文档介绍：Mathematics of Finance
Professor R. J. Williams
Mathematics Department,
University of California, San Diego,
La Jolla, CA 92093-0112 USA
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Chapter 1
Financial Markets and
Derivatives
Financial Markets
A financial market consists of tradable securities such as stocks, bonds, cur-
rencies, commodities, or even indexes (see the excerpt from the Wall Street
Journal). One reason for the existence of financial markets is that they facili-
tate the flow of capital. For example, if pany wants to finance the building
of a new production facility, it might sell shares of stock to investors who buy
the shares based on the anticipation of future rewards, such as dividends or a
rise in the stock price.
A variety of stochastic models is used in modeling the prices of securities. All
such models face the usual trade off, namely, plex models typically
provide a better fit to data (although there is the danger of overfitting), whereas
simpler models are generally more tractable and despite their simplicity can
sometimes provide useful qualitative insights. Finding a good balance between
a realistic and a tractable model is part of the art of stochastic modeling.
Both discrete and continuous time models will be considered here. The treat-
ment of discrete time models is not intended to be exhaustive, but is intended
as a means of introducing notions such as hedging and pricing by arbitrage in
a simpler setting to provide a bridge to the development in continuous time.
Binomial tree models will mostly be used in the discrete time setting.
An arbitrage opportunity is an opportunity for a risk free profit. A financial
market is said to be viable if there are no arbitrage opportunities. Typically,
liquid financial markets move rapidly to eliminate arbitrage opportunities. Ac-
cordingly, in this course, attention will focus on financial mark