108156.pdf

86 Mathematics for Finance

future prices of which may be random (except, of course, at maturity). We
shall also relax the assumption that an investment in a money market account
should be risk-free, with a view towards modelling random interest rates. In
this way we prepare the stage for a detailed study of derivative securities in
Chapters 6, 7 and 8, and random bond prices and the term structure of interest
rates in Chapters 10 and 11.
Securities of various kinds will be treated on a similar footing as stock in
Section 4.1. We shall denote byS 1 (n),...,Sm(n) the timenprices ofmdif-
ferent primary securities, typicallymdifferent stocks, though they may also
include other assets such as foreign currency, commodities or bonds of vari-
ous maturities. Moreover, the price of one distinguished primary security, the
money market account, will be denoted byA(n). In addition, we introducek
different derivative securities such as forwards, call and put options, or indeed
any other contingent claims, whose timenmarket prices will be denoted by
D 1 (n),...,Dk(n).
As opposed to stocks and bonds, we can no longer insist that the prices of all
derivative securities should be positive. For example, at the time of exchanging
a forward contract its value is zero, which may and often does become negative
later on because the holder of a long forward position may have to buy the
stock above its market price at delivery. The future pricesS 1 (n),...,Sm(n)and
A(n) of primary securities and the future pricesD 1 (n),...,Dk(n) of derivative
securities may be random forn=1, 2 ,..., but we do not rule out the possibility
that some of them, such as the prices of bonds at maturity, may in fact be
known in advance, being represented by constant random variables or simply
real numbers. All the current pricesS 1 (0),...,Sm(0),A(0),D 1 (0),...,Dk(0)
are of course known at time 0, that is, are also just real numbers.
The positions in primary securities, including the money market account,
will be denoted byx 1 ,...,xm andy, and those in derivative securities by
z 1 ,...,zk, respectively. The wealth of an investor holding such positions at
timenwill be

V(n)=

∑m

j=1

xjSj(n)+yA(n)+

∑k

i=1

ziDi(n),

which extends formula (4.1).
The assumptions in Section 4.1 need to be replaced by the following.

Assumption 4.1a (Randomness)

The asset pricesS 1 (n),...,Sm(n),A(n),D 1 (n),...,Dk(n) are random vari-
ables for anyn=1, 2 ,....