Classical financial mathematics deals first of all with basic financial instruments like stocks, foreign currencies and bonds. A derivative (derivative security or contingent claim) is a financial instrument whose value depends on the value of others, more basic underlying variables. In this chapter we consider forward contracts, futures contracts and options as well as some combinations.
Simple derivatives have been known on European stock exchanges
since the turn of the th century. While they lost popularity
between World War I and II, they revived in the seventies
accompanied by the work of Black, Scholes and Merton, who
developed a theoretical foundation to price such instruments.
Their entrepreneurial approach, which is not only applied to price
derivatives but everywhere in finance where the risk of complex
financial instruments is measured and controlled, was awarded the
Nobel price in economics in 1997. At the same time, it triggered
the development of modern financial mathematics whose basics we
describe in the first part of this book. Since we concentrate only
on the mathematical modelling ideas, we introduce the required
financial terminology as we pass by. We leave out numerous details
which are of practical importance but which are of no interest for
the mathematical modelling, and refer to,
for example, Hull (2000), Welcker et al. (1992).
Particularly simple derivative securities are forward and
future contracts. Both contracts are agreements involving two parties and
calling for future delivery of an asset at an agreed-upon price.
Stocks, currencies and bonds, as well as agricultural products
(grain, meat) and raw materials (oil, copper, electric energy) are
underlying in the contract.
At time , the value
of such a contract depends
on the current value of the underlying
, the time to
maturity
and of the parameters
,
specified in
the contract.
Contrary to forward and futures contracts where both parties are obligated to carry out the transaction, an option gives one party the right to buy or sell the security. Obviously, it's important to distinguish whether the buyer or seller of the option has the right to exercise. There are two types of options: call and put options. Furthermore, European options are delimited from American options. While European options are like forward contracts, American options can be exercised at any date before maturity. These terms are derived from historical, not geographical roots.
European put option
is an agreement which gives the holder the right to sell the
underlying asset at a specified date for a specified price
.
The holder of an American call
or put option has the
right to exercise the option at any time between and
.
Asian, lookback and knock-out options are path-dependent
derivatives. While the delivery price of an asian option
depends on the average value of the security of a certain period
of time, it depends in the case of a lookback option on the
minimum or maximum value of the security for a certain period of
time. Knock-out options expire worthless if the price level ever
hits a specified level.
To get used to forward and futures contracts, plain vanilla
options and simple combinations of them, it is convenient to have
a look at the payoff of an instrument,
i.e. the value of the derivative at maturity . The payoff of a
long position in a forward contract is just
, with
the security's spot price at expiration date
. The holder of
the contract pays
for the security and can sell it for
.
Thus, he makes a profit if the value of the security
at
expiration is greater than the delivery price
. Being short in
a forward contract implies a payoff
. Both payoff functions
are depicted in Figure 1.1.
The call option payoff function is denoted:
In contrast to forward and future contracts, options have to be
bought for a positive amount , called the option
price or option
prime. Often, the options profit
function is defined as
. However, this
definition adds cash flows of different points in time. The
correct profit is obtained by compounding the cash outflow in time
up to time
, since the investor could have invested the
option option at the risk-free interest rate
. Assuming
continuous compounding at a constant interest rate
, the profit
function of a call option is denoted:
.
Another fundamental financial instrument which is used in option pricing is a bond. Apart from interest yields, the bond holder possibly receives coupon payments at fixed points in time. In particular, we will consider zero-coupon bonds, i.e. bonds which promise a single payment at a fixed future date.
Buying a zero-coupon bond corresponds to lending money at a fixed
interest rate for a fixed period of time. Conversely, selling a
zero-coupon bond is equivalent to borrowing money at rate .
Since bonds are traded on an exchange, they can be sold prior to
maturity at price
, i.e.
plus accrued interest up to
time
.
In practice, interest rates are compounded at discrete points in
time, for example annually, semiannually or monthly. If the
interest rate is compounded annually, the initial investment
has
years later a value of
If it is compounded
times per annum (p.a.), the investment
pays an interest rate of
each
years,
and has a terminal value of
after
years. However, when options and other complex
derivatives are priced, continuous compounding is used, which
denoted for
. In this case, the initial
investment
grows in
years to
and
is called short rate. The
difference between discrete and continuous compounding is small
when
is large. While an investment of
EUR at a
yearly rate
% grows to 1100 EUR within a year when
annually compounded, it grows to 1105.17 EUR when continuously
compounded.
In light of this, the continuous compounded rate can be
modified to account for these deviations. Assuming annual
compounding at rate
, for both continuous and annual
compounding, a continuous compounded rate
has to be applied, in order to obtain the same terminal
value
If not stated otherwise, continuous compounding will be assumed
from here on. For comparing cash flows occurring at different
points in time, they have to be compounded or discounted to the
same point in time. That is, interest payments are added or
subtracted. With continuous compounding, an investment of in
time
in
is
Before finishing the chapter, some more financial terms will be introduced. A portfolio is a combination of one or more financial instruments - its value is considered as an individual financial instrument. One element of a portfolio is also called a position. An investor assumes a long position when he buys an instrument, and a short position when he sells it. A long call results from buying a call option, a long put from buying a put option, and a short forward from selling a forward contract.
An investor closes out a position of his portfolio by making the future portfolio performance independent of the instrument. If the latter is traded on an exchange, he can sell (e.g. a stock or a bond) or buy (e.g. borrowed money) it. Should the instrument not be traded, however, the investor can close out the position by adding to the portfolio the inverse instrument. Thus, both sum up to zero, and do not influence the portfolio performance any more.
Short selling is a trading strategy that involves selling financial instruments, for example stocks, which he does not own. At a later point in time, he buys back these objects. In practice, this requires the intervention of a broker who mediates another client owing the objects and willing to lend them to the investor. The short selling investor commits to pay to the client any foregone income, as dividends for example, that would be received in the meantime.