To illustrate how the binomial model gives increasingly refined estimates as the number of periods increases, consider a European call option that has one year until expiration and an exercise price of $100. Assume that the underlying stock trades for $100, with a standard deviation of 0.1. The risk-free rate of interest is 6%. The graph below shows how the binomial prices converge to the true option price of $7.42 as the number of periods increases. The binomial prices oscillate around the true price: for a single-period binomial model, the price is $7.86. With two periods, the binomial model gives a price of $7.18. With 20 periods the binomial price is 7.38, and with 100 periods the binomial price is 7.44.
In general, for an option of a given expiration, a greater pricing accuracy is obtained by dividing the option's life into a greater number of time periods in a binomial tree. As more time periods are added, the discrete-time binomial price converges to a stable value as though the option is being modeled in a continuous-time world.