O'Reilly logo

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

equation

where

equation

and

equation

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

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

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required