48 Mathematics for Finance
The current stock priceS(0) known to all investors is simply a positive
number, but it can be thought of as a constant random variable. The unknown
future pricesS(t)fort>0 are non-constant random variables. This means that
for eacht>0 there are at least two scenariosω, ̃ω∈Ωsuch thatS(t, ω)=
S(t,ω ̃).
We assume that time runs in a discrete manner,t = nτ,wheren =
0 , 1 , 2 , 3 ,...andτis a fixed time step, typically a year, a month, a week, a
day, or even a minute or a second to describe some hectic trading. Because we
take one year as the unit measure of time, a month corresponds toτ=1/12, a
week corresponds toτ=1/52, a day toτ=1/365, and so on.
To simplify our notation we shall writeS(0),S(1),S(2),...,S(n),...instead
ofS(0),S(τ),S(2τ),...,S(nτ),..., identifyingnwithnτ.This convention will
in fact be adopted for many other time-dependent quantities.
Example 3.1
Consider a market that can follow just two scenarios, boom or recession, de-
noted byω 1 andω 2 , respectively. The current share price of a certain stock is
$10, which may rise to $12 after one year if there is a boom or come down to
$7 in the case of recession. In these circumstancesΩ={ω 1 ,ω 2 }and, putting
τ=1,wehave
Scenario S(0) S(1)
ω 1 (boom) 10 12
ω 2 (recession) 10 7
Example 3.2
Suppose that there are three possible market scenarios,Ω={ω 1 ,ω 2 ,ω 3 },the
stock prices taking the following values over two time steps:
Scenario S(0) S(1) S(2)
ω 1 55 58 60
ω 2 55 58 52
ω 3 55 52 53These price movements can be represented as a tree, see Figure 3.1. It is con-
venient to identify the scenarios with paths through the tree leading from the
single node on the left (the ‘root’ of the tree) to the rightmost branch tips.
Such a tree structure of price movements, if found realistic and desirable,
can readily be implemented in a mathematical model.