Pricing American Options

In this lesson, we implemented our previous lessons on pricing European options to try to price American options.  This is somewhat more complicated than European options, because American options allow the owner to exercise early, so the option owner must re-evaluate their position at every time period t to see if it is advantageous to them to exercise early.

We must recall from a previous lesson the fact that it is a never optimal to exercise early an American call option on a non-dividend paying stock, so instead we will use a put option to model our pricing technique.

Consider an American put option with t=3 periods, K = $100, R =1.01, along with u = 1.07 and d = 1/1.07 = .9346.  The following is our binomial tree:

We can then, working backwards, input the fair price for the option at each node of the binomial tree.  This is shown below:

The values shown in blue above the stock prices are the values of the option at that point.  The 12.66 in orange is the value of exercising the option at that point in time: max(100-87.34,0).  The actual value of the option at that node is only 11.67.  At that node, you could keep the option, which has an expected value of $11.67 or you could strike on the option and receive $12.66.  It is obviously a better decision to strike the option at that node.  This is only place in the binomial tree where it would beneficial to exercise the option.  We can evaluate which choice would be better using the following formula:

Choice = max[K-S(t), [1/R *(q*S(t+1,u) + (1-q)*S(t+1,d))]]

S(t) is the stock price at the time period which you are making the decision.
S(t+1,u) is the stock price in the next period if the stock moves up in price.
S(t+1,d) is the stock price in the next period if the stock moves down in price.

For this particular example: