A risky business: how to price derivatives
Black-Scholes in the limit
This section contains some undergraduate level mathematics.
The Binomial model is a very simple model for understanding the ideas behind option pricing. However, so far the stock price can only take finitely many values and furthermore can only move at discrete time points. Both of these features are somewhat undesirable, and so in order to get around this we will look at the limit as the number of time periods tends to infinity. This will give us the celebrated Black-Scholes formula.
Firstly, we will adjust the expression for the value of the bond at time to . This isn't an obvious thing to do - it is itself the result of a limiting process. See Plus article Have we caught your interest for more details.
Before taking our limit, it is important that we choose our parameters so that they scale in the correct way. First of all, we fix a terminal time, , which is the expiry time of the option. Then set our to be given by . We will assume that:
We know now that we should base pricing on the formula given above. Therefore, we can now go about pricing the call option mentioned above. The price of this call option is: