Handbook of Multi-Commodity Markets and Products: Structuring, Trading and Risk Management

CHAPTER 11 An Introduction to Stochastic Calculus with Matlab® Examples

Laura Ballotta and Gianluca Fusai

Given the technical nature of the quantitative applications covered in Part II, we open with a comprehensive introduction to stochastic calculus. Readers who are competent in stochastic calculus are encouraged to proceed and explore the chapters that follow. Those less competent with stochastic calculus will find this chapter beneficial before progressing.

The aims of this chapter are:

1. To introduce the concept of Brownian motion.

   • What are W(t) and dW(t)?

   • What are the properties of W(t)?

2. To explain the meaning of the stochastic differential equation
   
3. To explain the meaning of the stochastic integral. For example:

   • To give meaning to

     

   • The properties of X(t): distribution, expected value, variance.

   • How to construct a (continuous) martingale.

4. To explain Itô's formula.

   • For example, how to relate

     

     to

     
5. Examples of common stochastic differential equations:

   • arithmetic Brownian motion;

   • geometric Brownian motion;

   • mean-reverting Gaussian model (Vasicek);

   • mean-reverting ...