Derivation

The derivation of this formula can be found in Hull's Options, Futures, and Other Derivatives (5th edition, pp. 202-203).

However, let us consider a numerical example so that we can gain an intuitive understanding of the binomial tree. Consider a stock XYZ which is trading at $50 and a call option with strike price $55. In 3 months, the stock price can be one of two values, $60 or $40. Our goal is to create a riskless portfolio, that is, one which has the same value regardless of whether the stock is at $60 or $40 in 3 months. We will do this by selling one call option and purchasing $\Delta shares$. So, fastforwarding 3 months, we calculate the value of the portfolio under the two possible conditions and making them equal to each other, thereby making it riskless.

$$60\Delta - 5 = 40\Delta$$ (5)

$$20\Delta = 5$$ (6)

$$\Delta = 1/4$$ (7)

Thus, our riskless portfolio contains .25 shares and one short call option. At expiration, this porfolio will equal

$$60 \times 0.25 - 5 = 10$$ (8)

Since this portfolio is riskless, its return should therefore be equal to the risk-free interest rate, which we will assume for the purposes of our example to be 12%. Therefore, the original investment should be worth

$$10e^{-rT} = 10e^{-0.12 \times 3/12} = \$9.70$$ (9)

Thus, our original investment should be $9.70, which will grow to $10 regardless of the movement in the stock price. Calculating the cost of our original investment and setting it equal to $9.70 will give us the option price, f:

$$50 \times 0.25 - f = \$9.70$$ (10)

$$12.50 - f = \$9.70$$ (11)

$$f = \$2.80$$ (12)

The value of the option is the value of the root node in this binomial tree.