4. Discrete Time Market Models..............................
Having discussed a number of different models of stock price dynamics, we
shall now generalise and pursue a little further some of the ideas introduced in
Chapter 1. In particular, we shall reformulate and extend the general notions
and assumptions underlying mathematical finance already mentioned in that
chapter.
As in Chapter 3, we assume that time runs in steps of fixed lengthτ.For
many time-dependent quantities we shall simplify the notation by writingnin
place of the timet=nτof thenth step.
4.1 Stock and Money Market Models...........................
Suppose thatmrisky assets are traded. These will be referred to as stocks. Their
prices at timen=0, 1 , 2 ,...are denoted byS 1 (n),...,Sm(n). In addition,
investors have at their disposal a risk-free asset, that is, an investment in the
money market. Unless stated otherwise, we take the initial level of the risk-
free investment to be one unit of the home currency,A(0) = 1. However, in
some numerical examples and exercises we shall often takeA(0) = 100 for
convenience. Because the money market account can be manufactured using
bonds (see Chapter 2), we shall frequently refer to a risk-free investment as a
position in bonds, finding it convenient to think ofA(n) as the bond price at
timen.
The risky positions in assets number 1,...,mwill be denoted byx 1 ,...,xm,
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