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##### Subject 2. Two-Period Binomial Model
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.

### User Contributed Comments8

User Comment
danlan2 Good extension of one-period case.
Shanax Good stuff!
KYC2016 From what I learn in B-school, the ability of exercising your option can only be a positive, since it's giving you more option when compare to a Euro-option. Everyone pays a premium in order to gain flexibility in the financial market (no free lunch). A large dividend (lets say worth 90% of stock value) would make most investor use there american option and exercise early. Also lets say if it was a binominal tree and the expected price of the second period is higher than the third period (no natural probability), then obviously the investor would exercise their option early and use there it in the second period.
dearcfa If exercising is valuable, then the option itself will be valuable! Yes more flexibility is good, and in that case I agree that having an American-style option is better, but the point I'm arguing is that the call option will not lose value once a large dividend is announced, nor will you miss the dividend if you don't exercise

1. Assume a stock is trading at \$60, and you hold an option with a strike of \$50, expiring in one month, probably worth about \$15.

2. The company announces a dividend of \$40 payble in two weeks.

3. What will happen to the stock price? What will happen to the call option price?

4. Stock price may go up as some people think they are getting free money, which they are not of course, but let us say the stock jumps to \$65.

5. What will happen to the call option price, trading currently at \$15? It will also jump to about \$20.

6. You may choose to exercise today, and hold the stock in your account if you choose to, but what are you gaining? I say nothing.

7. Let us say you don't exercise, so you keep on holding the call option, and now it's almost 4 pm on the day when tomorrow the stock goes ex-dividend. That means if the stock closes at \$65, tomorrow it will open at \$25.

8. What will you do seconds before 4 pm? What do you think the option price is at around that time?

Food for thought!
KYC2016 Well if you had an European option and it expires at 5 pm then it would be worth \$0 compare to a price of \$15 for the American option with a same expiry time. Your American call option that expires at 5 is worth something, but your European option is going to be worthless, assuming that people know that the price of the stock will be less than 50 after the dividend payment. It also proved the point that the ability to exercise an option earlier when it's in the money is something that's worth something. The whole point is if all else is the same American option has to be equal or great due to the added flexibility. Put call parity uses So. and we don't know what will happen in the future, so the price of stocks going up or down in the future has no effect on the current price of the call option (obviously this is unrealistic).

Also let's say you have an American option with 3 periods. In period 1 your stock is 15 and exercise price is 15. Next period there is 2 different payoffs. One is for \$30 the other is for 10. Let say in period 2 it was 30 dollars and the payoff for period 3 is 40 with a 10% probability and 5 dollars with a 90% probability (no natural probability). Since the expected payoff is 9.5 wouldn't you want to exercise your American option in period 2? If you had an European option you could only exercise it in period 3. Wouldn't the European option be worth less than the American option.
dearcfa I'm not arguing against the benefit of flexibility in some ways, and I don't really know first hand how European options work. My only argument is when the book says it's good to exercise early in order to get the dividend, implying that if you don't exercise you lose! That I don't believe, and I have seen it in reality, it makes no difference. The option you're holding is still worth the correct amount and you can always "sell" it and not lose anything.
KYC2016 Sorry had to sleep for a little (5 -6 am when I was typing yesterday).

Anyways an option is like a Binomial tree where you can expand (call) or sell (put) the business.

*Next part is absolutely useless for CFA, but food for thought.*

If you have an American option it means that you have more than one opportunity to expand your business. You can expand it at period 1, 2, 3. Had to do this for my undergrad MT. Basically you first calculate the business without expansion. After that you go to period 3 and see what happens to the firm value if you exercised your "expand" (call) option. The difference in price would be the value of an option at period 3. You then use both the options. The difference between the value of 1 option and 2 option is the value of an option in period 2. The is equal to the difference between an American and an European option. Obviously it could be 0 and the option are worthless (out of the money).

The Black Scholes is basically makes the Binomial tree into a model with infinite branches (that's why we use e^). From what I heard from my prof who went to Booth, in real life people will just use C++ to make an infinite period model. A lot more robust because they're allowed to control and change variables in each of the infinite periods.

Also using an option is just increase the leverage for a stock. Let say you pay 3 dollar for a call at a strike of 15. If it's less than 15 you have all of your capital -100% return. If it goes to 20 then your return would be 67% (5/3). If you just bough the stock at \$15 and the stock dropped by a \$1 your return would be 6.7% (-100% for option). If it were to increase to 20 your return is only 33%.

Also I think what you saw was an European option. They expire automatically and once it expires you just get the difference between strike place and stock price automatically. You don't have to "use" it since it's automated. Most of the option places I have seen don't actually give you the stock, but instead just give you a difference in strike and the market price. Easier to manage for these websites, they don't want to be the one liable to storing the stock and strike prices for it's customer.

I hope you can see it from my POV. Not trying to go against at al (arguing with you in two treads). If I offended you in anyway water under the bridge man. WATER UNDER THE BRIDGE.
shawnfuller for call, there's no difference since their lower bound is the same.

but for put, American options are worth more cos u can exercise it earlier and receive X.
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