This course introduces stochastic calculus based on Brownian motion and applies the theoretical concepts to finance, and especially, to option pricing within the Black-Scholes framework.

Stochastic ("Ito") calculus differs significantly from "ordinary" calculus because we want to integrate and differentiate with respect to the random Brownian motion process, which is not of bounded variation. It is essential for an understanding of the fundamental and advanced aspects of financial mathematics.

The course develops the basic concepts of:

• The Ito integral with an emphasis on martingales
• The Ito formula as a differentiation rule for stochastic processes
• The martingale representation theorem is derived
• The course continues with stochastic differential equations and develops the connection between them and "ordinary" partial differential equations.
• The modern finance theory of options pricing is developed and analysed using martingale methods and the techniques of stochastic integration theory.
• The renowned Black-Scholes formula is derived
• The course goes on to advanced options pricing techniques including a discussion of early exercise ("American") options.

Note: Graduate students attend joint classes with undergraduates but will be assessed separately.

On successful completion of this course, students will be able to:

Assessment is expected to be based on:

• Assignments (50%; LO 1-4)
• Final examination (50%; LO 1-4)