1. Introduction: A Simple Market Model......................
1.1 Basic Notions and Assumptions
Suppose that two assets are traded: one risk-free and one risky security. The
former can be thought of as a bank deposit or a bond issued by a government,
a financial institution, or a company. The risky security will typically be some
stock. It may also be a foreign currency, gold, a commodity or virtually any
asset whose future price is unknown today.
Throughout the introduction we restrict the time scale to two instants only:
today,t= 0, and some future time, say one year from now,t= 1. More refined
and realistic situations will be studied in later chapters.
The position in risky securities can be specified as the number of shares
of stock held by an investor. The price of one share at timetwill be denoted
byS(t). The current stock priceS(0) is known to all investors, but the future
priceS(1) remains uncertain: it may go up as well as down. The difference
S(1)−S(0) as a fraction of the initial value represents the so-calledrate of
return,orbrieflyreturn:
KS=S(1)−S(0)
S(0)
,
which is also uncertain. The dynamics of stock prices will be discussed in Chap-
ter 3.
The risk-free position can be described as the amount held in a bank ac-
count. As an alternative to keeping money in a bank, investors may choose to
invest in bonds. The price of one bond at timetwill be denoted byA(t). The
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