It is in the case of American options, allowing for the possibility of early exercise, that binomial approximations are useful. At each node we calculate the value of the option as a function of the next periods prices, and then check for the value exercising of exercising the option now

// file bin_am_call.cc // author: Bernt A Oedegaard // calculate the binomial option pricing formula for an American call #include <cmath> // standard mathematical library #include <algorithm> // defining the max() operator #include <vector> // STL vector templates double option_price_call_american_binomial( double S, // spot price double X, // exercice price double r, // interest rate double sigma, // volatility double t, // time to maturity int steps) { // no steps in binomial tree double R = exp(r*(t/steps)); // interest rate for each step double Rinv = 1.0/R; // inverse of interest rate double u = exp(sigma*sqrt(t/steps)); // up movement double uu = u*u; double d = 1.0/u; double p_up = (R-d)/(u-d); double p_down = 1.0-p_up; vector<double> prices(steps+1); // price of underlying vector<double> call_values(steps+1); // value of corresponding call prices[0] = S*pow(d, steps); // fill in the endnodes. for (int i=1; i<=steps; ++i) prices[i] = uu*prices[i-1]; for (int i=0; i<=steps; ++i) call_values[i] = max(0.0, (prices[i]-X)); // call payoffs at maturity for (int step=steps-1; step>=0; --step) { for (int i=0; i<=step; ++i) { call_values[i] = (p_up*call_values[i+1]+p_down*call_values[i])*Rinv; prices[i] = d*prices[i+1]; call_values[i] = max(call_values[i],prices[i]-X); // check for exercise }; }; return call_values[0]; };