Question

A stock price is currently $50. It is known that at the end of one year it will be either $40 and $60. The exercise price of a one-year European call option is $55. The risk-free interest rate is 5% per annum.

a. Construct a binomial tree to show the payoff of the call option at the expiration date.

b. Based on the binomial tree model, what is the value of the call option?

c. Address the relation between the binomial tree model and the Black-Scholes model.

Answer 1

c. the relation between the binomial tree model and the Black-Scholes model.

The same underlying assumptions regarding stock prices underpin both the binomial and Black-Scholes models: that stock prices follow a stochastic process described by geometric brownian motion. As a result, for European options, the binomial model converges on the Black-Scholes formula as the number of binomial calculation steps increases. In fact the Black-Scholes model for European options is really a special case of the binomial model where the number of binomial steps is infinite. In other words, the binomial model provides discrete approximations to the continuous process underlying the Black-Scholes model.

Whilst the Cox, Ross & Rubinstein binomial model and the Black-Scholes model ultimately converge as the number of time steps gets infinitely large and the length of each step gets infinitesimally small this convergence, except for at-the-money options, is anything but smooth or uniform. To examine the way in which the two models converge see the on-line Black-Scholes/Binomial convergence analysis calculator. This lets you examine graphically how convergence changes as the number of steps in the binomial calculation increases as well as the impact on convergence of changes to the strike price, stock price, time to expiration, volatility and risk free interest rate.