Financial Simulation Modeling in Excel

BLACK-SCHOLES OPTION PRICING METHOD

The Black-Scholes method is another common way of pricing options. However, instead of using an interest rate based model, Fischer Black and Myron Scholes modeled the stock price directly and assumed that its price can be adequately described as a geometric Brownian motion (equation 4.38)

Equation 4.38 by itself, along with the basic solution to S, has no sense of arbitrage or what a “fair” price is. We will not go into the details here, but the pricing equations we will mention a little later depend on the assumptions made on how an option should be priced, and those assumptions necessarily take arbitrage into account. Another thing to note here is that a geometric Brownian motion process is very different from a regular Brownian motion, which is shown in equation 4.4. Equation 4.38 can be rewritten to into equation 4.39:

If you recall from calculus that the integral of 1/x is the natural log of x, ln(x), then you can see that the natural log of S is normally distributed (since the right side of the equation is normally distributed). In summary regular Brownian motion produces normally distributed numbers while geometric Brownian motion produces lognormally distributed numbers. We will discuss this effect in more detail in the next chapter when ...