Binomial Convergence And Price Oscillation

The binomial tree prices an option by building a lattice of possible future prices and working backward, and as the number of steps $n$ in that lattice grows, the resulting price should converge to the continuous-time Black-Scholes value. It does converge — but not smoothly. Plot binomial price against $n$ and, instead of a clean line settling onto the Black-Scholes value, you see a zig-zag that oscillates around the true price, with the size of the wiggle shrinking only slowly as $n$ increases.

The oscillation comes from how the strike price lines up against the tree's terminal nodes. At some values of $n$, a tree node happens to land very close to the strike, giving an unusually accurate price for that step count; at other values of $n$, the nearest node misses the strike by nearly half a step, and the price is noticeably off in one direction. Whether the tree "gets lucky" with strike alignment at a given $n$ flips somewhat unpredictably as $n$ increases by one, which produces the zig-zag rather than a monotone approach.

The practical fix quants use is a smoothing technique — for example, averaging the price computed at $n$ steps with the price at $n+1$ steps, which largely cancels the oscillation and gives a much faster-converging estimate without needing an enormous $n$.