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This book is an elementary introduction to the basic concepts of
financial mathematics with a central focus on discrete models and
an aim to demonstrate simple but widely used financial derivatives
for managing market risks. Only a basic knowledge of probability,
real analysis, ordinary differential equations, linear algebra and
some common sense are required to utilise this book. Financial
mathematics is an application of advanced mathematical and
statistical methods to financial management and markets, with a
main objective to quantify and hedge risks. Since the book aims to
present the basics of financial mathematics to the reader, only
essential elements of probability and stochastic analysis are given
to explain ideas on derivative pricing and hedging. To keep the
reader intrigued and motivated, the book has a sandwich structure:
Probability and stochastics are given on the spot, at places where
mathematics can almost immediately be illustrated by an application
to finance. The first part of the book introduces one of the main
principles in finance - no arbitrage pricing.It also introduces
main financial instruments such as forward and futures contracts,
bonds and swaps, and options. This part is not mathematical. The
second part deals with pricing and hedging of European- and
American-type options in the discrete time setting. In addition,
the concept of complete and incomplete markets is discussed.
Elementary probability is briefly revised and discrete-time -
discrete-space stochastic processes used in financial modelling are
considered. The third part discusses stochastic analysis and
introduces the Wiener process, Ito integrals, and stochastic
differential equations. The main feature of this final part of the
book is the famous Black - Scholes formula for pricing European
options. Some guidance for further study of this exciting and
rapidly changing subject is given in the last chapter. The book has
approximately 100 exercises, for which most solutions have been
provided.
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