Notions of Mathematical Finance - Computational Methods for Quantitative Finance - page 4

Assets for which future prices are not known with certainty are called risky assets ,
while assets for which the future prices are known are called risk free .

Price Process The price at which a stock can be bought or sold at any given time
t on a stock exchange is called spot price and we shall denote it by S t . All possible
future prices S t as functions of t (together with probabilistic information on the
likelihood of a particular price history) constitute the price process S

}
of the asset. It is mathematically modelled by a stochastic process to be defined
below.

={ S t : t ≥ 0

Derivative Securities A derivative security , derivative for short, is a security
whose value depends on the value of one or several underlying assets and the de-
cisions of the investor. It is also called contingent claim . It is a financial contract
whose value at expiration time (or time of maturity ) T is determined by the price
process of the underlying assets up to time T . After choosing a price process for the
asset(s) under consideration, the task is to determine a price for the derivative secu-
rity on the asset. There are several types of derivatives: options, forwards, futures
and swaps. We focus exemplary on the pricing of options, since pricing other assets
leads to closely related problems.

Options An option is a derivative which gives its holder the right, but not the
obligation to make a specified transaction at or by a specified date at a specified
price. Options are sold by one party, the writer of the option, to another, the holder ,
of the option. If the holder chooses to make the transaction, he exercises the option.
There are many conditions under which an option can be exercised, giving rise to
different types of options. We list the main ones: Call options give the right (but
not the obligation) to buy, put options give the right (but not an obligation) to sell
the underlying at a specified price, the so-called strike price K . The simplest op-
tions are the European call and put options. They give the holder the right to buy
(resp., sell) exactly at maturity T . Since they are described by very simple rules,
they are also called plain vanilla options. Options with more sophisticated rules
than those for plain vanillas are called exotic options . A particular type of exotic
options are American options which give the holder the right (but not the obliga-
tion) to buy (resp., sell) the underlying at any time t at or before maturity T .For
European options the price does not depend on the path of the underlying, but only
on the realisation at maturity T . There are also so-called path dependent options,
like Asian, lookback or barrier contracts. The value of Asian options depends on the
average price of the option's underlying over a period, lookback options depend on
the maximum or minimum asset price over a period, and barrier options depend on
particular price level(s) being attained over a period.

Payoff The payoff of an option is its value at the time of exercise T . For a Euro-
pean call with strike price K , the payoff g is

S T −

K if S T >K,

g(S T ) = (S T − K) + =

0 else.