Introduction to Probability and Stochastic Processes with Applications

CHAPTER 12

INTRODUCTION TO MATHEMATICAL FINANCE

Mathematical finance is the study of financial markets and is one of the rapidly growing subjects in applied mathematics. This is due to the fact that in recent years mathematical finance has become an indispensable tool for risk managers and investors. The fundamental problem in the mathematics of financial derivatives is the pricing and hedging. During the years 1950–1960, the research focus was basically on the resolution of problems in economics and statistics. However, many researchers were concerned with the problem that was initiated in the early 20th century by Bachelier, the father of modern mathematical finance: what is the fair price for an option on a particular stock? The answer to this question was given in 1973 by researchers Fisher Black and Myron Scholes. They argued that a rational investor does not wait passively until the expiry of the contract but, in contrast, invests in a portfolio consisting of a risk-free asset and a risky stock, so that the value of this portfolio will be equal to the value of the option. Therefore, the fair price of the option is the present value of the portfolio. To obtain this value they generated a partial differential equation whose solution is known as the “Black-Scholes formula”.

The initial focus on the valuations of financial derivatives was to describe the appropriate movements of stock prices and solving partial differential equations. However, another purely probabilistic approach ...