دانلود کتاب Financial Mathematics An Introduction to Derivatives Pricing

This book is intended as a guide to some elements of the mathematics of
finance. Had we been a bit bolder it would have been entitled ‘Mathematics
for Money Makers’ since it deals with derivatives, one of the most notorious
ways to make (or lose) a lot of money. Our main goal in the book is to
develop the basics of the theory of derivative pricing, as derived from the
so-called ‘no arbitrage condition’. In doing so, we also introduce a number
of mathematical tools that are of interest in their own right. At the end of it
all, while you may not be a millionaire, you should understand how to avoid
‘breaking the bank’ with a few bad trades.
In order to motivate the study of derivatives, we begin the book with a
discussion of the financial markets, the instruments that are traded on them
and how arbitrage opportunities can occur if derivatives are mispriced. We
then arrive at a problem that inevitably arises when dealing with physical
systems such as the financial markets: how to deal with the ‘flow of time’.
There are two primary means of parametrizing time—the discrete time pa-
rameterization, where time advances in finite steps; and the continuous time
parameterization, where time varies smoothly. We initially choose the former
method, and develop a simple discrete time model for the movements of asset
prices and their associated derivatives. It is based on an idealised Casino,
where betting on the random outcome of a coin toss replaces the buying and
selling of an asset. Once we have seen the basic ideas in this context, we then
expand the model and interpret it in a language that brings out the analogy
with a stock market. This is the binomial model for a stock market, where
time is discrete and stock prices move in a random fashion. In the second
half of the notes, we make the transition from discrete to continuous time
models, and derive the famous Black-Scholes formula for option pricing, as
well as a number of interesting extensions of this result.
Throughout the book we emphasise the use of modern probabilistic meth-
ods and stress the novel financial ideas that arise alongside the mathematical
innovations. Some more advanced topics are covered in the final sections—
stocks which pay dividends, multi-asset models and one of the great simpli-
fications of derivative pricing, the Girsanov transformation.
This book ia based on a series of lectures given by L.P. Hughston at
King’s College London in 1997. The material in appendix D was provided by
Professor R.F. Streater, whom we thank for numerous helpful observations
on the structure and layout of the material in these notes.