Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 42

BLACK-SCHOLES OPTION PRICING MODELS

The Black-Scholes model for pricing stock options is a revolutionary result in mathematical finance. Many of the pricing techniques and models used today in finance are rooted in the methods and ideas from the Black-Scholes model. In this chapter, we present the Black-Scholes model and relevant results.

42.1 Basic Concepts and Facts

Definition 42.1 (Continuous Market). Let be an n-dimensional Brownian motion on some filtered probability space be an adapted interest rate process. A continuous market is a (d + 1)-dimensional stochastic process

where

and

Here the mean rate of return vector and the volatility matrix are also adapted processes.

Definition 42.2 (Black-Scholes Market). Let {B_t : t ≥ 0} be a Brownian motion on some probability space ...