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Setup.

Let us start by reviewing the setup. The basic assumption used is about the stochastic process governing the price of the underlying asset the option is written on. In the following discussion we will use the standard example of a stock option, but the theory is not only relevant for stock options.

The price of the underlying asset, $S$, is assumed to follow a geometric Brownian Motion process, conveniently written in either of the shorthand forms

\begin{displaymath}dS = \mu S dt + \sigma S dZ \end{displaymath}

or

\begin{displaymath}\frac{dS}{S} = \mu dt + \sigma dZ \end{displaymath}

where $\mu$ and $\sigma$ are constants, and $Z$ is Brownian motion.

Using Ito's lemma, the assumption of no arbitrage, and the ability to trade continuously, Black and Scholes showed that the price of any contingent claim written on the underlying must solve the following partial differential equation:

\begin{displaymath}\frac{\partial f}{\partial S}rS +
\frac{\partial f}{\partial...
...\frac{1}{2}
\frac{\partial^2 f}{\partial S^2}\sigma^2S^2 = rf \end{displaymath}

For any particular contingent claim, the terms of the claim will give a number of boundary conditions that determines the form of the pricing formula.

We will start by discussing the original example solved by Black, Scholes, Merton: European call and put options.


next up previous contents index
Next: European call and put Up: Basic Option Pricing, analytical Previous: Basic Option Pricing, analytical   Contents   Index
Bernt Arne Odegaard
1999-09-09