3. Risky Assets................................................
The future prices of any asset are unpredictable to a certain extent. In this
chapter we shall typically be concerned with common stock, though any security
such as foreign currency, a commodity, or even a partially unpredictable future
cash flow can be considered. Market prices depend on the choices and decisions
made by a great number of agents acting under conditions of uncertainty. It
is therefore reasonable to treat the prices of assets as random. However, little
more can be said in a fully general situation. We shall therefore impose specific
conditions on asset prices, motivated by a need for the mathematical model to
be realistic and relevant on the one hand, and tractable on the other hand.
3.1 Dynamics of Stock Prices..................................
The price of stock at timetwill be denoted byS(t). It is assumed to be strictly
positive for allt.We taket= 0 to be the present time,S(0) being the current
stock price, known to all investors. The future pricesS(t)fort>0remain
unknown, in general. Mathematically,S(t) can be represented as a positive
random variable on a probability spaceΩ,thatis,
S(t):Ω→(0,∞).The probability spaceΩconsists of all feasible price movement ‘scenarios’ω∈
Ω. We shall writeS(t, ω) to denote the price at timetif the market follows
scenarioω∈Ω.
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