Here we will walk through a stochastic programming example of a six-period American put-option model. The holder of an American put option has the right to sell a specified (underlying) stock at any time (the feature of American options) between now and a specified expiration date at a specified strike price.  The holder makes a profit in the period of exercise if the strike price exceeds the market price of the stock at the time of sale, otherwise the profit is 0.  The objective will be to maximize the expected present value of the option in period 0, which also corresponds to the maximum amount we should be willing to pay for the option.

In this example, the underlying stock's initial price is $100, with the option's strike price equal to $99.  This initially places the option "out of the money" and make its value $0.  However, should the stock's price decline over subsequent periods to the point where it becomes less than the strike price, then the option will have a positive value.

We will also use a risk free interest rate of 3% for discounting revenue back to the initial period.

The price behavior of the underlying stock is the stochastic element in this example.  In any given period, we have determined that there are four equally likely outcomes for the stock's price:

Using the following reasoning, a casual observer might analyze the above data and declare the option as being worthless:

The stock is presently out of the money by $1.  Before it will move into the money, the stock must decline by at least $1.  However, this seems highly unlikely given that the expected return in any period is positive +0.75% = .25 * (9.0+3.0-1.0-8.0)%.  On average, the stock will just go up in value, and the option will never go into the money.  Given this, we should not be willing to pay anything for this option.

We will see below if our casual observer in on the right track or not.

The complete LINGO model can be found in the LINGO samples folder under the name SPPUTOPT1.LG4.