2 Mathematics for Finance
current bond priceA(0) is known to all investors, just like the current stock
price. However, in contrast to stock, the priceA(1) the bond will fetch at time 1
is also known with certainty. For example,A(1) may be a payment guaranteed
by the institution issuing bonds, in which case the bond is said to mature at
time 1 with face valueA(1). The return on bonds is defined in a similar way
as that on stock,
KA=A(1)−A(0)
A(0).
Chapters 2, 10 and 11 give a detailed exposition of risk-free assets.
Our task is to build a mathematical model of a market of financial securi-
ties. A crucial first stage is concerned with the properties of the mathematical
objects involved. This is done below by specifying a number of assumptions,
the purpose of which is to find a compromise between the complexity of the
real world and the limitations and simplifications of a mathematical model,
imposed in order to make it tractable. The assumptions reflect our current
position on this compromise and will be modified in the future.
Assumption 1.1 (Randomness)
The future stock priceS(1) is arandom variablewith at least two different
values. The future priceA(1) of the risk-free security is a known number.
Assumption 1.2 (Positivity of Prices)
All stock and bond prices are strictly positive,
A(t)>0andS(t)>0fort=0, 1.The total wealth of an investor holdingxstock shares andybonds at a
time instantt=0,1is
V(t)=xS(t)+yA(t).
The pair (x, y) is called aportfolio,V(t)beingthevalueof this portfolio or, in
other words, thewealthof the investor at timet.
The jumps of asset prices between times 0 and 1 give rise to a change of
the portfolio value:
V(1)−V(0) =x(S(1)−S(0)) +y(A(1)−A(0)).This difference (which may be positive, zero, or negative) as a fraction of the
initial value represents the return on the portfolio,
KV=