomial tree ($7.976).
 G. Barone–Adesi and R. Whaley, "Efficient Analytic Approximation of American Option Values," Journal of Finance, 42(2), 1987 pp. 301–320.