Abstract

Economic behavior and market evolution present notoriously difficult complex systems, where physical interacting particles become purpose-pursuing interacting agents, thus providing a kind of a bridge between physics and social sciences.

We systematically develop the mathematical content of the basic theory of financial economics that can be presented rigorously using elementary probability and calculus, that is, the notions of discrete and absolutely continuous random variables, their expectation, notions of independence and of the law of large numbers, basic integration – differentiation, ordinary differential equations and (only occasionally) the method of Lagrange multipliers. We do not assume any knowledge of finance, apart from an elementary understanding of the idea of compound interest, which can be of two types: (i) simple compounding with rate r and a fixed period of time means your capital in this period is multiplied by (1 + r); (ii) continuous compounding with rate r means your capital in a period of time of length t is multiplied by ert.

This chapter is based on several lecture courses for statistics and mathematics students at the University of Warwick and on invited mini-courses presented by the author at various other places. Sections 6.2 and 6.3 are developed from the author's booklet [9]. The chapter is written in a rather concise (but comprehensive) style in attempt to pin down as clear as possible the mathematical relations that govern the laws of financial economics. Numerous heavy volumes are devoted to the detail discussion of the economic content of these mathematical relations, see e.g. [5], [6], [8], [15], [17].