The principles we developed for the one-period binomial model also apply to a two-period framework. Over two periods, the underlying price must follow one of four patterns: up-up, up-down, down-up, and down-down. Assuming fixed down and up percentages, the up-down and down-up sequences result in the same terminal underlying price. For each terminal stock price, a call option has a specific value.

To clarify the notation, S⁺⁺ indicates the terminal underlying price if the price goes up in both periods, and c⁺⁺ is the resulting call price at expiration when the underlying price goes up in both periods. Other patterns are defined accordingly.

The valuation process is iterative, starting at each final node and working backwards through the tree to the first node (valuation date), where the calculated result is the value of the option.

Suppose the price of a stock is $100. It can either rise or fall by 10% in any period. Assume the exercise price of a call option on the stock is $100. The call is two periods from expiration. The risk-free rate is 6%.

The formula for π (pi) is still the same: π = (1 + r - d) / (μ - d) = (1.06 - 0.9) / (1.1 - 0.9) = 0.8.

The stock price at expiration will be:

• S⁺⁺ = 100 x 1.1 x 1.1 = $121
• S^+- = S^-+ = 100 x 1.1 x 0.9 = $99

• S^-- = 100 x 0.9 x 0.9 = $81

When the option expires, it will be worth:

• c⁺⁺ = Max (0, S⁺⁺ - 100) = $21
• c^+- = c^-+ = Max (0, S+- - 100) = $0
• c^-- = Max (0, S^-- - 100) = $0

Now we move forward to the end of the first period. Suppose we are at the point where the stock price is S⁺. There is one period to go and two outcomes. The call price is c⁺ and can go up to c⁺⁺ or down to c^+-. Using what we know from the one-period model, the call price must be:

c⁺ = [π c⁺⁺ + (1 - π) c^+-] / (1 + r) = (0.8 x 21 + 0.2 x 0) / 1.06 = $15.85.
c^- = ... = $0

Now we step back to the starting point and find that the option price is still given as c = [π c⁺ + (1 - π) c^-] / (1 + r) = (0.8 x 15.85 + 0.2 x 0) / 1.06 = $11.96

Recall that the hedge ratio, n, was the differences in the next two call prices divided by the differences in the next two underlying prices. This will be true in all cases throughout the binomial tree. Hence, we have different hedge ratios at each time point:

• n = (c⁺ - c^-) / (S⁺ - S^-)
• n⁺ = (c⁺⁺ - c^+-) / (S⁺⁺ - S^+-)
• n^- = (c^-+ - c^--) / (S^-+ - S^--)

Valuing American Style Options

American-style options can be exercised early. If the underlying stock is non-dividend-paying:

• American-style call options are worth the same as European-style call options, because it makes no economic sense to exercise these early.
• American-style put options may be worth more than European-style call options, because the proceeds from early exercise can generate extra risk-free returns. The early exercise feature has a significant impact on both the hedge ratio and the put option value.

In general, dividends lower the price of stocks and call options. They may encourage early exercise of American options, in order for the investor to receive the dividends.

When early exercise has value, the no-arbitrage approach is the only way to value American-style options.