Joe Willenborg

Tutorial for Valuing an American Put Option

This tutorial will describe how to value an American put option over two time periods using the binomial method. The binomial method of valuing options is a discrete-time options valuation method that assumes there an underlying asset will be one of exactly two specific prices over a given time period. Each of these prices is weighted by a risk neutral probability to determine the value at the preceding node. An American option is an option that can be called any time between the present and a specified exercise date. This is more complicated than the typical European option which cannot be exercised before the exercise date because the American option has more times at which a decision to exercise must be made.

Solving the Problem:
One must begin solving this problem by working backward from the latest time nodes to the present.

For each set of three nodes where S0 is the current market price, u is the ratio of the next period’s stock price to the current price in the up state, and d is the ratio of the next period’s stock price to the current price in the down state:

binomial_triode_graphic

In the diagram above, the value of the stock is above the node, and the value of the option is below the node.

The risk-neutral probability can be found by the following equation:

(1)
\begin{align} \pi = \frac{1+{r }_{f }-d }{u-d } \end{align}

The value of the call option at a give time thus is:

(2)
\begin{align} { p}_{0 } = \frac{ \pi* max \left[K-u {S }_{0 }, 0 \right ] + \left ( 1- \pi \right )*max \left[ K-d {S }_{ 0},0 \right ] }{1+ { r}_{f } } \end{align}

Since this is an American put, there is the possibility of exercising the option early. The value produced by the calculation above is for when the investor holds onto the investment. To solve this problem, you must compare the value of the risk neutral discounting with the value of exercising the option at a given time. The value of the early exercise of the option will be K - S at that node. If the value of the early exercise of the option is greater than the value of waiting one more time period, then the option should be exercised and this will be the value of the option at the decision node. This decision is denoted with the max[0,K-S] in equation (2) shown above.

This process should be followed iteratively for each set of three nodes in the diagram as you work from later time periods to the present. Once the root node of the tree is reached, you have the value of the option at the present time.