Notions of Mathematical Finance - Computational Methods for Quantitative Finance - page 3

Chapter 1
Notions of Mathematical Finance
The present notes deal with topics of computational finance, with focus on the analy-
sis and implementation of numerical schemes for pricing derivative contracts. There
are two broad groups of numerical schemes for pricing: stochastic (Monte Carlo)
type methods and deterministic methods based on the numerical solution of the
Fokker-Planck (or Kolmogorov) partial integro-differential equations for the price
process. Here, we focus on the latter class of methods and address finite difference
and finite element methods for the most basic types of contracts for a number of
stochastic models for the log returns of risky assets. We cover both, models with
(almost surely) continuous sample paths as well as models which are based on
price processes with jumps. Even though emphasis will be placed on the (partial
integro)differential equation approach, some background information on the market
models and on the derivation of these models will be useful particularly for readers
with a background in numerical analysis.
Accordingly, we collect synoptically terminology, definitions and facts about
models in finance. We emphasise that this is a collection of terms, and it can, of
course, in no sense claim to be even a short survey over mathematical modelling
in finance. Readers who wish to obtain a perspective on mathematical modelling
principles for finance are referred to the monographs of Mao [120], Øksendal [131],
Gihman and Skorohod [71-73], Lamberton and Lapeyre [109], Shiryaev [152], as
well as Jacod and Shiryaev [97].
1.1 Financial Modelling
Stocks Stocks are shares in a company which provide partial ownership in the
company, proportional with the investment in the company. They are issued by a
company to raise funds. Their value reflects both the company's real assets as well
as the estimated or imagined company's earning power. Stock is the generic term for
assets held in the form of shares. For publicly quoted companies, stocks are quoted
and traded on a stock exchange. An index tracks the value of a basket of stocks .