This section will consider the pricing of a vanilla option using a Binomial Tree. In the first stages our model will be inaccurate, but as we add complexity the model will become more realistic. Our first model will consist of an asset (such as a stock), valued today at $S$, which is only allowed to take two future values. $S$ is worth 100 today (day 1) and can be worth either 110 or 90 tomorrow (day 2). We will denote these two states: $U$ for up and $D$ for down. This can be written formally as:

\begin{eqnarray*} S_1 = 10, S_2 \in \{90,110\} \end{eqnarray*}Now we can imagine that we are based in an insurance firm and that somebody wishes to purchase a call option (essentially an insurance policy) on $S$, with strike $K = 100$ from us. In this simplistic model we are going to ignore interest rates and set them to be zero, for the time being. The purchaser of our option will only exercise the option if $S$ reaches 110 tomorrow, in state U. The purchaser will then make 10, as they can purchase $S$ for 100 and immediately sell it for 110 on the market. If $S$ drops to 90 tomorrow, in state D, our purchaser will not exercise and will not make any money.

How does this affect us in the insurance firm? If state $U$ occurs, we are forced to sell our stock at a price below market rate to our option purchaser, so we lose 10. If $D$ occurs, we do not have to sell anything as the option expired *out of the money*. From this we can determine that if we were to put a price $C$ on our call option, it would have to lie between 0 and 10. Formally:

The task of the insurance firm is to eliminate, or hedge, the risk of selling this option to the purchaser, so that whatever happens the insurance firm will not lose any money. Since we are unable to predict the future, we are unable to simply buy the stock if $U$ occurs or do nothing if $D$ occurs. Our hedging strategy must not be concerned with what happens tomorrow - we must make use only of today's information in order to price our option.

If we employ the Binomial Model, then our task will be put a price on the option via three methods: *Hedging*, *risk neutrality* and *replication*. We will show that in a two-state (i.e. a "two-time") model, they all lead to the same price.

The next article discusses hedging with a Binomial Model ».